VECTORCALCULUS

Prepared byEngr. Mark Angelo C. Purio

Differential Length, Area, and Volume

Differential Length, Area, and Volume

Differential elements in length, area, andvolume are useful in vector calculus.

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

systems

Differential Length, Area, and Volume

A. Cartesian Coordinates1. Differential Displacement

2. Differential Normal Area

3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in the Right-handedCartesian Coordinate System

Differential Length, Area, and Volume

Differential Normal Areas inCartesian Coordinates

Differential Length, Area, and Volume

A. Cylindrical Coordinates1. Differential Displacement

2. Differential Normal Area

3. Differential Volume

Differential Length, Area, and Volume

Differential Elements inCylindrical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas inCylindrical Coordinates

Differential Length, Area, and Volume

A. Spherical Coordinates1. Differential Displacement

2. Differential Normal Area

3. Differential Volume

Differential Length, Area, and Volume

Differential Elements inSpherical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas inSpherical Coordinates

EXAM

PLE 3

.1 Consider the objectshown. Calculate:a) The distance BCb) The distance CDc) The surface area

ABCDd) The surface area ABOe) The surface area

AOFDf) The volume ABDCFO

a) 10b) 2.5 πc) 25 π

d) 6.25 πe) 50f) 62.5 π

EXER

CISE

3.1 Disregard the differential

lengths and imagine that theobject is part of a sphericalshell. It may be describe as3 ≤ ≤ 5, 60° ≤ ≤ 90°,45° ≤ ∅ ≤ 60°where surface= 3 is the same as ,

surface = 60° is , andsurface ∅ = 45° is .

Calculatea) The distance DHb) The distance FGc) The surface area AEHDd) The surface area ABDCe) The volume of the

object

a) 0.7854b) 2.618c) 1.179

d) 4.189e) 4.276

Line, Surface, and Volume Integrals

Line, Surface, and Volume Integrals

By a line we mean the path along a curvein space.

Line, curve, and contour can be usedinterchangeably.

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

L.

Line, Surface, and Volume Integrals

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

integral

as the line integral ofA around L.

Path of integration of avector field

Line, Surface, and Volume Integrals

If the path of integration is a closed curve suchas abcba

Becomes a closed contour integral

which is called circulation of A around L

Line, Surface, and Volume IntegralsGiven a vector fieldA, continuous in aregion containing

the smooth surfaceS, we define the

surface integral orthe flux of Athrough S as The flux of a vector field A

through surface S

Line, Surface, and Volume Integralswhere at any pointon S , is the unit

normal to S.For a closed

surface (defining avolume):

The flux of a vector field Athrough surface S

Which is referred to a thenet outward flux of A from

S.

Line, Surface, and Volume IntegralsWe define

as the volume integral of the scalar overthe volume .

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

physicsl quantity represented by A or .

EXAM

PLE 3

.2 Given = − − ,calculate the circulation of F around a

closed path shown.

• = −

EXER

CISE

3.2 Calculate the circulation of= cos ∅ + sin ∅

around the edge L of thewedge defined by 0 ≤ ≤ 2,0 ≤ ∅ ≤ 60°, = 0 and

shown.

• =

DEL OPERATOR

Del Operator

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

In Cartesian coordinates,

a.k.a gradient operator

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, ( × )4. The Laplacian of a scalar V, ( )

Del Operator

In Cylindrical coordinates,

In Spherical coordinates,

Gradient of a Scalar

The gradient of ascalar field V is a

vector that representsboth the magnitudeand the direction ofthe maximum spacerate of increase of V.

Gradient of a ScalarIn Cartesiancoordinates,

In Cylindrical coordinates,

Gradient of a ScalarIn Spherical coordinates,

Gradient of a ScalarThe followingcomputationformulas on

gradient,which are

easilyproved,

should benoted:

where U and V arescalars and n is an

integer

EXAM

PLE 3

.3 Find the gradient of the followingscalar fields:a) =b) =c) =

EXER

CISE

3.3 Determine the gradient of the

following scalar fields:a) = +b) = + ∅ +c) = + ∅

EXAM

PLE 3

.4 Given = + ,compute and the direction

derivative / in the direction+4 + 12 at (2, -1, 0).

−

EXER

CISE

3.4 Given = + + . Find

gradient at point (1, 2, 3) and thedirectional derivative of at the same

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

+ + ,

EXAM

PLE 3

.5 Find the angle at which line= =intersects the ellipsoid+ + 2 = 10.

= .

EXER

CISE

3.5 Calculate the angle between the

normal to the surfaces+ = 3 and log − = −4at the point of intersection (-1, 2, 1)

. °

Divergence of a Vector and Divergence Theorem

The divergence of A at a given point P isthe outward flux per unit volume as the

volume about P.

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

Divergence of a Vector and Divergence Theorem

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

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

c) A vector field has zero divergence at P.

Divergence of a Vector and Divergence Theorem

In Cartesian coordinates,

In Cylindrical coordinates,

Divergence of a Vector and Divergence Theorem

In Spherical coordinates,

Note the following properties of thedivergence of a vector field:

Divergence of a Vector and Divergence Theorem

The divergence theorem states that thetotal outward flux of a vector field A

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

A.

Otherwise known asGauss-Ostrogradsky theorem

EXAM

PLE 3

.6 Find the divergence of these vectorfields:a) = +b) = + ∅ +c) = + +∅

EXER

CISE

3.6 Determine the divergence of the

following vector fields and evaluatethem at the specified points,:

EXAM

PLE 3

.7 If = 10 ( + ),determine the flux of G outof the entire surface of the

cylinder= 1, 0 ≤ ≤ 1.Confirm the result usingthe divergence theorem.

Ψ =

EXER

CISE

3.7 Determine the flux of= cos + sin

over the closed surface of the cylinder0 ≤ ≤ 1, = 4.Verify the divergence theorem for this case.

Curl of a Vector and Stoke’s TheoremThe curl of A is an axial (or rotational)

vector whose magnitude is themaximum circulation of A per unit area

as the area tends to zero and whosedirection is the normal direction of thearea when the area is oriented so as to

make a circulation maximum.

Curl of a Vector and Stoke’s Theorem

In Cartesiancoordinates

Curl of a Vector and Stoke’s Theorem

In Cylindricalcoordinates

Curl of a Vector and Stoke’s TheoremIn Spherical coordinates

Curl of a Vector and Stoke’s Theorem

Note the following properties of the curl:

Curl of a Vector and Stoke’s Theorem

The curl provides the maximum value ofthe circulation of the field per unit area

(or circulation density) and indicates thedirection to which this maximum value

occurs.

Measures the circulation of how muchthe field curls around P

Curl of a Vector and Stoke’s Theorem

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

Curl of a Vector and Stoke’s TheoremStokes’s theorem statesthat the circulation of avector field A around a(closed) path L is equalto the surface integral

of the curl of A over theopen surface Sbounded by L

provided that A and× are continuouson S.

EXAM

PLE 3

.8 Determine the curl of these vectorfields:a) = +b) = + ∅ +c) = + +∅

EXER

CISE

3.8 Determine the curl of the following

vector fields and evaluate them at thespecified points,:

EXAM

PLE 3

.9 If = + ∅, evaluate∮ • around the path shown.Confirm using Stokes’s theorem.

4.941

EXER

CISE

3.9 Confirm the circulation of= cos ∅ + sin ∅

around the edge L of the wedgedefined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, =0 and shown using Stoke’s Theorem

• =

EXAM

PLE 3

.10 For the vector field A,show explicitly that

× = 0;that is, the divergence of the curl of

any vector field is zero.

EXER

CISE

3.10 For a scalar field V, show that× = 0;

that is, the curl of the gradient ofany scalar field vanishes.

Laplacian of a ScalarThe Laplacian of a scalar field V, writtenas , is the divergence of the gradientof V.In Cartesian:

Laplacian of a Scalar

In Cylindrical coordinates,

In Spherical coordinates,

EXAM

PLE 3

.11Find the Laplacian of the followingscalar fields:a) =b) =c) =

EXER

CISE

3.11 Determine the Laplacian of the

following scalar fields:a) = +b) = + ∅ +c) = + ∅

Classification of Vector FieldsA vector field is uniquely characterized by its

divergence and curl.

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

field.

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

curl

Classification of Vector Fields

Classification of Vector FieldsA vector field A is said to be solenoidal

(or divergenceless) if • =Examples:• incompressible fluids,• magnetic fields• conduction current density

under steady state conditions.

Classification of Vector FieldsA vector field A is said to be irrotational

(or potential) if × =Also known as conservative field.

Examples:• electrostatic field• gravitational field

SUMMARY

Reference:Elements of Electromagnetics

by Matthew N. O. Sadiku