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ContentsArticles

Vector calculus 1Gradient 5Divergence 11Curl (mathematics) 14Laplace operator 22Gradient theorem 27Green's theorem 28Stokes' theorem 31Divergence theorem 37Green's function 42

ReferencesArticle Sources and Contributors 48Image Sources, Licenses and Contributors 49

Article LicensesLicense 50

Vector calculus 1

Vector calculusVector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration ofvector fields, primarily in 3 dimensional Euclidean space The term "vector calculus" is sometimes used as asynonym for the broader subject of multivariable calculus, which includes vector calculus as well as partialdifferentiation and multiple integration. Vector calculus plays an important role in differential geometry and in thestudy of partial differential equations. It is used extensively in physics and engineering, especially in the descriptionof electromagnetic fields, gravitational fields and fluid flow.Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end ofthe 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson intheir 1901 book, Vector Analysis. In the traditional form using cross products, vector calculus does not generalize tohigher dimensions, while the alternative approach of geometric algebra, which uses exterior products doesgeneralize, as discussed below.

Basic objectsThe basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valuedfunctions). These are then combined or transformed under various operations, and integrated. In more advancedtreatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fieldsand scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vectorfield is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. Thisdistinction is clarified and elaborated in geometric algebra, as described below.

Vector operations

Algebraic operationsThe basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being definedfor a vector space and then globally applied to a vector field, and consist of:scalar multiplication

multiplication of a scalar field and a vector field, yielding a vector field: ;vector addition

addition of two vector fields, yielding a vector field: ;dot product

multiplication of two vector fields, yielding a scalar field: ;cross product

multiplication of two vector fields, yielding a vector field: .There are also two triple products: the scalar triple product and the vector triple product, but these are less used.

Vector calculus 2

Differential operationsVector calculus studies various differential operators defined on scalar or vector fields, which are typically expressedin terms of the del operator ( ). The four most important differential operations in vector calculus are:

Operation Notation Description Domain/Range

Gradient Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.

Curl Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vectorfields.

Divergence Measures the magnitude of a source or sink at a given point in avector field.

Maps vector fields to scalar fields.

Laplacian A composition of the divergence and gradient operations. Maps scalar fields to scalar fields.

where the curl and divergence differ because the former uses a cross product and the latter a dot product, and fdenotes a scalar field and F denotes a vector field. A quantity called the Jacobian is useful for studying functionswhen both the domain and range of the function are multivariable, such as a change of variables during integration.

TheoremsLikewise, there are several important theorems related to these operators which generalize the fundamental theoremof calculus to higher dimensions:

Theorem Statement Description

Gradienttheorem

The line integral through a gradient (vector) field equals thedifference in its scalar field at the endpoints of the curve L.

Green'stheorem

The integral of the scalar curl of a vector field over some region inthe plane equals the line integral of the vector field over the closed

curve bounding the region.

Stokes'theorem

The integral of the curl of a vector field over a surface in equalsthe line integral of the vector field over the closed curve bounding

the surface.

Divergencetheorem

The integral of the divergence of a vector field over some solidequals the integral of the flux through the closed surface bounding

the solid.

Generalizations

Different 3-manifolds

Vector calculus is initially defined for Euclidean 3-space, which has additional structure beyond simply being a3-dimensional real vector space, namely: an inner product (the dot product), which gives a notion of length (andhence angle), and an orientation, which gives a notion of left-handed and right-handed. These structures give rise to avolume form, and also the cross product, which is used pervasively in vector calculus.The gradient and divergence only require the inner product, while the curl and the cross product also requires thehandedness of the coordinate system to be taken into account (see cross product and handedness for more detail).Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or moregenerally a symmetric nondegenerate form) and an orientation; note that this is less data than an isomorphism toEuclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vectorcalculus is invariant under rotations (the special orthogonal group SO(3)).

Vector calculus 3

More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold, or moregenerally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an innerproduct (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is asymmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms oftangent vectors at each point.

Other dimensionsMost of the analytic results are easily understood, in a more general form, using the machinery of differentialgeometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as dothe gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross productdo not generalize as directly.From a general point of view, the various fields in (3-dimensional) vector calculus are uniformly seen as beingk-vector fields: scalar fields are 0-vector fields, vector fields are 1-vector fields, pseudovector fields are 2-vectorfields, and pseudoscalar fields are 3-vector fields. In higher dimensions there are additional types of fields(scalar/vector/pseudovector/pseudoscalar corresponding to 0/1/n−1/n dimensions, which is exhaustive in dimension3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector fieldis a scalar function, but only in dimension 3 and 7[1] (and, trivially, dimension 0) is the curl of a vector field a vectorfield, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities eitherrequire vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetricbilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl:Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the specialorthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because thedimensions differ - there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4dimensions (and more generally dimensions of rotations in n dimensions).There are two important alternative generalizations of vector calculus. The first, geometric algebra, uses k-vectorfields instead of vector fields (in 3 or fewer dimensions, every k-vector field can be identified with a scalar functionor vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3dimensions, taking in two vector fields and giving as output a vector field, with the exterior product, which exists inall dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yieldsClifford algebras as the algebraic structure on vector spaces (with an orientation and nondegenerate form).Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and iswidely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, inparticular yielding Hodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, anddiv correspond to the differential of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vectorcalculus are all special cases of the general form of Stokes' theorem.From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinctobjects, which makes the presentation more elegant but the underlying mathematical structure and generalizationsless clear. From the point of view of geometric algebra, vector calculus implicitly identifies k-vector fields withvector fields or scalar functions: 0-vectors and 3-vectors with scalars, 1-vectors and 2-vectors with vectors. From thepoint of view of differential forms, vector calculus implicitly identifies k-forms with scalar fields or vector fields:0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturallytakes as input a vector field, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), whichis then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in thecurl of a vector field in higher dimensions not having as output a vector field.

Vector calculus 4

References• Michael J. Crowe (1967). A History of Vector Analysis : The Evolution of the Idea of a Vectorial System. Dover

Publications; Reprint edition. ISBN 0-486-67910-1.• H. M. Schey (2005). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company.

ISBN 0-393-92516-1.• J.E. Marsden (1976). Vector Calculus. W. H. Freeman & Company. ISBN 0-7167-0462-5.• Chen-To Tai (1995). A historical study of vector analysis [2]. Technical Report RL 915, Radiation Laboratory,

University of Michigan.

External links• Vector Calculus Video Lectures [3] from University of New South Wales on Academic Earth• A survey of the improper use of ∇ in vector analysis [4] (1994) Tai, Chen• Expanding vector analysis to an oblique coordinate system [5]

• Vector Analysis: [6] A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures ofWillard Gibbs) by Edwin Bidwell Wilson, published 1902.

• Earliest Known Uses of Some of the Words of Mathematics: Vector Analysis [7]

References[1] http:/ / www. springerlink. com/ content/ r3p3602pq2t10036/[2] http:/ / deepblue. lib. umich. edu/ handle/ 2027. 42/ 7868[3] http:/ / academicearth. org/ courses/ vector-calculus[4] http:/ / hdl. handle. net/ 2027. 42/ 7869[5] http:/ / www. mc. maricopa. edu/ ~kevinlg/ i256/ Nonortho_math. pdf[6] http:/ / books. google. com/ books?id=R5IKAAAAYAAJ& printsec=frontcover[7] http:/ / www. economics. soton. ac. uk/ staff/ aldrich/ vector%20analysis. htm

Gradient 5

GradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate ofincrease of the scalar field, and whose magnitude is the greatest rate of change.A generalization of the gradient for functions on a Euclidean space which have values in another Euclidean space isthe Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.

In the above two images, the scalar field is in black and white, black representinghigher values, and its corresponding gradient is represented by blue arrows.

Interpretations

Consider a room in which the temperature isgiven by a scalar field, , so at each point

the temperature is (we will assume that the temperature doesnot change in time). At each point in theroom, the gradient of at that point willshow the direction the temperature risesmost quickly. The magnitude of the gradientwill determine how fast the temperaturerises in that direction.

Consider a surface whose height above sea level at a point is . The gradient of at a point is avector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point isgiven by the magnitude of the gradient vector.The gradient can also be used to measure how a scalar field changes in other directions, rather than just the directionof greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. If a road goes directlyup the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle(the gradient vector), then it will have a shallower slope. For example, if the angle between the road and the uphilldirection, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is40% times the cosine of 60°.This observation can be mathematically stated as follows. If the hill height function is differentiable, then thegradient of dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely,when is differentiable, the dot product of the gradient of with a given unit vector is equal to the directionalderivative of in the direction of that unit vector.

Gradient 6

Definition

The gradient of the function f(x,y) = −(cos2x + cos2y)2 depicted as a vector field on thebottom plane

The gradient (or gradient vector field) of a scalar function is denoted or where(the nabla symbol) denotes the vector differential operator, del. The notation is also used for the

gradient. The gradient of f is defined to be the vector field whose components are the partial derivatives of . Thatis:

Here the gradient is written as a row vector, but it is often taken to be a column vector. When a function also dependson a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

The gradient of a vector is

or the transpose of the Jacobian matrix .

It is a second-rank tensor.More generally, the gradient may be defined using the exterior derivative:

Here and are the musical isomorphisms.

Gradient 7

Expression in 3-dimensional rectangular coordinatesThe form of the gradient depends on the coordinate system used. In Cartesian coordinates, the above expressionexpands to

which is often written using the standard vectors :

ExampleFor example, the gradient of the function in Cartesian coordinates

is:

Gradient and the derivative or differential

Linear approximation to a function

The gradient of a function from the Euclidean space to at any particular point x0 in characterizes thebest linear approximation to f at x0. The approximation is as follows: for

close to , where is the gradient of f computed at , and the dot denotes the dot product on .This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of f at x0.

Differential or (exterior) derivative

The best linear approximation to a function at a point in is a linear map from to which is often denoted by or and called the differential or (total) derivative of at . Thegradient is therefore related to the differential by the formula for any . Thefunction , which maps to , is called the differential or exterior derivative of and is an example of adifferential 1-form.If is viewed as the space of (length ) column vectors (of real numbers), then one can regard as the rowvector

so that is given by matrix multiplication. The gradient is then the corresponding column vector, i.e.,.

Gradient 8

Gradient as a derivativeLet U be an open set in Rn. If the function f:U → R is differentiable, then the differential of f is the (Fréchet)derivative of f. Thus is a function from U to the space R such that

where • is the dot product.As a consequence, the usual properties of the derivative hold for the gradient:LinearityThe gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a∈Rn, and αand β are two constants, then αf+βg is differentiable at a, and moreover

Product ruleIf f and g are real-valued functions differentiable at a point a∈Rn, then the product rule asserts that the product (fg)(x)= f(x)g(x) of the functions f and g is differentiable at a, and

Chain ruleSuppose that f:A→R is a real-valued function defined on a subset A of Rn, and that f is differentiable at a point a.There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametriccurve; that is, a function g : I → Rn maps a subset I ⊂ R into Rn. If g is differentiable at a point c ∈ I such that g(c) =a, then

More generally, if instead I⊂Rk, then the following holds:

where (Dg)T denotes the transpose Jacobian matrix.For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h isdifferentiable at the point c = f(a) ∈ I. Then

Transformation propertiesAlthough the gradient is defined in term of coordinates, it is contravariant under the application of an orthogonalmatrix to the coordinates. This is true in the sense that if A is an orthogonal matrix, then

which follows by the chain rule above. A vector transforming in this way is known as a contravariant vector, and sothe gradient is a special type of tensor.The differential is more natural than the gradient because it is invariant under all coordinate transformations (ordiffeomorphisms), whereas the gradient is only invariant under orthogonal transformations (because of the implicituse of the dot product in its definition). Because of this, it is common to blur the distinction between the twoconcepts using the notion of covariant and contravariant vectors. From this point of view, the components of thegradient transform covariantly under changes of coordinates, so it is called a covariant vector field, whereas thecomponents of a vector field in the usual sense transform contravariantly. In this language the gradient is thedifferential, as a covariant vector field is the same thing as a differential 1-form.[1]

[1] Unfortunately this confusing language is confused further by differing conventions. Although the components of a differential 1-formtransform covariantly under coordinate transformations, differential 1-forms themselves transform contravariantly (by pullback) under

Gradient 9

diffeomorphism. For this reason differential 1-forms are sometimes said to be contravariant rather than covariant, in which case vector fieldsare covariant rather than contravariant.

Further properties and applications

Level sets

If the partial derivatives of f are continuous, then the dot product of the gradient at a point x with avector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f isorthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation ofthe form F(x, y, z) = c. The gradient of F is then normal to the surface.More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the formF(P) = 0 such that dF is nowhere zero. The gradient of F is then normal to the hypersurface.Let us consider a function f at a point P. If we draw a surface through this point P and the function has the samevalue at all points on this surface,then this surface is called a 'level surface'.

Conservative vector fieldsThe gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vectorfield: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradienttheorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vectorfield is always the gradient of a function.

Riemannian manifoldsFor any smooth function f on a Riemannian manifold (M,g), the gradient of f is the vector field such that forany vector field ,

where denotes the inner product of tangent vectors at x defined by the metric g and (sometimesdenoted X(f)) is the function that takes any point x∈M to the directional derivative of f in the direction X, evaluated atx. In other words, in a coordinate chart from an open subset of M to an open subset of Rn, is givenby:

where Xj denotes the jth component of X in this coordinate chart.So, the local form of the gradient takes the form:

Generalizing the case M=Rn, the gradient of a function is related to its exterior derivative, since. More precisely, the gradient is the vector field associated to the differential 1-form

df using the musical isomorphism (called "sharp") defined by the metric g. The relationbetween the exterior derivative and the gradient of a function on Rn is a special case of this in which the metric is theflat metric given by the dot product.

Gradient 10

Non-Cartesian Coordinate SystemsIn cylindrical coordinates, the gradient is given by (Schey 1992, pp. 139–142):

where is the azimuthal angle, is the axial coordinate, and eρ, eθ and ez are unit vectors pointing along thecoordinate directions.In spherical coordinates (Schey 1992, pp. 139–142):

where is the azimuth angle and is the zenith angle.

References• Korn, Theresa M.; Korn, Granino Arthur (2000), Mathematical Handbook for Scientists and Engineers:

Definitions, Theorems, and Formulas for Reference and Review, New York: Dover Publications, pp. 157–160,ISBN 0-486-41147-8, OCLC 43864234.

• Schey, H.M. (1992), Div, Grad, Curl, and All That (2nd ed.), W.W. Norton, ISBN 0-393-96251-2,OCLC 25048561.

External links• Kuptsov, L.P. (2001), "Gradient" (http:/ / eom. springer. de/ G/ g044680. htm), in Hazewinkel, Michiel,

Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104• Weisstein, Eric W., " Gradient (http:/ / mathworld. wolfram. com/ Gradient. html)" from MathWorld.

Divergence 11

DivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at agiven point, in terms of a signed scalar. More technically, the divergence represents the volume density of theoutward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it isheated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heatedin a region it will expand in all directions such that the velocity field points outward from that region. Therefore thedivergence of the velocity field in that region would have a positive value, as the region is a source. If the air coolsand contracts, the divergence is negative and the region is called a sink.

Definition of divergenceIn physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flowbehaves like a source or a sink at a given point. It is a local measure of its "outgoingness"—the extent to which thereis more exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point then theremust be a source or sink at that position[1] . (Note that we are imagining the vector field to be like the velocity vectorfield of a fluid (in motion) when we use the terms flow, sink and so on.)More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across thesmooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

where |V | is the volume of V, S(V) is the boundary of V, and the integral is a surface integral with n being theoutward unit normal to that surface. The result, div F, is a function of the location p. From this definition it alsobecomes explicitly visible that div F can be seen as the source density of the flux of F.In light of the physical interpretation, a vector field with constant zero divergence is called incompressible orsolenoidal – in this case, no net flow can occur across any closed surface.The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region ismade precise by the divergence theorem.

Application in Cartesian coordinatesLet x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be thecorresponding basis of unit vectors.The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valuedfunction:

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physicalinterpretation suggests.The common notation for the divergence ∇·F is a convenient mnemonic, where the dot denotes an operationreminiscent of the dot product: take the components of ∇ (see del), apply them to the components of F, and sum theresults. As a result, this is considered an abuse of notation.

Divergence 12

Decomposition theoremIt can be shown that any stationary flux v(r) which is at least two times continuously differentiable in andvanishes sufficiently fast for |r| → ∞ can be decomposed into an irrotational part E(r) and a source-free part B(r).Moreover, these parts are explicitly determined by the respective source-densities (see above) and circulationdensities (see the article Curl):For the irrotational part one has

with

The source-free part, B, can be similarly written: one only has to replace the scalar potential Φ(r) by a vectorpotential A(r) and the terms −∇Φ by +∇×A, and finally the source-density div v by the circulation-density ∇×v.This "decomposition theorem" is in fact a by-product of the stationary case of electrodynamics. It is a special case ofthe more general Helmholtz decomposition which works in dimensions greater than three as well.

PropertiesThe following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, thedivergence is a linear operator, i.e.

for all vector fields F and G and all real numbers a and b.There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

or in more suggestive notation

Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl andreads as follows:

or

The Laplacian of a scalar field is the divergence of the field's gradient.The divergence of the curl of any vector field (in three dimensions) is equal to zero:

If a vector field F with zero divergence is defined on a ball in R3, then there exists some vector field G on the ballwith F = curl(G). For regions in R3 more complicated than this, the latter statement might be false (see Poincarélemma). The degree of failure of the truth of the statement, measured by the homology of the chain complex

(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham

Divergence 13

cohomology.

Relation with the exterior derivativeOne can establish a parallel between the divergence and a particular case of the exterior derivative, when it takes a2-form to a 3-form in R3. If we define:

its exterior derivative is given by

Thus, using the exterior derivative, the divergence can be expressed as:

Here the superscript is one of the two musical isomorphisms, and is the Hodge dual.

GeneralizationsThe divergence of a vector field can be defined in any number of dimensions. If

in a Euclidean coordinate system where and , define

The appropriate expression is more complicated in curvilinear coordinates.For any n, the divergence is a linear operator, and it satisfies the "product rule"

for any scalar-valued function .The divergence can be defined on any manifold of dimension n with a volume form (or density) e.g. aRiemannian or Lorentzian manifold. Generalising the construction of a two form for a vectorfield on , on such amanifold a vectorfield X defines a n-1 form obtained by contracting X with . The divergence is thenthe function defined by

Standard formulas for the Lie derivative allow us to reformulate this as

This means that the divergence measures the rate of expansion of a volume element as we let it flow with thevectorfield.On a Riemannian or Lorentzian manifold the divergence with respect to the metric volume form can be computed interms of the Levi Civita connection

where the second expression is the contraction of the vectorfield valued 1 -form with itself and the lastexpression is the traditional coordinate expression used by physicists.Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector isgiven by

Divergence 14

where is the covariant derivative. Equivalently, some authors define the divergence of any mixed tensor byusing the "musical notation #":If T is a (p,q)-tensor,(p for the contravariant vector and q for the covariant one), then we define the divergence of T tobe the (p,q-1)-tensor

,that is we trace the covariant derivative on the first two covariant indices.

Notes[1] DIVERGENCE of a Vector Field (http:/ / musr. phas. ubc. ca/ ~jess/ hr/ skept/ Gradient/ node4. html)

References1. Brewer, Jess H. (1999-04-07). "DIVERGENCE of a Vector Field" (http:/ / musr. phas. ubc. ca/ ~jess/ hr/ skept/

Gradient/ node4. html). Vector Calculus. Retrieved 2007-09-28.2. Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions,

Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160.ISBN 0-486-41147-8.

External links• The idea of divergence and curl (http:/ / www. math. umn. edu/ ~nykamp/ m2374/ readings/ divcurl)

Curl (mathematics)In vector calculus, the curl (or rotor) is a vector operator that describes the infinitesimal rotation of a 3-dimensionalvector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length anddirection) characterize the rotation at that point.The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl isthe magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is thecirculation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form ofdifferentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem,which relates the surface integral of the curl of a vector field to the line integral of the vector field around theboundary curve.The alternative terminology rotor or rotational and alternative notations rot F and ∇×F are often used (the formerespecially in many European countries, the latter using the del operator and the cross product) for curl and curl F.Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations arepossible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This is asimilar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇× for thecurl.The name "curl" was first suggested by James Clerk Maxwell in 1871.[1]

Curl (mathematics) 15

DefinitionThe curl of a vector field F, denoted curl F or ∇×F, at a point is defined in terms of its projection onto various linesthrough the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value ofa closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to thepoint, divided by the area enclosed.As such, the curl operator maps C1 functions from R3 to R3 to C0 functions from R3 to R3.

Convention for vector orientation of the line integral

Implicitly, curl is defined by:[2]

Here is a line integral along the boundary of the area in question, and A is the magnitude of the area. If isan outward pointing in-plane normal, whereas is the unit-vector perpendicular to the plane (see caption at right),then the orientation of C is chosen so that a vector tangent to C is positively oriented if and only if formsa positively oriented basis for R3 (right-hand rule).The above formula means that the curl of a vector field is defined as the infinitesimal area-density of the circulationof that field. To this definition fit naturally (i) the Kelvin-Stokes theorem, as a global formula corresponding to thedefinition, and (ii) the following "easy to memorize" definition of the curl in orthogonal curvilinear coordinates, e.g.in cartesian coordinates, spherical, or cylindrical, or even elliptical or parabolical coordinates:

If (x1,x2,x3) are the Cartesian coordinates and (u1,u2,u3) are the curvilinear coordinates, then is the

length of the coordinate vector corresponding to . The remaining two components of curl result from cyclicindex-permutation: 3,1,2 → 1,2,3 → 2,3,1.

Intuitive interpretationSuppose the vector field describes the velocity field of a fluid flow (maybe a large tank of water or gas) and a smallball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a roughsurface it will be made to rotate by the fluid flowing past it. The rotation axis (oriented according to the right handrule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation ishalf the value of the curl at this point.[3]

Curl (mathematics) 16

UsageIn practice, the above definition is rarely used because in virtually all cases, the curl operator can be applied usingsome set of curvilinear coordinates, for which simpler representations have been derived.The notation ∇×F has its origins in the similarities to the 3 dimensional cross product, and it is useful as a mnemonicin Cartesian coordinates if we take ∇ as a vector differential operator del. Such notation involving operators iscommon in physics and algebra. If certain coordinate systems are used, for instance, polar-toroidal coordinates(common in plasma physics) using the notation ∇×F will yield an incorrect result.Expanded in Cartesian coordinates (see: Del in cylindrical and spherical coordinates for spherical and cylindricalcoordinate representations), ∇×F is, for F composed of [Fx, Fy, Fz]:

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[4]

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes butthe result inverts under reflection.In a general coordinate system, the curl is given by[2]

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einsteinsummation convention implies that repeated indices are summed over. Equivalently,

where ek are the coordinate vector fields. Equivalently, using the exterior derivative, the curl can be expressed as:

Here and are the musical isomorphisms, and is the Hodge dual. This formula shows how to calculate the curlof F in any coordinate system, and how to extend the curl to any oriented three dimensional Riemannian manifold.Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed,then the direction of the curl is also reversed.

Curl (mathematics) 17

Examples

A simple vector fieldTake the vector field, which depends on x and y linearly:

Its plot looks like this:

Simply by visual inspection, we can see that the field is rotating. If we stick a paddle wheel anywhere, we seeimmediately its tendency to rotate clockwise. Using the right-hand rule, we expect the curl to be into the page. If weare to keep a right-handed coordinate system, into the page will be in the negative z direction. The lack of x and ydirections is analogous to the cross product operation.If we calculate the curl:

Which is indeed in the negative z direction, as expected. In this case, the curl is actually a constant, irrespective ofposition. The "amount" of rotation in the above vector field is the same at any point (x, y). Plotting the curl of F isnot very interesting:

Curl (mathematics) 18

A more involved exampleSuppose we now consider a slightly more complicated vector field:

Its plot:

We might not see any rotation initially, but if we closely look at the right, we see a larger field at, say, x=4 than atx=3. Intuitively, if we placed a small paddle wheel there, the larger "current" on its right side would cause thepaddlewheel to rotate clockwise, which corresponds to a curl in the negative z direction. By contrast, if we look at apoint on the left and placed a small paddle wheel there, the larger "current" on its left side would cause thepaddlewheel to rotate counterclockwise, which corresponds to a curl in the positive z direction. Let's check out ourguess by doing the math:

Curl (mathematics) 19

Indeed the curl is in the positive z direction for negative x and in the negative z direction for positive x, as expected.Since this curl is not the same at every point, its plot is a bit more interesting:

Curl of F with the x=0 plane emphasized in dark blue

We note that the plot of this curl has no dependence on y or z (as it shouldn't) and is in the negative z direction forpositive x and in the positive z direction for negative x.

Identities

Consider the example ∇ × [ v × F ]. Using Cartesian coordinates, it can be shown that

In the case where the vector field v and ∇ are interchanged:

which introduces the Feynman subscript notation ∇F, which means the subscripted gradient operates only on thefactor F.

Another example is ∇ × [ ∇ × F ]. Using Cartesian coordinates, it can be shown that:

which can be construed as a special case of the previous example with the substitution v → ∇.The curl of the gradient of any scalar field is always the zero vector:

If is a scalar valued function and F is a vector field, then

Curl (mathematics) 20

Descriptive examples• In a vector field describing the linear velocities of each part of a rotating disk, the curl has the same value at all

points.• Of the four Maxwell's equations two, Faraday's law and Ampère's law can be compactly expressed using curl.

Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of themagnetic field, while Ampère's law relates the curl of the magnetic field to the current and rate of change of theelectric field.

GeneralizationsThe vector calculus operations of grad, curl, and div are most easily generalized and understood in the context ofdifferential forms, which involves a number of steps. In a nutshell, they correspond to the derivatives of 0-forms,1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifyingbivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra of infinitesimal rotations (incoordinates, skew-symmetric 3×3 matrices), while representing rotations by vectors corresponds to identifying1-vectors (equivalently, 2-vectors) and these all being 3-dimensional spaces.

Differential forms

In 3 dimensions, a differential 0-form is simply a function ; a differential 1-form is a linear combinationof three functions a differential 2-form is a linear combination of three functions

and a differential 3-form is defined by a single function:The exterior derivative of a k-form is a -form, and denoting the space of k-forms by

and the exterior derivative by d yields a sequence:

Here is the space of sections of the exterior algebra vector bundle over Rn, whose dimension isthe binomial coefficient note that for or Writing only dimensions, one obtainsa row of Pascal's triangle:

the 1-dimensional fibers correspond to functions, and the 3-dimensional fibers to vector fields, as described below.Note that modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond tograd, curl, and div.Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without anynotion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-formscan be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian metric gives anisomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (anisomorphism between vectors and covectors), there is an isomorphism between k-vectors and -vectors; inparticular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an orientedpseudo-Riemannian manifold, one can interchange k-forms, k-vector fields, -forms, and -vectorfields; this is known as Hodge duality. Concretely, on this is given by:• 1-forms and 1-vector fields: the 1-form corresponds to the vector field • 1-forms and 2-forms: one replaces by (i.e., omit dx), and likewise, taking care of orientation:

corresponds to and corresponds to Thus corresponds to

Thus, identifying 0-forms and 3-forms with functions, and 1-forms and 2-forms with vector fields:• grad takes a function (0-form) to a vector field (1-form);

Curl (mathematics) 21

• curl takes a vector field (1-form) to a vector field (2-form);• div takes a vector field (2-form) to a function (3-form).Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation,because the spaces of 0-forms and n-forms is always (fiberwise) 1-dimensional and can be identified with scalarfunctions, while the spaces of 1-forms and -forms are always fiberwise n-dimensional and can beidentified with vector fields.Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensionsthe dimensions are

so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which is fiberwise 6-dimensional andcannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a3-vector field ( ), as taking the differential twice yields zero ( ). Thus there is no curl functionfrom vector fields to vector fields in other dimensions arising in this way.However, one can define a curl of a vector field as a 2-vector field in general, as described below.

Curl geometrically

2-vectors correspond to the exterior power in the presence of an inner product, in coordinates these are theskew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra ofinfinitesimal rotations. This has dimensions, and allows one to interpret the differential of a1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) is which is the most elegant and common case. is In 2 dimensions the curl of a vector field is not a vector field but afunction, as 2-dimensional rotations are given by an angle (a scalar - an orientation is required to choose whether onecounts clockwise or counterclockwise rotations as positive); note that this is not the div, but is rather perpendicular toit. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vectorfield is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is,geometrically, at each point an element of the 6-dimensional Lie algebra Note also that the curl of a 3-dimensional vector field which only depends on 2 coordinates (say x, y) is simply avertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in theexamples on this page.Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus andassociated physics to higher dimensions.[5]

See also• Del• Gradient• Divergence• Nabla in cylindrical and spherical coordinates• Vorticity• Cross product• Helmholtz decomposition

Curl (mathematics) 22

Notes[1] Proceedings of the London Mathematical Society, March 9th, 1871 (http:/ / www. clerkmaxwellfoundation. org/

MathematicalClassificationofPhysicalQuantities_Maxwell. pdf)[2] Weisstein, Eric W., " Curl (http:/ / mathworld. wolfram. com/ Curl. html)" from MathWorld.[3] Gibbs, Josiah Willard; Wilson, Edwin Bidwell (1902), Vector analysis (http:/ / books. google. com/ books?id=R5IKAAAAYAAJ&

printsec=frontcover),[4] Arfken, p. 43.[5] Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions (http:/ / arxiv. org/ abs/ hep-ph/ 0609260), A.W.

McDavid, C.D. McMullen, 2006

References• Arfken, George B. and Hans J. Weber. Mathematical Methods For Physicists, Academic Press; 6 edition (June

21, 2005). ISBN 978-0120598762.• Korn, Granino Arthur and Theresa M. Korn. Mathematical Handbook for Scientists and Engineers: Definitions,

Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160.ISBN 0-486-41147-8.

External links• The idea of divergence and curl (http:/ / www. math. umn. edu/ ~nykamp/ m2374/ readings/ divcurl)

Laplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradientof a function on Euclidean space. It is usually denoted by the symbols Δ, ∇2 or ∇·∇. The Laplacian Δƒ(p) of afunction ƒ at a point p, up to a constant depending on the dimension, is the rate at which the average value of ƒ overspheres centered at p deviates from ƒ(p) as the radius of the sphere grows. In a Cartesian coordinate system, theLaplacian is given by sum of all the (unmixed) second partial derivatives of the function. In other coordinate systemssuch as cylindrical and spherical coordinates, the Laplacian also has a useful form.The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who firstapplied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the massdensity when it is applied to a given gravitational potential. Solutions of the equation Δƒ = 0, now called Laplace'sequation, are the so-called harmonic functions, and represent the possible gravitational fields in free space.It occurs in the differential equations that describe many physical phenomena, such as electric and gravitationalpotentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacianrepresents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolvedin a fluid moves towards or away from some point is proportional to the Laplacian of the chemical concentration atthat point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it isextensively used in the sciences for modelling all kinds of physical phenomena. The Laplacian is the simplest ellipticoperator, and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing andcomputer vision, the Laplacian operator has been used for various tasks such as blob and edge detection.

Laplace operator 23

DefinitionThe Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as thedivergence (∇·) of the gradient (∇ƒ). Thus if ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ isdefined by

   (1)Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates

:

   (2)

As a second-order differential operator, the Laplace operator maps Ck-functions to Ck−2-functions for k ≥ 2. Theexpression (1) (or equivalently (2)) defines an operator Δ : Ck(Rn) → Ck−2(Rn), or more generally an operator Δ :Ck(Ω) → Ck−2(Ω) for any open set Ω.

Motivation

DiffusionIn the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematicaldescription of equilibrium.[1] Specifically, if u is the density at equilibrium of some quantity such as a chemicalconcentration, then the net flux of u through the boundary of any smooth region V is zero, provided there is nosource or sink within V:

where n is the outward unit normal to the boundary of V. By the divergence theorem,

Since this holds for all smooth regions V, it can be shown that this implies

The left-hand side of this equation is the Laplace operator. The Laplace operator itself has a physical interpretationfor non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, ina sense made precise by the diffusion equation.

Density associated to a potentialIf φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is givenby the Laplacian of φ:{{{}}}

(1)

{{{}}}This is a consequence of Gauss's law. Indeed, if V is any smooth region, then by Gauss's law the flux of theelectrostatic field E is equal to the charge enclosed (in appropriate units):

Laplace operator 24

where the first equality uses the fact that the electrostatic field is the gradient of the electrostatic potential. Thedivergence theorem now gives

and since this holds for all regions V, (1) follows.The same approach implies that the Laplacian of the gravitational potential is the mass distribution. Often the charge(or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject tosuitable boundary conditions is equivalent to solving Poisson's equation.

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to in a region U are functionsthat make the Dirichlet energy functional stationary:

To see this, suppose is a function, and is a function that vanishes on the boundary of U.Then

where the last equality follows using Green's first identity. This calculation shows that if , then E isstationary around f. Conversely, if E is stationary around f, then by the fundamental lemma of calculus ofvariations.

Coordinate expressions

Two dimensionsThe Laplace operator in two dimensions is given by

where x and y are the standard Cartesian coordinates of the xy-plane.In polar coordinates,

Three dimensionsIn three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.In Cartesian coordinates,

In cylindrical coordinates,

In spherical coordinates:

Laplace operator 25

(here θ represents the azimuthal angle and φ the zenith angle).

In general curvilinear coordinates ( ):

where summation over the repeated indices is implied.

N dimensionsIn spherical coordinates in N dimensions, with the parametrization x = rθ ∈ RN with r representing a positive realradius and θ an element of the unit sphere SN−1,

where is the Laplace–Beltrami operator on the (N−1)-sphere, known as the spherical Laplacian. The tworadial terms can be equivalently rewritten as

As a consequence, the spherical Laplacian of a function defined on SN−1 ⊂ RN can be computed as the ordinaryLaplacian of the function extended to RN\{0} so that it is constant along rays, i.e., homogeneous of degree zero.

Spectral theoryThe spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒwith

If Ω is a bounded domain in Rn then the eigenfunctions of the Laplacian are an orthonormal basis in the Hilbertspace L2(Ω). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied tothe inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem).[2] Itcan also be shown that the eigenfunctions are infinitely differentiable functions.[3] More generally, these results holdfor the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichleteigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the n-sphere,the eigenfunctions of the Laplacian are the well-known spherical harmonics.

Generalizations

D'AlembertianThe Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, orultrahyperbolic.In the Minkowski space the Laplacian becomes the d'Alembert operator or d'Alembertian:

The D'Alembert operator is also known as the wave operator, because it is the differential operator appearing in the four-dimensional wave equation. It is also the leading part of the Klein–Gordon equation. The signs in front of the spatial derivatives are negative, while they would have been positive in the Euclidean space. The additional factor of c is needed if space and time are measured in different units; a similar factor would be required if, for example, the x

Laplace operator 26

direction were measured in meters while the y direction were measured in centimeters. Indeed, physicists usuallywork in units such that c=1 in order to simplify the equation.

Laplace–Beltrami operatorThe Laplacian can also be generalized to an elliptic operator called the Laplace–Beltrami operator defined on aRiemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannianmanifolds. The Laplace–Beltrami operator, when applied to a function, trace of the function's Hessian:

where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator can also begeneralized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similarformula.Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exteriorderivative, in terms of which the Laplacian is expressed as

Here d∗ is the codifferential, which can also be expressed using the Hodge dual. More generally, the Laplacian isdefined on differential forms α by

This is known as the Laplace–de Rham operator, which is related Laplace–Beltrami operator by the Weitzenböckidentity.

Notes[1] Evans 1998, §2.2[2] Gilbarg & Trudinger 2001, Theorem 8.6[3] Gilbarg & Trudinger 2001, Corollary 8.11

References• Evans, L (1998), Partial Differential Equations, American Mathematical Society, ISBN 978-0821807729.• Feynman, R, Leighton, R, and Sands, M (1970), "Chapter 12: Electrostatic Analogs", The Feynman Lectures on

Physics, Volume 2, Addison-Wesley-Longman.• Gilbarg, D.; Trudinger, N. (2001), Elliptic partial differential equations of second order, Springer,

ISBN 978-3540411604.• Schey, H. M. (1996), Div, grad, curl, and all that, W W Norton & Company, ISBN 978-0393969979.• M.A. Shubin (2001), "Laplace operator" (http:/ / eom. springer. de/ / l/ l057510. htm), in Hazewinkel, Michiel,

Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104.

External links• Weisstein, Eric W., " Laplacian (http:/ / mathworld. wolfram. com/ Laplacian. html)" from MathWorld.

Gradient theorem 27

Gradient theoremThe gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a lineintegral through a gradient field (any conservative vector field can be expressed as a gradient) can be evaluated byevaluating the original scalar field at the endpoints of the curve:

It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generallyn-dimensional) rather than just the real line.The gradient theorem implies that line integrals through irrotational vector fields are path independent. In physicsthis theorem is one of the ways of defining a "conservative" force. By placing as potential, is a conservativefield. Work done by conservative forces does not depend on the path followed by the object, but only the end points,as the above equation shows.

ProofLet be a 0-form (scalar field).Let L be a 1-segment (curve) from p to q.By Stokes' theorem:

But because ,

Restricting the curve to Euclidean space and expanding in Cartesian coordinates:

Green's theorem 28

Green's theoremIn mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C anda double integral over the plane region D bounded by C. It is the two-dimensional special case of the more generalStokes' theorem, and is named after British mathematician George Green.Let C be a positively oriented, piecewise smooth, simple closed curve in the plane 2, and let D be the regionbounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partialderivatives there, then

For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in theintegral symbol.In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluidoutflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In planegeometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of planefigures solely by integrating over the perimeter.

Proof when D is a simple region

If D is a simple region with its boundary consisting of the curves C1, C2, C3, C4,Green's theorem can be demonstrated.

The following is a proof of the theorem forthe simplified area D, a type I region whereC2 and C4 are vertical lines. A similar proofexists for when D is a type II region whereC1 and C3 are straight lines. The generalcase can be deduced from this special caseby approximating the domain D by a unionof simple domains.

If it can be shown that

and

Green's theorem 29

are true, then Green's theorem is proven in the first case.Define the type I region D as pictured on the right by:

where g1 and g2 are continuous functions on [a, b]. Compute the double integral in (1):

Now compute the line integral in (1). C can be rewritten as the union of four curves: C1, C2, C3, C4.With C1, use the parametric equations: x = x, y = g1(x), a ≤ x ≤ b. Then

With C3, use the parametric equations: x = x, y = g2(x), a ≤ x ≤ b. Then

The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively(counterclockwise). On C2 and C4, x remains constant, meaning

Therefore,

Combining (3) with (4), we get (1). Similar computations give (2).

Relationship to the Stokes theoremGreen's theorem is a special case of Stokes' theorem, when applied to a region on the xy-plane:We can augment the two-dimensional field into a three-dimensional field with a z-component that is always 0:

.Start with the left side of Green's theorem:

Then by Stokes' Theorem:

The surface is just the region in the plane , with the unit normals pointing up (in +z direction) to match the"positive orientation" definitions for both theorems.The expression inside the integral becomes

Thus we get the right side of Green's theorem

Green's theorem 30

Relationship to the divergence theoremConsidering only two-dimensional vector fields, Green's theorem is equivalent to the following two-dimensionalversion of the divergence theorem:

where is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal in the right side of the equation. Since is a vector pointingtangential along a curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary,an outward normal would be a vector which points 90° to the right, which would be . The length of thisvector is . So

Now let the components of . Then the right hand side becomes

which by Green's theorem becomes

Area CalculationGreen's theorem can be used to compute area by line integral.[1] The area of D is given by:

Provided we choose L and M such that:

Then the area is given by:

Possible formulas for the area of D include[1] :

External links• Green's Theorem on MathWorld [2]

Reference[1] Stewart, James. Calculus (6th ed.). Thomson, Brooks/Cole.[2] http:/ / mathworld. wolfram. com/ GreensTheorem. html

Stokes' theorem 31

Stokes' theoremIn differential geometry, Stokes' theorem (or Stokes's theorem, also called the generalized Stokes' theorem) is astatement about the integration of differential forms on manifolds, which both simplifies and generalizes severaltheorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokesin July 1850.[1] [2] Stokes set the theorem as a question on the 1854 Smith's Prize exam [3], which led to the resultbearing his name.[2]

IntroductionThe fundamental theorem of calculus states that the integral of a function f over the interval [a, b] can be calculatedby finding an antiderivative F of f:

Stokes' theorem is a vast generalization of this theorem in the following sense.• By the choice of F, . In the parlance of differential forms, this is saying that f(x) dx is the exterior

derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies tohigher differential forms instead of F.

• A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its boundary is the setconsisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on ahigher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and theform has to be compactly supported in order to give a well-defined integral.

• The two points a and b form the boundary of the open interval. More generally, Stokes' theorem applies tooriented manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientationfrom that of the manifold. For example, the natural orientation of the interval gives an orientation of the twoboundary points. Intuitively, a inherits the opposite orientation as b, as they are at opposite ends of the interval.So, "integrating" F over two boundary points a, b is taking the difference F(b) − F(a).

So the fundamental theorem reads:

General formulationLet be an oriented smooth manifold of dimension n and let be an n-differential form that is compactlysupported on . First, suppose that α is compactly supported in the domain of a single, oriented coordinate chart{U, φ}. In this case, we define the integral of over as

i.e., via the pullback of α to Rn.More generally, the integral of over is defined as follows: Let {ψi} be a partition of unity associated with alocally finite cover {Ui, φi} of (consistently oriented) coordinate charts, then define the integral

where each term in the sum is evaluated by pulling back to Rn as described above. This quantity is well-defined; thatis, it does not depend on the choice of the coordinate charts, nor the partition of unity.

Stokes' theorem 32

Stokes' theorem reads: If is an (n − 1)-form with compact support on and denotes the boundary of withits induced orientation, then

An integration manifold known as"normal" (here instead of called

D) for the special case n=2

Here is the exterior derivative, which is defined using the manifold structure only. On the r.h.s., a circle issometimes used within the integral sign to stress the fact that the (n-1)-manifold is closed. [4] The r.h.s. of theequation is often used to formulate integral laws; the l.h.s. then leads to equivalent differential formulations (seebelow).The theorem is often used in situations where is an embedded oriented submanifold of some bigger manifold onwhich the form is defined.A proof becomes particularly simple if the submanifold is a so-called "normal manifold", as in the figure on ther.h.s., which can be segmented into vertical stripes (e.g. parallel to the xn direction), such that after a partialintegration concerning this variable, nontrivial contributions come only from the upper and lower boundary surfaces(coloured in yellow and red, respectively), where the complementary mutual orientations are visible through thearrows.

Topological reading; integration over chainsLet M be a smooth manifold. A smooth singular k-simplex of M is a smooth map from the standard simplex in Rk toM. The free abelian group, Sk, generated by singular k-simplices is said to consist of singular k-chains of M. Thesegroups, together with boundary map, ∂, define a chain complex. The corresponding homology (resp. cohomology) iscalled the smooth singular homology (resp. cohomology) of M.On the other hand, the differential forms, with exterior derivative, d, as the connecting map, form a cochain complex,which defines de Rham cohomology.Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to Rk. Extending bylinearity allows one to integrate over chains. This gives a linear map from the space of k-forms to the k-th group inthe singular cochain, Sk*, the linear functionals on Sk. In other words, a k-form defines a functional

on the k-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology;the exterior derivative, d, behaves like the dual of ∂ on forms. This gives a homomorphism from de Rhamcohomology to singular cohomology. On the level of forms, this means:1. closed forms, i.e., , have zero integral over boundaries, i.e. over manifolds that can be written as

, and

Stokes' theorem 33

2. exact forms, i.e., , have zero integral over cycles, i.e. if the boundaries sum up to the empty set:

.

De Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to 1 and 2 above holdtrue. In other words, if {ci} are cycles generating the k-th homology group, then for any corresponding real numbers,{ai}, there exist a closed form, , such that:

and this form is unique up to exact forms.

Underlying principle

To simplify these topological arguments, it isworthwhile to examine the underlying principle byconsidering an example for d = 2 dimensions. Theessential idea can be understood by the diagram on theleft, which shows that, in an oriented tiling of amanifold, the interior paths are traversed in oppositedirections; their contributions to the path integral thuscancel each other pairwise. As a consequence, only thecontribution from the boundary remains. It thus suffices

to prove Stokes' theorem for sufficiently fine tilings (or, equivalently, simplices), which usually is not difficult.

Special casesThe general form of the Stokes theorem using differential forms is more powerful and easier to use than the specialcases. Because in Cartesian coordinates the traditional versions can be formulated without the machinery ofdifferential geometry they are more accessible, older and have familiar names. The traditional forms are oftenconsidered more convenient by practicing scientists and engineers but the non-naturalness of the traditionalformulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindricalcoordinates. There is potential for confusion in the way names are applied, and the use of dual formulations.

Stokes' theorem 34

Kelvin–Stokes theorem

An illustration of the Kelvin–Stokes theorem,with surface , its boundary and the

"normal" vector n.

This is a (dualized) 1+1 dimensional case, for a 1-form (dualizedbecause it is a statement about vector fields). This special case is oftenjust referred to as the Stokes' theorem in many introductory universityvector calculus courses and as used in physics and engineering. It isalso sometimes known as the curl theorem.

The classical Kelvin–Stokes theorem:

which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the lineintegral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once weidentify a vector field with a 1 form using the metric on Euclidean three-space. The curve of the line integral, ∂Σ,must have positive orientation, meaning that dr points counterclockwise when the surface normal, dΣ, points towardthe viewer, following the right-hand rule.One consequence of the formula is that the field lines of a vector field with zero curl cannot be closed contours.The formula can be rewritten as:

  

where P, Q and R are the components of F.These variants are frequently used:

  

  

Stokes' theorem 35

In electromagnetism

Two of the four Maxwell equations involve curls of 3-D vector fields and their differential and integral forms arerelated by the Kelvin–Stokes theorem. Caution must be taken to avoid cases with moving boundaries: the partialtime derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integrationand differentiation introduces terms related to boundary motion not included in the results below:

Name Differential form Integral form (using Kelvin–Stokes theorem plus relativistic invariance, )

Maxwell-Faraday equationFaraday's law of induction: (with C and S not necessarily

stationary)Ampère's law(with Maxwell'sextension): (with C and S not necessarily stationary)

The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in SI units. In othersystems of units, such as CGS or Gaussian units, the scaling factors for the terms differ. For example, in Gaussianunits, Faraday's law of induction and Ampère's law take the forms[5] [6]

respectively, where c is the speed of light in vacuum.

Divergence theoremLikewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss's theorem)

is a special case if we identify a vector field with the n−1 form obtained by contracting the vector field with theEuclidean volume form.

Green's theoremGreen's theorem is immediately recognizable as the third integrand of both sides in the integral in terms of P, Q, andR cited above.

Notes[1] Olivier Darrigol,Electrodynamics from Ampere to Einstein, p. 146,ISBN 0198505930 Oxford (2000)[2] Spivak (1965), p. vii, Preface.[3] http:/ / www. clerkmaxwellfoundation. org/ SmithsPrizeExam_Stokes. pdf[4] For mathematicians this fact is known, therefore the circle is redundant and often left away. However, one should keep in mind here that in

thermodynamics, where frequently expressions as appear (wherein the total derivative, see below, should not be

mixed-up with the exterior one), the integration path W is a one-dimensional closed line on a much higher-dimensional manifold. I.e. in athermodynamic application, where U is a function of the temperature , the volume and the electrical polarization

of the sample, one has and the circle is really necessary, e.g. if one considers the

differential consequences of the integral postulate

[5] J.D. Jackson, Classical Electrodynamics, 2nd Ed (Wiley, New York, 1975).

Stokes' theorem 36

[6] M. Born and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, Cambridge, 1980).

Further reading• Joos, Georg. Theoretische Physik. 13th ed. Akademische Verlagsgesellschaft Wiesbaden 1980. ISBN

3-400-00013-2• Katz, Victor J. (May 1979), "The History of Stokes' Theorem" (http:/ / links. jstor. org/

sici?sici=0025-570X(197905)52:3<146:THOST>2. 0. CO;2-O), Mathematics Magazine 52 (3): 146–156• Marsden, Jerrold E., Anthony Tromba. Vector Calculus. 5th edition W. H. Freeman: 2003.• Lee, John. Introduction to Smooth Manifolds. Springer-Verlag 2003. ISBN 978-0-387-95448-6• Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X• Spivak, Michael (1965), Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced

Calculus, HarperCollins, ISBN 978-0-8053-9021-6• Stewart, James. Calculus: Concepts and Contexts. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001.• Stewart, James. Calculus: Early Transcendental Functions. 5th ed. Brooks/Cole, 2003.

External links• Proof of the Divergence Theorem and Stokes' Theorem (http:/ / higheredbcs. wiley. com/ legacy/ college/

hugheshallett/ 0471484822/ theory/ hh_focusontheory_sectionm. pdf)• Proof of general Stokes theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=4370) on

PlanetMath• Differential Forms and Stokes' Theorem Jerrold E. Marsden Control and Dynamical Systems, Caltech (https:/ /

www. cds. caltech. edu/ help/ uploads/ wiki/ files/ 177/ Diff_Forms_pauses. pdf)• Calculus 3 - Stokes Theorem from lamar.edu (http:/ / tutorial. math. lamar. edu/ classes/ calcIII/ stokestheorem.

aspx) - an expository explanation

Divergence theorem 37

Divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem (Carl Friedrich Gauss), Ostrogradsky'stheorem (Mikhail Vasilievich Ostrogradsky), or Gauss–Ostrogradsky theorem is a result that relates the flow (thatis, flux) of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equalto the volume integral of the divergence on the region inside the surface. Intuitively, it states that the sum of allsources minus the sum of all sinks gives the net flow out of a region.The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics andfluid dynamics.The theorem is a special case of the more general Stokes' theorem, which generalizes the fundamental theorem ofcalculus.[1]

IntuitionIf a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area,then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vectorfield, and the vector field's divergence at a given point describes the strength of the source or sink there. So,integrating the field's divergence over the interior of the region should equal the integral of the vector field over theregion's boundary. The divergence theorem says that this is true.The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, thevolume integral of the divergence, is equal to the net flow across the volume's boundary.[2]

Mathematical statement

A region V bounded by the surface S=∂V with the surface normal n

Suppose V is a subset of Rn (in the case of n = 3, Vrepresents a volume in 3D space) which is compact andhas a piecewise smooth boundary S. If F is acontinuously differentiable vector field defined on aneighborhood of V, then we have

Divergence theorem 38

The divergence theorem can be used to calculate a fluxthrough a closed surface that fully encloses a volume,

like any of the surfaces on the left. It can not be used tocalculate the flux through surfaces with boundaries,

like those on the right. (Surfaces are blue, boundariesare red.)

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of thevolume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and nis the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.) By thesymbol within the two integrals it is stressed once more that ∂V is a closed surface. In terms of the intuitivedescription above, the left-hand side of the equation represents the total of the sources in the volume V, and theright-hand side represents the total flow across the boundary ∂V.

CorollariesBy applying the divergence theorem in various contexts, other useful identities can be derived (cf. vector identities).• Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

A special case of this is , in which case the theorem is the basis for Green's identities.• Applying the divergence theorem to the cross-product of two vector fields , the result is

• Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, thefollowing theorem can be proven:[3]

.

Divergence theorem 39

• Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, thefollowing theorem can be proven:[3]

Example

A vector field on a sphere (this is not the field inthe example).

Suppose we wish to evaluate

where S is the unit sphere defined by and F is the vector field

The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using thedivergence theorem:

Since the functions and are odd on (which is a symmetric set with respect to the coordinate planes), one has

Therefore,

because the unit sphere W has volume .

Divergence theorem 40

Applications

Differential form and integral form of physical lawsAs a result of the divergence theorem, a host of physical laws can be written in both a differential form (where onequantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface isequal to another quantity). Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, andGauss's law for gravity.

Continuity equations

Continuity equations offer more examples of laws with both differential and integral forms, related to each other bythe divergence theorem. In fluid dynamics, electromagnetism, quantum mechanics, relativity theory, and a number ofother fields, there are continuity equations that describe the conservation of mass, momentum, energy, probability, orother quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equalto the distribution of sources or sinks of that quantity. The divergence theorem states that any such continuityequation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).

Inverse-square lawsAny inverse-square law can instead be written in a Gauss' law-type form (with a differential and integral form, asdescribed above). Two examples are Gauss' law (in electrostatics), which follows from the inverse-square Coulomb'slaw, and Gauss' law for gravity, which follows from the inverse-square Newton's law of universal gravitation. Thederivation of the Gauss' law-type equation from the inverse-square formulation (or vice-versa) is exactly the same inboth cases; see either of those articles for details.

HistoryThe theorem was first discovered by Joseph Louis Lagrange in 1762, then later independently rediscovered by CarlFriedrich Gauss in 1813, by George Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gavethe first proof of the theorem. Subsequently, variations on the Divergence theorem are called Gauss's Theorem,Green's theorem, and Ostrogradsky's theorem.

ExamplesVerify the planar variant of the divergence theorem for a region R, with F(x,y)=2yi + 5xj, where R is the regionbounded by the circle

Solution: The boundary of R is the unit circle, C, that can be represented parametrically by:

such that where s units is the length arc from the point s = 0 to the point P on C. Then a vector equation ofC is

At a point on C, . Therefore,

Divergence theorem 41

Because , , and because , . Thus

Notes[1] Stewart, James (2008), "Vector Calculus", Calculus: Early Transcendentals (6 ed.), Thomson Brooks/Cole, ISBN 9780495011668[2] Byron, Frederick; Fuller, Robert (1992), Mathematics of Classical and Quantum Physics, Dover Publications, p. 22, ISBN 9780486671642[3] MathWorld (http:/ / mathworld. wolfram. com/ DivergenceTheorem. html)

External links• Differential Operators and the Divergence Theorem (http:/ / www. mathpages. com/ home/ kmath330/ kmath330.

htm) at MathPages• The Divergence (Gauss) Theorem (http:/ / demonstrations. wolfram. com/ TheDivergenceGaussTheorem/ ) by

Nick Bykov, Wolfram Demonstrations Project.• Weisstein, Eric W., " Divergence Theorem (http:/ / mathworld. wolfram. com/ DivergenceTheorem. html)" from

MathWorld.

This article was originally based on the GFDL article from PlanetMath at http:/ / planetmath. org/ encyclopedia/Divergence. html

Green's function 42

Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subjectto specific initial conditions or boundary conditions. Under many-body theory, the term is also used in physics,specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types ofcorrelation functions, even those that do not fit the mathematical definition.Green's functions are named after the British mathematician George Green, who first developed the concept in the1830s. In the modern study of linear partial differential equations, Green's functions are studied largely from thepoint of view of fundamental solutions instead.

Definition and usesA Green's function, G(x, s), of a linear differential operator L = L(x) acting on distributions over a subset of theEuclidean space Rn, at a point s, is any solution of{{{}}}

(1)

{{{}}}where is the Dirac delta function. This property of a Green's function can be exploited to solve differentialequations of the form{{{}}}

(2)

{{{}}}If the kernel of L is non-trivial, then the Green's function is not unique. However, in practice, some combination ofsymmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function. Also,Green's functions in general are distributions, not necessarily proper functions.Green's functions are also a useful tool in solving wave equations, diffusion equations, and in quantum mechanics,where the Green's function of the Hamiltonian is a key concept, with important links to the concept of density ofstates. As a side note, the Green's function as used in physics is usually defined with the opposite sign; that is,

This definition does not significantly change any of the properties of the Green's function.If the operator is translation invariant, that is when L has constant coefficients with respect to x, then the Green'sfunction can be taken to be a convolution operator, that is,

In this case, the Green function is the same as the impulse response of linear time-invariant system theory.

Green's function 43

MotivationLoosely speaking, if such a function G can be found for the operator L, then if we multiply the equation (1) for theGreen's function by f(s), and then perform an integration in the s variable, we obtain;

The right hand side is now given by the equation (2) to be equal to L u(x), thus:

Because the operator L=L(x) is linear and acts on the variable x alone (not on the variable of integration s), we cantake the operator L outside of the integration on the right hand side, obtaining;

And this suggests;{{{}}}

(3)

{{{}}}Thus, we can obtain the function u(x) through knowledge of the Green's function in equation (1), and the source termon the right hand side in equation (2). This process has resulted from the linearity of the operator L.In other words, the solution of equation (2), u(x), can be determined by the integration given in equation (3).Although f(x) is known, this integration cannot be performed unless G is also known. The problem now lies infinding the Green's function G that satisfies equation (1). For this reason, the Green's function is also sometimescalled the fundamental solution associated to the operator L.Not every operator L admits a Green's function. A Green's function can also be thought of as a right inverse of L.Aside from the difficulties of finding a Green's function for a particular operator, the integral in equation (3), may bequite difficult to evaluate. However the method gives a theoretically exact result.This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over δ(x − s)) and asuperposition of the solution on each projection. Such an integral equation is known as a Fredholm integral equation,the study of which constitutes Fredholm theory.

Green's functions for solving inhomogeneous boundary value problemsThe primary use of Green's functions in mathematics is to solve non-homogeneous boundary value problems. Inmodern theoretical physics, Green's functions are also usually used as propagators in Feynman diagrams (and thephrase Green's function is often used for any correlation function).

FrameworkLet L be the Sturm–Liouville operator, a linear differential operator of the form

and let D be the boundary conditions operator

Let f(x) be a continuous function in [0,l]. We shall also suppose that the problem

Green's function 44

is regular (i.e., only the trivial solution exists for the homogeneous problem).

TheoremThere is one and only one solution u(x) which satisfies

and it is given by

where G(x,s) is a Green's function satisfying the following conditions:1. G(x,s) is continuous in x and s2. For , 3. For , 4. Derivative "jump": 5. Symmetry: G(x, s)=G(s, x)

Finding Green's functions

Eigenvalue expansions

If a differential operator L admits a set of eigenvectors (i.e., a set of functions and scalars such that) that are complete, then it is possible to construct a Green's function from these eigenvectors and

eigenvalues.Complete means that the set of functions satisfies the following completeness relation:

Then the following holds:

where * represents complex conjugation.Applying the operator L to each side of this equation results in the completeness relation, which was assumed true.The general study of the Green's function written in the above form, and its relationship to the function spacesformed by the eigenvectors, is known as Fredholm theory.

Green's function 45

Green's functions for the LaplacianGreen's functions for linear differential operators involving the Laplacian may be readily put to use using the secondof Green's identities.To derive Green's theorem, begin with the divergence theorem (otherwise known as Gauss's theorem):

Let and substitute into Gauss' law. Compute and apply the chain rule for the operator:

Plugging this into the divergence theorem produces Green's theorem:

Suppose that the linear differential operator L is the Laplacian, , and that there is a Green's function G for theLaplacian. The defining property of the Green's function still holds:

Let in Green's theorem. Then:

Using this expression, it is possible to solve Laplace's equation or Poisson's equation, subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for

everywhere inside a volume where either (1) the value of is specified on the bounding surface of thevolume (Dirichlet boundary conditions), or (2) the normal derivative of is specified on the bounding surface(Neumann boundary conditions).Suppose the problem is to solve for inside the region. Then the integral

reduces to simply due to the defining property of the Dirac delta function and we have:

This form expresses the well-known property of harmonic functions that if the value or normal derivative is knownon a bounding surface, then the value of the function inside the volume is known everywhere.

In electrostatics, is interpreted as the electric potential, as electric charge density, and the normalderivative as the normal component of the electric field.If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that

vanishes when either x or x' is on the bounding surface; conversely, if the problem is to solve a Neumannboundary value problem, the Green's function is chosen such that its normal derivative vanishes on the boundingsurface. Thus only one of the two terms in the surface integral remains.With no boundary conditions, the Green's function for the Laplacian (Green's function for the three-variable Laplaceequation) is:

Green's function 46

Supposing that the bounding surface goes out to infinity, and plugging in this expression for the Green's function,this gives the familiar expression for electric potential in terms of electric charge density (in the CGS unit system) as

ExampleGiven the problem

Find the Green's function.First step: The Green's function for the linear operator at hand is defined as the solution to

If , then the delta function gives zero, and the general solution is

For , the boundary condition at implies

The equation of is skipped because if and

For , the boundary condition at implies

The equation of is skipped for similar reasons.To summarize the results thus far:

Second step: The next task is to determine and .Ensuring continuity in the Green's function at implies

One can also ensure proper discontinuity in the first derivative by integrating the defining differential equation fromto and taking the limit as goes to zero:

The two (dis)continuity equations can be solved for and to obtain

So the Green's function for this problem is:

Green's function 47

Further examples• Let n = 1 and let the subset be all of R. Let L be d/dx. Then, the Heaviside step function H(x − x0) is a Green's

function of L at x0.• Let n = 2 and let the subset be the quarter-plane { (x, y) : x, y ≥ 0 } and L be the Laplacian. Also, assume a

Dirichlet boundary condition is imposed at x=0 and a Neumann boundary condition is imposed at y=0. Then theGreen's function is

References• S. S. Bayin (2006), Mathematical Methods in Science and Engineering, Wiley, Chapters 18 and 19.• Eyges, Leonard, The Classical Electromagnetic Field, Dover Publications, New York, 1972. ISBN

0-486-63947-9. (Chapter 5 contains a very readable account of using Green's functions to solve boundary valueproblems in electrostatics.)

• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition),Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman &Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

External links• Weisstein, Eric W., "Green's Function [1]" from MathWorld.• Green's function for differential operator [2] at PlanetMath.• Green's function [3] on PlanetMath• GreenFunctionsAndConformalMapping [4] on PlanetMath• Introduction to the Keldysh Nonequilibrium Green Function Technique [5] by A. P. Jauho• Tutorial on Green's functions [6]

• Boundary Element Method (for some idea on how Green's functions may be used with the boundary elementmethod for solving potential problems numerically) [7]

• At Citizendium [8]

References[1] http:/ / mathworld. wolfram. com/ GreensFunction. html[2] http:/ / planetmath. org/ encyclopedia/ GreensFunctionForDifferentialOperator. html[3] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=6355[4] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=7963[5] http:/ / nanohub. org/ resources/ 1877[6] http:/ / www. boulder. nist. gov/ div853/ greenfn/ tutorial. html[7] http:/ / www. ntu. edu. sg/ home/ mwtang/ bemsite. htm[8] http:/ / en. citizendium. org/ wiki/ Green%27s_function

Article Sources and Contributors 48

Article Sources and ContributorsVector calculus  Source: http://en.wikipedia.org/w/index.php?oldid=399252982  Contributors: Akerans, Andrei Stroe, Andy M. Wang, Anonymous Dissident, Ap, Arabani, ArielGold,Asyndeton, AxelBoldt, Azmiriam, Charles Matthews, ChristopherWillis, Cometstyles, Conversion script, Correogsk, Damian Yerrick, DavidBlackwell, DavidCary, Discospinster, Dysprosia,Edgerck, Foxjwill, Giftlite, Gombang, Gurch, Hadal, Heathhunnicutt, Hellisp, Herebo, Hotstreets, Hypernurb, Iantresman, Icedragoniii, Icekiss, JDX, Jacolston, Jhausauer, Jim.belk, Jitse Niesen,Joey-das-WBF, JohnBlackburne, Jossi, KGasso, Kristensson, Kyle1278, L Kensington, L-H, Lavaka, Linuxlad, Logitech111, M1ss1ontomars2k4, MFNickster, Maksim-e, MathKnight, Meier99,Metacomet, MrOllie, Msh210, NETTKNUT, Nanmus, Nbarth, Oldway1, Oleg Alexandrov, Orhanghazi, Paolo.dL, Paul August, Phantomsteve, Phyguy03, PlatypeanArchcow, RMFan1,Randomblue, Realworth, Rgdboer, Rossami, Rst20xx, Sammy1339, Shadow1, Sholtar, Siddhant, Silly rabbit, Simanchalajena, Snotchstar!, Squirreljml, Srinivas.zinka, Stepshep, Symon,Sławomir Biała, TLSupper, TStein, Tarquin, Template namespace initialisation script, That Guy, From That Show!, The Thing That Should Not Be, Tobias Hoevekamp, WhiteHatLurker,WikiMaster64, Woohookitty, 110 ,لیقع فشاک anonymous edits

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Divergence  Source: http://en.wikipedia.org/w/index.php?oldid=406657676  Contributors: 16@r, 24.176.164.xxx, A. B., ALittleSlow, Ae77, Akriasas, Anakata, Andres, Andres Agudelo,Anonymous Dissident, Apankrat, Arthur Rubin, AutomaticWriting, AvicAWB, AxelBoldt, Azmiriam, Benjaminevans82, Bryan Derksen, Caiyu, Camiflower, Cataphract, Charles Matthews,Closedmouth, Conversion script, Dnwq, DogFog, Dysprosia, Edgerck, Eigenlambda, Emote, Epbr123, Erudecorp, Etola, Etxrge, Fresheneesz, Giftlite, Greensan, Headbomb, Helixblue, Hellisp,Holmansf, Hypernurb, Icairns, Innumerate1979, JJ Harrison, Jeff3000, Kamix, Kiteinthewind, Kri, Loodog, Lyctc, MFNickster, Mah159, Martynas Patasius, Matthew Yeager, Mo-Al, Ms2ger,Msreeharsha, Mxn, OlEnglish, Oleg Alexandrov, Omenge, Patrick, Paul Matthews, Paulck, Pax85, Pj.de.bruin, Quadell, RAult, Rama's Arrow, Rjwilmsi, Roastytoast, Rob Hooft, Sam Hocevar,Sct72, Solarapex, StradivariusTV, Sverdrup, Sławomir Biały, TStein, Tarquin, Tbsmith, Tcnuk, Template namespace initialisation script, The Anome, Thehotelambush, Threshold, Tim Starling,Tobias Hoevekamp, Trovatore, Tschong, Voidxor, Wile E. Heresiarch, Wolfnix, WriterHound, Wwoods, Yecril, Zeroparallax, 120 anonymous edits

Curl (mathematics)  Source: http://en.wikipedia.org/w/index.php?oldid=403160945  Contributors: 345Kai, Aleriterra, Alksentrs, Andrei Stroe, Attilios, AxelBoldt, Bluethroat, Brews ohare,Bushido Hacks, CDiPoce, Caiyu, Charles Matthews, CiaPan, Commentor, Conversion script, Crystallina, Cvaneg, David Tombe, Dburghoff, Disdero, Diza, Dysprosia, Ed Cormany, Ehrenkater,El C, Elwikipedista, Enok.cc, Fatsamsgrandslam, FilipeS, Gauge, Giftlite, Gilliam, Gillis, Gvozdet, Hellisp, Herbythyme, Holmansf, Icairns, Incnis Mrsi, JJ Harrison, JRSpriggs, Jim.belk,JohnBlackburne, Jopo, Josh Grosse, Kallikanzarid, Kamix, Kinu, Kri, LOL, Loodog, MFNickster, Maltelauridsbrigge, Marmelad, Mboverload, Michael C Price, Michael Hardy, Mormegil,Mwtoews, Nbarth, Oleg Alexandrov, Pamputt, Patrick, Paul August, Peter Harriman, Pne, Poor Yorick, ProperFraction, Robinh, Rrjanbiah, Skittleys, Solarapex, Stevenj, StewartMH, StuartH,Sławomir Biały, TStein, Tango, Tarquin, Ted Longstaffe, Tegla, Teply, Thefox258, Tim Starling, Tim1357, Tkuvho, Tobias Bergemann, Tobias Hoevekamp, Wile E. Heresiarch, WriterHound,Wwoods, Xaonon, YordanGeorgiev, ZICO, 133 anonymous edits

Laplace operator  Source: http://en.wikipedia.org/w/index.php?oldid=405710434  Contributors: Adoniscik, Albert einstein2010, AlekseyFy, Andregomez3, Andrei Stroe, Andres, Ariya,Azmiriam, Barak Sh, Bdmy, Ben pcc, Benzh, CBM, Cgwaldman, Charles Matthews, ChriS, Comech, Crowsnest, Cyp, Daniel.Cardenas, Delaszk, Djozwebo, Dkf11, Donarreiskoffer, Dysprosia,FinalRapture, Fredrik, Gabriele Dini Ciacci, Gauge, Giftlite, Hankwang, Haseldon, Heptor, J.Voss, JYOuyang, Jacobolus, JasonSaulG, Jjauregui, Jorge Stolfi, Just Another Dan, KirilSimeonovski, KittySaturn, Kpym, KrisK, Lantonov, Lazarus666, Leobh, Lethe, Linas, Lupin, Lysdexia, Lzur, MarSch, Marc Venot, MarcelB612, Martynas Patasius, Metacomet, Michael Hardy,Mo-Al, NathanielEP, Nuwewsco, Oleg Alexandrov, Ollivier, Ostrouchov, PAR, PMajer, Patrick, Pilotguy, PowerWill500, RayAYang, RexNL, Rich Farmbrough, Riick, RobHar, Romanm, RuudKoot, SPat, Shadowjams, Siddhant, Silly rabbit, Skittleys, Slaniel, Slawekb, Spartaz, StevenJohnston, Suruena, Sławomir Biały, TStein, TakuyaMurata, Tarquin, Tbsmith, Tpl, Ulner,Whosasking, Wikeithpedia, Willkn, Wolfkeeper, Wwoods, XJamRastafire, Xihr, Zhanna leuven, 106 anonymous edits

Gradient theorem  Source: http://en.wikipedia.org/w/index.php?oldid=381073195  Contributors: AeroSpace, Alexfusco5, Amalas, Barak Sh, Brews ohare, CBM, Giftlite, J.delanoy, Jigdo,Lurcher66, Michael Hardy, Oore, Pavel Jelinek, Prcnarhet arthas, QTCaptain, Richard L. Peterson, SkZ, Spoon!, TStein, Tcnuk, Y0u, 14 anonymous edits

Green's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=403925918  Contributors: Abdull, AeroSpace, Amiruchka, Aphexer, AxelBoldt, BeteNoir, Caiyu, Caltas, CharlesMatthews, Cmullins10, Cronholm144, DantheCowMan, Darth Panda, DearPrudence, Dysprosia, EconoPhysicist, Elwikipedista, Evercat, Giftlite, Gwaihir, Helloher, Hydrogen Iodide, JJHarrison, JabberWok, Jay Gatsby, Jmath666, Kiensvay, Klapi, Lambiam, Liftarn, Looxix, MathMartin, Melchoir, Michael Hardy, Nicegom, Nijdam, No Worries, Obradovic Goran, OlegAlexandrov, Patrick, Perfecto, Physic sox, Pomte, Poor Yorick, Prcnarhet arthas, Randomblue, Salgueiro, Shim'on, Silly rabbit, SimonArlott, Spoon!, Sverdrup, Sławomir Biały, T-rithy, TStein,Tapir Terrific, Tcnuk, Temporaluser, TheObtuseAngleOfDoom, Thumperward, Tristanreid, User A1, Waldo, WriterHound, Yecril, 91 anonymous edits

Stokes' theorem  Source: http://en.wikipedia.org/w/index.php?oldid=403414775  Contributors: 198.135.118.xxx, A. Pichler, AdamSmithee, AeroSpace, Aetheling, Arthena, AxelBoldt, B41988,Ben pcc, BenFrantzDale, BeteNoir, Bkell, Brews ohare, Bryan Derksen, Buka, Bzz42, C S, CBM, Camfordwiki, Can't sleep, clown will eat me, Charles Matthews, Chip McShoulder, Conversionscript, Cronholm144, Curps, Custos0, Danielsimonjr, DefLog, Discospinster, Dysprosia, Ebony Jackson, EconoPhysicist, Fintor, Fropuff, Fullmetal2887, GCarty, Gareth Jones, Geometry guy,Geremia, Giftlite, Gvozdet, HHHEB3, Hakkasberra, Hao2lian, Helder.wiki, Hess88, HorsePunchKid, Ht686rg90, Huangjs, Izno, J.Rohrer, Jacobolus, Jakob.scholbach, Jhwilliams, JonathanWebley, KSmrq, Kimchi.sg, Kwiki, Laurentius, Lseixas, Luis Sanchez, Lupin, M simin, MarSch, Marcoii, MathMartin, MathematicsNerd, Mav, Mct mht, Metacomet, Mgummess, Mhartl,Michael Hardy, MoraSique, Naddy, Nbarth, Nick Wilson, Nightstallion, Nistra, Nossac, Oleg Alexandrov, Omnipaedista, Paolo Giarrusso, Patrick, Paul August, Peko (usurped), Pillyp, Point-settopologist, Pomte, Python eggs, RG2, Randomblue, Rodhullandemu, RogierBrussee, Sean D Martin, Shadowjams, Silly rabbit, Simsea, Spoon!, Stevenj, Stevertigo, StradivariusTV, Syndicate,TStein, Tarquin, Template namespace initialisation script, Tesseran, That Guy, From That Show!, The Anome, Theneokid, Thumperward, Tosha, User A1, Whaa?, XJamRastafire, Yecril, ZhouYu, Zinoviev, 165 anonymous edits

Divergence theorem  Source: http://en.wikipedia.org/w/index.php?oldid=404951036  Contributors: 4C, A. Pichler, Ariya, AxelBoldt, BeteNoir, Caiyu, Camw, Charles Matthews, Cronholm144,DacodaNelson, DragonflySixtyseven, Duae Quartunciae, Dysprosia, EconoPhysicist, Eric119, Extendo-Brain, Gala.martin, Giftlite, Gregbard, Griffgruff, Hao2lian, Hirak 99, Hvn0413, Irishguy,J.Rohrer, JL 09, Jesin, Jim.belk, Jleto, JoergenB, Karada, LTPR22, Lethe, Light current, Lucio, Lupin, MSGJ, Materialscientist, Menta78, Mhym, Michael Hardy, Oleg Alexandrov, Pakaran,Patrick, Paul Matthews, Pfalstad, Pleasantville, Rdrosson, Revolver, SJP, Saghi-002, Sbyrnes321, Sciyoshi, Siddhant, Silly rabbit, Spinoza1989, StradivariusTV, TStein, Tarquin, The Anome,Tim Starling, Tosha, User A1, V81, Villamota, Voidxor, Whaa?, XJamRastafire, Yecril, 115 anonymous edits

Green's function  Source: http://en.wikipedia.org/w/index.php?oldid=399994111  Contributors: Aajaja, Adam majewski, Adselsum, Aliotra, Andrei Polyanin, Andrewwall, Andywall, Apchar,Beatnik8983, BenFrantzDale, Billlion, Bkonrad, Brews ohare, Bryan Derksen, Cederal, Chaoma1988, Charles Matthews, Complexica, Compsonheir, Davy p, Dclader, Dicklyon, Docu, Drusus 0,Dysprosia, EconoPhysicist, Fhrouet, GeoGreg, Giftlite, Grafen, GreatWhiteNortherner, Hannes Eder, HappyCamper, Hashproduct, HowiAuckland, Ibison, Ikh, Jitse Niesen, Jiy, Jmnbatista, JonasOlson, Lavanyaravi, Lethe, Linas, Lumidek, Magnesium, MathKnight, MathMartin, Mathieu Perrin, Mel Etitis, Mhuzefa, Michael Hardy, Middayexpress, Molitorppd22, Nbarth, No-body, OlegAlexandrov, Paranoidhuman, Parkyere, Pdenapo, Phe, Phys, Pjacobi, Pooryamcgill, Ppablo1812, Quibik, Silly rabbit, Simetrical, Stedder, Stevvers, StewartMH, Technopilgrim, That Guy, FromThat Show!, Thenub314, ThomasStrohmann, ThorinMuglindir, Thumperward, TobinFricke, User A1, W.F.Galway, Warbler271, Wavelength, Wolfch, WriterHound, Wshun, Yecril, 124anonymous edits

Image Sources, Licenses and Contributors 49

Image Sources, Licenses and ContributorsImage:Gradient2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Gradient2.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: 555, Cweiske, Darapti,Deerstop, PsychoMessiah, 3 anonymous editsImage:Gradient99.png  Source: http://en.wikipedia.org/w/index.php?title=File:Gradient99.png  License: Creative Commons Attribution 3.0  Contributors: SimiprofImage:Curlorient.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Curlorient.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:MarmeladImage:Uniform curl.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Uniform_curl.svg  License: Creative Commons Attribution 2.5  Contributors: Original uploader was Loodog aten.wikipediaImage:Curl of uniform curl.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Curl_of_uniform_curl.JPG  License: Creative Commons Attribution 2.5  Contributors: Originaluploader was Loodog at en.wikipediaImage:Nonuniformcurl.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Nonuniformcurl.JPG  License: Creative Commons Attribution 2.5  Contributors: Original uploader wasLoodog at en.wikipediaImage:Curl of nonuniform curl.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Curl_of_nonuniform_curl.JPG  License: Creative Commons Attribution 2.5  Contributors: Originaluploader was Loodog at en.wikipediaImage:Green's-theorem-simple-region.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Green's-theorem-simple-region.svg  License: Creative Commons Attribution-Sharealike 2.5 Contributors: Cronholm144, 1 anonymous editsImage:Stokes-patch.png  Source: http://en.wikipedia.org/w/index.php?title=File:Stokes-patch.png  License: Public Domain  Contributors: User:NcneverImage:Stokes' Theorem.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Stokes'_Theorem.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors:User:Cronholm144File:Divergence theorem.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Divergence_theorem.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors:User:Cronholm144File:SurfacesWithAndWithoutBoundary.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SurfacesWithAndWithoutBoundary.svg  License: Creative Commons Attribution 3.0 Contributors: User:Geek3, User:Sbyrnes321Image:Vector sphere.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Vector_sphere.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: User:Cronholm144

License 50

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/