3.1 VECTOR CALCULUS.pdf

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VECTOR

CALCULUS

Prepared by

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Differential Length, Area, and Volume

Differential elements in length, area, and volume are useful in vector calculus.

They are defined in the Cartesian, Cylindrical, and Spherical coordinate

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Differential Length, Area, and Volume

A. Cartesian Coordinates

1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

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Differential Length, Area, and Volume

Differential Elements in the Right-handed Cartesian Coordinate System

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Differential Length, Area, and Volume

Differential Normal Areas in Cartesian Coordinates

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Differential Length, Area, and Volume

A. Cylindrical Coordinates

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Differential Length, Area, and Volume

Differential Elements in Cylindrical Coordinates

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Differential Length, Area, and Volume

Differential Normal Areas in Cylindrical Coordinates

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Differential Length, Area, and Volume

A. Spherical Coordinates

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Differential Length, Area, and Volume

Differential Elements in Spherical Coordinates

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Differential Length, Area, and Volume

Differential Normal Areas in Spherical Coordinates

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EX

AMPLE

3.1

Consider the object shown. Calculate:

a) The distance BC b) The distance CD c) The surface area

ABCD

d) The surface area ABO e) The surface area

AOFD

f) The volume ABDCFO

a) 10 b) 2.5 π c) 25 π d) 6.25 π e) 50 f) 62.5 π

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EXERCISE

3.1

Disregard the differential lengths and imagine that the

object is part of a spherical shell. It may be describe as 3 ≤ ≤ 5, 60° ≤ ≤ 90°, 45° ≤ ∅ ≤ 60° where surface = 3 is the same as , surface = 60° is , and surface ∅ = 45° is . Calculate a) The distance DH b) The distance FG

c) The surface area AEHD d) The surface area ABDC e) The volume of the

object a) 0.7854 b) 2.618 c) 1.179 d) 4.189 e) 4.276

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Line, Surface, and Volume Integrals

By a line we mean the path along a curve in space.

Line, curve, and contour can be used interchangeably.

The line integral ∫ • is the of the tangential component of A along curve

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Line, Surface, and Volume Integrals

Given a vector field A and a curve L, we define the

integral

as the line integral of A around L.

Path of integration of a vector field

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Line, Surface, and Volume Integrals

If the path of integration is a closed curve such as abcba

Becomes a closed contour integral

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Line, Surface, and Volume Integrals

Given a vector field A, continuous in a

region containing the smooth surface

S, we define the surface integral or

the flux of A

through S as The flux of a vector field A

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Line, Surface, and Volume Integrals

where at any point on S , is the unit

normal to S. For a closed

surface (defining a volume):

The flux of a vector field A through surface S

Which is referred to a the net outward flux of A from

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Line, Surface, and Volume Integrals

We define

as the volume integral of the scalar over the volume .

The physical meaning of the line, surface, or volume integral depends on the nature of the

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EX

AMPLE

3.2

calculate the circulation of F around aGiven = − − , closed path shown.

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EXERCISE

3.2

Calculate the circulation of= cos ∅ + sin ∅ around the edge L of the wedge defined by 0 ≤ ≤ 2,

0 ≤ ∅ ≤ 60°, = 0 and shown.

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Del Operator

The del operator, written as , is a vector differential operator.

In Cartesian coordinates,

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Del Operator

It is not a vector in itself,

Useful in defining the following:

1. The gradient of a scalar V, ( )

2. The divergence of a vector A,( • ) 3. The curl of a vector A, ( × )

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Del Operator

In Cylindrical coordinates,

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Gradient of a Scalar

The gradient of a scalar field V is a

vector that represents both the magnitude

and the direction of the maximum space rate of increase of V.

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Gradient of a Scalar

In Cartesiancoordinates,

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Gradient of a Scalar

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Gradient of a Scalar

The following computation formulas on gradient, which are easily proved, should be noted:

where U and V are scalars and n is an

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EX

AMPLE

3.3

Find the gradient of the following

scalar fields:

a)

=

b)

=

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EXERCISE

3.3

Determine the gradient of thefollowing scalar fields:

a)

= +

b)

= + ∅ +

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EX

AMPLE

3.4

computeGiven =and the direction+ , derivative _{/ in the direction}

+4 + 12 at (2, -1, 0).

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EXERCISE

3.4

gradient at point (1, 2, 3) and theGiven = + + . Find directional derivative of at the same

point in the direction toward point (3, 4, 4).

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EX

AMPLE

3.5

Find the angle at which line= = intersects the ellipsoid

+ + 2 = 10.

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EXERCISE

3.5

Calculate the angle between thenormal to the surfaces

+ = 3 and log − = −4

at the point of intersection (-1, 2, 1)

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Divergence of a Vector and Divergence Theorem The divergence of A at a given point P is

the outward flux per unit volume as the volume about P.

where ∆ is the volume enclosed by the closed surface S in which P is located

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Divergence of a Vector and Divergence Theorem

a) The divergence of a vector field at point P is positive because the vector diverges (spreads out) at P.

b) A vector field has negative divergence (convergence) at P

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Divergence of a Vector and Divergence Theorem In Cartesian coordinates,

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Divergence of a Vector and Divergence Theorem In Spherical coordinates,

Note the following properties of the divergence of a vector field:

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Divergence of a Vector and Divergence Theorem The divergence theorem states that the

total outward flux of a vector field A

through a closed surface S is the same as the volume integral of the divergence of

A.

Otherwise known as

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EX

AMPLE

3.6

Find the divergence of these vector fields:

a) = +

b) = + _∅ +

c) = + +

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EXERCISE

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EX

AMPLE

3.7

If _{= 10} ₍ _{+ ),}

determine the flux of G out of the entire surface of the

cylinder

= 1, 0 ≤ ≤ 1.

Confirm the result using the divergence theorem.

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EXERCISE

3.7

= cosDetermine the flux of+ sin

over the closed surface of the cylinder 0 ≤ ≤ 1, = 4.

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Curl of a Vector and Stoke’s Theorem

The curl of A is an axial (or rotational) vector whose magnitude is the

maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to

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Curl of a Vector and Stoke’s Theorem

In Cartesian

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Curl of a Vector and Stoke’s Theorem

In Cylindrical

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Curl of a Vector and Stoke’s Theorem

In Spherical coordinates

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Curl of a Vector and Stoke’s Theorem

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Curl of a Vector and Stoke’s Theorem

The curl provides the maximum value of the circulation of the field per unit area (or circulation density) and indicates the

direction to which this maximum value occurs.

Measures the circulation of how much the field curls around P

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Curl of a Vector and Stoke’s Theorem

a) Curl at P points out of the page b) Curl at P is zero.

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Curl of a Vector and Stoke’s Theorem

Stokes’s theorem states

that the circulation of a vector field A around a (closed) path L is equal

to the surface integral of the curl of A over the

open surface S bounded by L

provided that A and × are continuous

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EX

AMPLE

3.8

Determine the curl of these vector fields:

a) = +

b) = + _∅ +

c) = + +

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EXERCISE

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EX

AMPLE

3.9

If = + _∅, evaluate

∮ • around the path shown. Confirm using Stokes’s theorem.

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EXERCISE

3.9

Confirm the circulation of= cos ∅ + sin ∅

around the edge L of the wedge

defined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, = 0 and shown using Stoke’s Theorem

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EX

AMPLE

3.10

For the vector field A, show explicitly that

 _{× = 0;}

that is, the divergence of the curl of any vector field is zero.

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EXERCISE

3.10

For a scalar field V, show that

×

= 0;

that is, the curl of the gradient of

any scalar field vanishes.

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Laplacian of a Scalar

The Laplacian of a scalar field V, written as , is the divergence of the gradient of V.

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Laplacian of a Scalar

In Cylindrical coordinates,

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EX

AMPLE

3.11

Find the Laplacian of the following scalar fields:

a)

=

b)

=

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EXERCISE

3.11

Determine the Laplacian of the following scalar fields:

a)

= +

b)

= + ∅ +

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Classification of Vector Fields

A vector field is uniquely characterized by its divergence and curl.

Neither the divergence nor curl of a vector field is sufficient to completely describe a

field.

All vector fields can be classified in terms of their vanishing or non vanishing divergence or

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Classification of Vector Fields

A vector field A is said to be solenoidal (or divergenceless) if • =

Examples:

• incompressible fluids, • magnetic fields

• conduction current density

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Classification of Vector Fields

A vector field A is said to be irrotational (or potential) if × =

Also known as conservative field. Examples:

• electrostatic field • gravitational field

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Reference:

Elements of Electromagnetics by Matthew N. O. Sadiku