VECTOR CALCULUS16

VECTOR CALCULUSHere, we define two operations that:

Can be performed on vector fields.

Play a basic role in the applications of vector calculus to fluid flow, electricity, and magnetism.

VECTOR CALCULUSEach operation resembles differentiation.

However, one produces a vector field whereas the other produces a scalar field.

16.5 Curl and DivergenceVECTOR CALCULUSIn this section, we will learn about:The operations of curl and divergence and how they can be used to obtain vector forms of Greens Theorem.

CURLSuppose:

F = P i + Q j + R k is a vector field on .

The partial derivatives of P, Q, and R all exist

CURLThen, the curl of F is the vector field on defined by: Equation 1

CURLAs a memory aid, lets rewrite Equation 1 using operator notation.

We introduce the vector differential operator (del) as:

CURLIt has meaning when it operates on a scalar function to produce the gradient of f :

CURLIf we think of as a vector with components /x, /y, and /z, we can also consider the formal cross product of with the vector field F as follows.

CURL

CURLThus, the easiest way to remember Definition 1 is by means of the symbolic expressionEquation 2

CURLIf F(x, y, z) = xz i + xyz j y2 kfind curl F.

Using Equation 2, we have the following result.Example 1

CURLExample 1

CURLMost computer algebra systems (CAS) have commands that compute the curl and divergence of vector fields.

If you have access to a CAS, use these commands to check the answers to the examples and exercises in this section.

CURLRecall that the gradient of a function f of three variables is a vector field on .So, we can compute its curl.

The following theorem says that the curl of a gradient vector field is 0.

GRADIENT VECTOR FIELDSIf f is a function of three variables that has continuous second-order partial derivatives, thenTheorem 3

GRADIENT VECTOR FIELDSBy Clairauts Theorem,Proof

GRADIENT VECTOR FIELDSNotice the similarity to what we know from Section 12.4:

a x a = 0 for every three-dimensional (3-D) vector a.

CONSERVATIVE VECTOR FIELDSA conservative vector field is one for which

So, Theorem 3 can be rephrased as: If F is conservative, then curl F = 0.

This gives us a way of verifying that a vector field is not conservative.

CONSERVATIVE VECTOR FIELDSShow that the vector field F(x, y, z) = xz i + xyz j y2 k is not conservative.

In Example 1, we showed that: curl F = y(2 + x) i + x j + yz k

This shows that curl F 0.

So, by Theorem 3, F is not conservative. Example 2

CONSERVATIVE VECTOR FIELDSThe converse of Theorem 3 is not true in general.

The following theorem, though, says that it is true if F is defined everywhere.

More generally, it is true if the domain is simply-connectedthat is, has no hole.

CONSERVATIVE VECTOR FIELDSTheorem 4 is the 3-D version of Theorem 6 in Section 16.3

Its proof requires Stokes Theorem and is sketched at the end of Section 16.8

CONSERVATIVE VECTOR FIELDSIf F is a vector field defined on all of whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field.Theorem 4

CONSERVATIVE VECTOR FIELDSShow that F(x, y, z) = y2z3 i + 2xyz3 j + 3xy2z2 k is a conservative vector field.

Find a function f such that . Example 3

CONSERVATIVE VECTOR FIELDSAs curl F = 0 and the domain of F is , F is a conservative vector field by Theorem 4.Example 3 a

CONSERVATIVE VECTOR FIELDSThe technique for finding f was given in Section 16.3 We have: fx(x, y, z) = y2z3 fy(x, y, z) = 2xyz3 fz(x, y, z) = 3xy2z2E. g. 3 bEqns. 5-7

CONSERVATIVE VECTOR FIELDSIntegrating Equation 5 with respect to x, we obtain: f(x, y, z) = xy2z3 + g(y, z)E. g. 3 bEqn. 8

CONSERVATIVE VECTOR FIELDSDifferentiating Equation 8 with respect to y, we get: fy(x, y, z) = 2xyz3 + gy(y, z)

So, comparison with Equation 6 gives: gy(y, z) = 0

Thus, g(y, z) = h(z) and fz(x, y, z) = 3xy2z2 + h(z)Example 3 b

CONSERVATIVE VECTOR FIELDSThen, Equation 7 gives: h(z) = 0

Therefore, f(x, y, z) = xy2z3 + KExample 3 b

CURLThe reason for the name curl is that the curl vector is associated with rotations.

One connection is explained in Exercise 37.

Another occurs when F represents the velocity field in fluid flow (Example 3 in Section 16.1).

CURLParticles near (x, y, z) in the fluid tend to rotate about the axis that points in the direction of curl F(x, y, z).

The length of this curl vector is a measure of how quickly the particles move around the axis.

F = 0 (IRROTATIONAL CURL)If curl F = 0 at a point P, the fluid is free from rotations at P.

F is called irrotational at P.

That is, there is no whirlpool or eddy at P.

F = 0 & F 0If curl F = 0, a tiny paddle wheel moves with the fluid but doesnt rotate about its axis.

If curl F 0, the paddle wheel rotates about its axis.

We give a more detailed explanation in Section 16.8 as a consequence of Stokes Theorem.

DIVERGENCEIf F = P i + Q j + R k is a vector field on and P/x, Q/y, and R/z exist, the divergence of F is the function of three variables defined by:Equation 9

CURL F VS. DIV FObserve that:

Curl F is a vector field.

Div F is a scalar field.

DIVERGENCEIn terms of the gradient operator

the divergence of F can be written symbolically as the dot product of and F:Equation 10

DIVERGENCEIf F(x, y, z) = xz i + xyz j y2 k find div F.

By the definition of divergence (Equation 9 or 10) we have: Example 4

DIVERGENCEIf F is a vector field on , then curl F is also a vector field on .

As such, we can compute its divergence.

The next theorem shows that the result is 0.

DIVERGENCEIf F = P i + Q j + R k is a vector field on and P, Q, and R have continuous second-order partial derivatives, then div curl F = 0Theorem 11

DIVERGENCEBy the definitions of divergence and curl,

The terms cancel in pairs by Clairauts Theorem.Proof

DIVERGENCENote the analogy with the scalar tripleproduct: a . (a x b) = 0

DIVERGENCEShow that the vector field F(x, y, z) = xz i + xyz j y2 kcant be written as the curl of another vector field, that is, F curl G

In Example 4, we showed that div F = z + xz and therefore div F 0.Example 5

DIVERGENCEIf it were true that F = curl G, then Theorem 11 would give: div F = div curl G = 0

This contradicts div F 0.

Thus, F is not the curl of another vector field.Example 5

DIVERGENCEAgain, the reason for the name divergence can be understood in the context of fluid flow.

If F(x, y, z) is the velocity of a fluid (or gas), div F(x, y, z) represents the net rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point (x, y, z) per unit volume.

INCOMPRESSIBLE DIVERGENCEIn other words, div F(x, y, z) measures the tendency of the fluid to diverge from the point (x, y, z).

If div F = 0, F is said to be incompressible.

GRADIENT VECTOR FIELDSAnother differential operator occurs when we compute the divergence of a gradient vector field .

If f is a function of three variables, we have:

LAPLACE OPERATORThis expression occurs so often that we abbreviate it as .

The operator is called the Laplace operator due to its relation to Laplaces equation

LAPLACE OPERATORWe can also apply the Laplace operator to a vector field F = P i + Q j + R k in terms of its components:

VECTOR FORMS OF GREENS THEOREMThe curl and divergence operators allow us to rewrite Greens Theorem in versions that will be useful in our later work.

VECTOR FORMS OF GREENS THEOREMWe suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypotheses of Greens Theorem.

VECTOR FORMS OF GREENS THEOREMThen, we consider the vector field F = P i + Q j

Its line integral is:

VECTOR FORMS OF GREENS THEOREMRegarding F as a vector field on with third component 0, we have:

VECTOR FORMS OF GREENS THEOREMTherefore,

VECTOR FORMS OF GREENS TH.Hence, we can now rewrite the equation in Greens Theorem in the vector formEquation 12

VECTOR FORMS OF GREENS TH.Equation 12 expresses the line integral of the tangential component of F along C as the double integral of the vertical component of curl F over the region D enclosed by C.

We now derive a similar formula involving the normal component of F.

VECTOR FORMS OF GREENS TH.If C is given by the vector equationr(t) = x(t) i + y(t) j a t b

then the unit tangent vector (Section 13.2) is:

VECTOR FORMS OF GREENS TH.You can verify that the outward unit normal vector to C is given by:

VECTOR FORMS OF GREENS TH.Then, from Equation 3 in Section 16.2, by Greens Theorem, we have:

VECTOR FORMS OF GREENS TH.

VECTOR FORMS OF GREENS TH.However, the integrand in that double integral is just the divergence of F.

So, we have a second vector form of Greens Theoremas follows.

VECTOR FORMS OF GREENS TH.This version says that the line integral of the normal component of F along C is equal to the double integral of the divergence of F over the region D enclosed by C. Equation 13