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VECTOR CALCULUS AND LINEAR ALGEBRA

Presented by

Upendra Talar-140410117055

Jahnvi Trivedi-140410117056

Keyur Trivedi-140410117057

Maharsh Trivedi-140410117058

Divya Upadhyay-140410117059

Guided by Jaimin Patel sir

CONTENTS

oPHYSICAL INTERPRETATION OF GRADIENToCURLoDIVERGENCEoSOLENOIDAL AND IRROTATIONAL FIELDSoDIRECTIONAL DERIVATIVE

GRADIENT OF A SCALAR FIELD

The gradient of a scalar function f(x1 x2 x3 xn) is denoted by nablaf or where nabla (the nabla symbol) denotes the vector differential operator del The notation grad(f) is also commonly used for the gradient

The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v That is

In 3-dimensional cartesian coordinate system it is denoted by

f f ff

x y z

f f f

x y z

i j k

i j k

PHYSICAL INTERPRETATION OF GRADIENT One is given in terms of the graph of

some function z = f(x y) where f is a reasonable function ndash say with continuous first partial derivatives In this case we can think of the graph as a surface whose points have variable heights over the x y ndash plane

An illustration is given below If say we place a marble at some point (x y) on this graph with zero initial

force its motion will trace out a path on the surface and in fact it will choose the direction of steepest descent

This direction of steepest descent is given by the negative of the gradient of f One takes the negative direction because the height is decreasing rather than increasing

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then thE gradient gives the direction in which the temperature is increasing most rapidly

CURL In vector calculus the curl is a vector

operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) 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 is the magnitude of that rotation

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

Conservative field If F is a vector force field then line integral

Represents the work done around a closed path If it is zero then the field is said to be conservative

Irrotational field is always conservative If F is irrotational then the scalar potential of F is obtained by the formula

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

Equation 9

div P Q R

x y z

F

deg 3

DIVERGENCE In terms of the gradient operator

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

x y z

i j k

div F F

Equation 10

DIVERGENCE

If F(x y z) = xz i + xyz j ndash y2 k find div F

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

2

div

xz xyz yx y z

z xz

F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

THEOREM

FIND THE DIRECTIONAL DERIVATIVE OF F AT THE GIVEN POINT IN THE DIRECTION INDICATED BY THE ANGLE

4)12()( 432 yyxyxf


CONTENTS


PHYSICAL INTERPRETATION OF GRADIENT

Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18

Find the directional derivative of f at the given point in the

CONTENTS

oPHYSICAL INTERPRETATION OF GRADIENToCURLoDIVERGENCEoSOLENOIDAL AND IRROTATIONAL FIELDSoDIRECTIONAL DERIVATIVE





f f ff

x y z

f f f

x y z

i j k

i j k













Definition


POINTS TO BE NOTED









DIVERGENCE



DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18






f f ff

x y z

f f f

x y z

i j k

i j k













Definition


POINTS TO BE NOTED









DIVERGENCE



DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18














Definition


POINTS TO BE NOTED









DIVERGENCE



DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18


POINTS TO BE NOTED









DIVERGENCE



DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18





DIVERGENCE



DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18


DIVERGENCE


Equation 9

div P Q R

x y z

F

deg 3



x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18




x y z

i j k

div F F

Equation 10

DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18


DIVERGENCE



2

div

xz xyz yx y z

z xz

F F









THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18










THEOREM


4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18



4)12()( 432 yyxyxf


CONTENTS



Slide 5

CURL

Slide 7

Points to be noted

Slide 9

DIVERGENCE

DIVERGENCE (2)

DIVERGENCE (3)

DIVERGENCE (4)


Slide 15

Slide 16

THEOREM

Slide 18


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