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Differential Calculus: . Del, Divergence and Curl. f ( x ) : a function of a variable ( x ). Differential Calculus: f ( x ) : a function of a variable ( x ). Differential Calculus: f ( x ) : a function of a variable ( x ). Differential Calculus: f ( x ) : a function of a variable ( x ).

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slide1

Differential Calculus:

Del, Divergence and Curl

f(x) : a function of a variable (x)

slide2

Differential Calculus:

f(x) : a function of a variable (x)

slide3

Differential Calculus:

f(x) : a function of a variable (x)

slide4

Differential Calculus:

f(x) : a function of a variable (x)

f(x)

x

slide5

Differential Calculus:

f(x) : a function of a variable (x)

  • Change in f(x) w.r.t x
  • Slope of a graph f vs x
  • df/dx: Ordinary derivative

f(x)

x

slide6

Differential Calculus:

f(x,y,z) : function of more than one variables

slide7

Differential Calculus:

f(x,y,z) : function of more than one variables

slide8

Differential Calculus:

f(x,y,z) : function of more than one variables

slide9

Differential Calculus:

f(x,y,z) : function of more than one variables

slide10

Differential Calculus:

f(x,y,z) : function of more than one variables

slide11

Differential Calculus:

f(x,y,z) : function of more than one variables

slide12

Differential Calculus:

f(x,y,z) : function of more than one variables

slide13

Differential Calculus:

f(x,y,z) : function of more than one variables

slide14

Differential Calculus:

f(x,y,z) : function of more than one variables

  • Del is a vector operator
slide15

The Del Operator

The collection of partial derivative operators

is commonly called the del operator.

slide16

The Del Operator

The collection of partial derivative operators

is commonly called the del operator.

  • Aa a vector A can be multiply
  • by a scalar ‘a’ aA
  • by another vectorBB, via the dot product
  • A.B
  • by another vectorBB, via the cross product
              • AxB
slide17

The Del Operator

The collection of partial derivative operators

is commonly called the del operator.

  • Similarly Del Operator can operate in 3 ways:
  • Operate on a scalar function “f”
slide18

The Del Operator

The collection of partial derivative operators

is commonly called the del operator.

  • Similarly Del Operator can operate in 3 ways:
  • Operate on a scalar function “f”
  • Operate on a vector VB, via the dot product
slide19

The Del Operator

The collection of partial derivative operators

is commonly called the del operator.

  • Similarly Del Operator can operate in 3 ways:
  • Operate on a scalar function “f”
  • Operate on a vector VB, via the dot product
  • Operate on a vector VB, via the cross product
slide20

The Gradient

  • the gradient ‘del f’ is not a
  • multiplication of del & f,
slide21

The Gradient

the gradient ‘del f’

slide22

The Gradient

the gradient ‘del f’

For a fix dl, at θ=0,

slide23

The Gradient

the gradient ‘del f’

For a unit change in dl, at θ=0,

The gradient is a vector operation which operates on a scalar function to produce a vector whose magnitude is the maximum rate of change of the function at the point of the gradient and which is pointed in the direction of that maximum rate of change.

slide24

Gradient in various Coordinates

Cartesian Coordinate system

Cylindrical Coordinate system

Spherical Coordinate system

slide25

There are three ways the operator can act:

  • On a scalar function T
  • (the gradient)
  • 2. On a vector function V, via the dot product
  • (the divergence)
slide26

The divergence of a vector field

in rectangular coordinates is defined as the scalar product of the del operator and the function

The divergence is a scalar quantity. We can’t take divergence of a scalar field. a The divergence theorem is an important mathematical tool in electricity and magnetism.

slide27

Physical significance of divergence of vector

The divergence of vector V is a measure of how much the vector V spreads out i.e. diverges from the point in question.

slide28

The divergence of F at a given point P is the outward flux per unit volume as the volume shrinks about P.

P

P

P

The divergence of a vector field can be viewed as the limit of the field’s source strength per unit volume i.e. it is positive at a source point in the field, negative at a sink point, or zero where there is neither source or sink.

slide29

Divergence in various Coordinates

In rectangular coordinates:

In cylindrical polar coordinates:

in spherical polar coordinates:

slide30

There are three ways the operator can act:

  • On a scalar function T
  • (the gradient)
  • 2. On a vector function V, via the dot product
  • (the divergence)
  • 3. On a vector function V, via the cross product :
  • (the curl)
slide31

The Curl

The curl of a vector function is the vector product of the del operator with a vector function:

where i,j,k are unit vectors in the x, y, z directions. It can also be expressed in determinant form:

slide32

Physical Significance of Curl of vector

The curl of vector V is a measure of how much the vector V “curls around” the point in question. Curl is also called circular rotation.

If ,V is called as irrotational vector.

slide33

Curl in various Coordinates

In rectangular coordinates:

In cylindrical polar coordinates:

in spherical polar coordinates:

slide34

Operator In Cartesian Coordinate System

Line element:

Volume Element:

Gradient:

gradf: points the direction of maximum increase of the function f.

Mag(del f) : the slope (rate of increase) along this maximum direction.

Divergence:

Curl:

slide35

Operator In Cylindrical Coordinate System

Line Element:

Volume Element:

Gradient:

Divergence:

Curl:

slide36

Operator In Spherical Coordinate System

Line Element:

Volume Element:

Gradient :

Divergence:

Curl:

basic vector calculus
Basic Vector Calculus

Divergence or Gauss’ Theorem

The divergence theorem states that the total outward flux of a vector field F through the closed surface S is the same as the volume integral of the divergence of F.

Closed surface S, volume V, outward pointing normal

Conversion of volume integral to surface integral and vice verse.

slide38

Stokes’ Theorem

Stokes’s theorem states that the circulation of a vector field F around a closed path L is equal to the surface integral of the curl of F over the open surface S bounded by L

Oriented boundary L

Conversion of surface integral to line integral and vice verse.

slide39

EXAMPLE 1: Use the Divergence theorem to find the outward flux of F = (y - x) i + (x - y) j + (y – x)across the boundary of the cube bounded by the planes x = ± 1, y = ± 1, and z = ± 1.

SOLUTION: Let M = y - x, N = z - y, and P = y – x

slide40

EXAMPLE 2: Use the Divergence theorem to find the outward flux of = x 2 i – 2vbbvvvvvvbvxy j + 3 across the boundary cut from the first octant by the sphere x 2 + y 2 + z 2 = 4.

slide41

EXAMPLE 3: Use the Divergence theorem to find the outward flux of

across the boundary of the region 1 ≤x 2 + y 2 + z 2 ≤ 4.

SOLUTION:

slide42

EXAMPLE 4: Use the Divergence theorem to find the outward flux of

across the boundary of the region D which is the thick-walled cylinder

1 ≤ x 2 + y 2 ≤ 2, -1 ≤ z ≤ 2.

example 1 suppose check stoke s theorem for the square surface shown
EXAMPLE 1.Suppose Check Stoke’s theorem for the square surface shown.
  • z
  • (iii)
  • 1
  • (ii)
    • (i) 1 y

SOLUTION: Here,

For line integral , we break up into four segments:

x

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