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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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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)

f(x)

x


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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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


Differential Calculus:

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

  • Del is a vector operator


The Del Operator

The collection of partial derivative operators

is commonly called the del operator.


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


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”


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


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


The Gradient

  • the gradient ‘del f’ is not a

  • multiplication of del & f,


The Gradient

the gradient ‘del f’


The Gradient

the gradient ‘del f’

For a fix dl, at θ=0,


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.


Gradient in various Coordinates

Cartesian Coordinate system

Cylindrical Coordinate system

Spherical Coordinate system



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.


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.


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.


Divergence in various Coordinates per unit volume as the volume shrinks about P.

In rectangular coordinates:

In cylindrical polar coordinates:

in spherical polar coordinates:


  • There are three ways the operator can act: per unit volume as the volume shrinks about P.

  • 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)


The Curl per unit volume as the volume shrinks about P.

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:


Physical Significance of Curl of vector per unit volume as the volume shrinks about P.

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.


Curl in various Coordinates per unit volume as the volume shrinks about P.

In rectangular coordinates:

In cylindrical polar coordinates:

in spherical polar coordinates:


Operator In per unit volume as the volume shrinks about P. 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:


Operator In per unit volume as the volume shrinks about P. Cylindrical Coordinate System

Line Element:

Volume Element:

Gradient:

Divergence:

Curl:


Operator In per unit volume as the volume shrinks about P. Spherical Coordinate System

Line Element:

Volume Element:

Gradient :

Divergence:

Curl:


Basic vector calculus
Basic Vector Calculus per unit volume as the volume shrinks about P.

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.


Stokes’ Theorem per unit volume as the volume shrinks about P.

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.


EXAMPLE 1 per unit volume as the volume shrinks about P. : 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


EXAMPLE 2: per unit volume as the volume shrinks about P. 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.


EXAMPLE 3 per unit volume as the volume shrinks about P. : 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:


EXAMPLE 4 per unit volume as the volume shrinks about P. : 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 per unit volume as the volume shrinks about P. 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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