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Vector Calculus

Basic Vector Algebra

- Scalars are quantities having only a magnitude.
- Length, mass, temperature etc.

- Vectors are quantities having both a magnitude

and a direction. - Force, velocity, acceleration etc.

Coordinate System

Rectangular or Cartesian Coordinate

Coordinate System

Cylindrical Coordinate

Coordinate System

Spherical Coordinate

Vectors in Cartesian Coordinate System

A A1i A2j A3k (A1, A2, A3)

i, j, and k are unit vectors pointing in the

positive x, y, and z directions

A1, A2 and A3 are called x, y, and z component of

vector A

Vectors in Cartesian Coordinate System

Magnitude of A

Direction of A

It is a unit directional vector !

Equality If A B, it means A B and also

Does the equality of two vectors necessarily

imply that they are identical?

Addition of Vectors

Adding corresponding components

A (A1, A2, A3)

A B (A1 B1, A2 B2, A3 B3)

B (B1, B2, B3)

Geometrical representation

(a) A pair of vectors A and B

(b) Added by the head-to-tail method

(c) Added by parallelogram law

(d) B is subtracted from A

Multiplication of a Vector by a Scalar

aA (aA1, aA2, aA3),

a is a real number

multiply the length of the vector by a the

direction unchanged

If a gt 0,

What happens if a lt 0?

multiply the length of the vector by a the

direction changed by 180

If a lt 0,

Basic Properties of the Above Algebraic

1. Commutative law A B B A. 2. Associate

law (A B) C A (B C). 3. Zero vector

(0, 0, 0) A (0, 0, 0) A. 4. Negative

vector -A (-A1, -A2, -A3). 5. a(A B) aA

aB. 6. (aß)A a(ßA) 7. (a ß)A aA ßA.

Dot Product

If A A1i A2j A3k and B B1i B2j B3k

It's Scalar, NOT Vector!

A . B A1B1 A2B2 A3B3

Another name scalar product.

Example If A (1, -3, 2) and B (4, 5, -8),

the dot product of A and B is -27.

Basic properties of the dot product

- A . B B . A
- (A B) . C A . C B . C
- A . (0, 0, 0) (0, 0, 0)
- A . A A2 A2

Geometric Interpretation of Dot Product

A . B ABcosq ABcosq

If two nonzero vectors A . B 0, then

?

cosq 0

q 90

Perpendicular

Cross Product

If A A1i A2j A3k and B B1i B2j B3k

Vector Product!

Geometric Interpretation

If two nonzero vectors A B 0, then

?

sinq 0

q 0or 180

Parallel

Example

Let A (1, -3, 2) and B (4, 5, -8), then

Basic properties of cross product

- A B -B A
- (A B) C A C B C
- A (B C) (A . C)B (A . B)C

Vector and Scalar Functions

A vector valued function A(t) is a rule that

associates with each real number t a vector A(t).

A(t) A1(t)i A2(t)j A3(t)k

For example, f(t) t3 2t 4 is a scalar

function of a single variable t, while A(t) cos

ti sin tj tk is a vector function of t.

Vector Differentiation

A vector function A(t) is differentiable at a

point t if

exists, and A'(t) is called the derivative of

A(t), written as

A'(t) A1'(t)i A2'(t)j A3'(t)k

Calculate the derivative of each component!

Example Let A(t) cos ti sin tj tk. Find

the derivative of A(t). Solution

A'(t) -sin ti cos tj k

Rules of Vector Differentiation

if A constant.

Vector Integration

Let A(t) A1(t)i A2(t)j A3(t)k and suppose

that the component functions A1(t), A2(t) and

A3(t) are integrable. Then the indefinite

integral of A(t) is defined by

Calculate the integral of each component!

If A1(t), A2(t) and A3(t) are integrable over the

interval t1, t2, then the definite integral of

A(t) is defined by

Example

Let A(t) cos ti sin tj tk. Find

Solution

Line Integral of Vector Functions

dl dxi dyj dzk

For a closed loop, i.e. ABCA,

circulation of P around L

Line given by L(x(s), y(s), z(s)), s parametric

variable

Always take the differential element dl as

positive and insert the integral limits according

to the paths!!!

Example

For F yi xj, calculate the circulation of F

along the two paths as shown below.

Solution

dl dxi dyj dzk

Along path C2

Example - Continue

Along path C1

Using x as the parametric variable, the path

equations are given as

Therefore,

and

Example - Continue

The vector field defined by F in a given domain

is non-conservative. The line integral is

dependent on the integration path!

is work done on an object along path C if F

force !!

Is the static Electrical field conservative?

Yes, because the work done when we move a charge

from one point to another is independent of the

path but determined by the potential difference

between these two points.

Surface Integral

Surface integral or the flux of P across the

surface S is

is the outward unit vector normal to the

surface.

For closed surface,

net outward flux of P.

Example

If F xi yj (z2 1)k, calculate the flux of

F across the surface shown in the figure.

Solution

Volume Integral

Evaluation choose a suitable integration order

and then find out the suitable lower and upper

limits for x, y and z respectively.

Example Let F 2xzi xj y2k. Evaluate

where V is the region bounded by the surface x

0, x 2, y 0, y 6, z 0, z 4.

Volume Integral

Solution

In electromagnetic,

Total charge within the volume

where ?v volume charge density (C/m3)

Scalar Field

Every point in a region of space is assigned a

scalar value obtained from a scalar function f(x,

y, z), then a scalar field f(x, y, z) is defined

in the region, such as the pressure in atmosphere

and mass density within the earth, etc.

Partial Derivatives

Mixed second partials

Example

Let f x2 2y2. Calculate

and

Solution

Gradient

Del operator

Gradient

Gradient characterizes maximum increase. If at a

point P the gradient of f is not the zero vector,

it represents the direction of maximum space rate

of increase in f at P.

Example

Given potential function V x2y xy2 xz2, (a)

find the gradient of V, and (b) evaluate it at

(1, -1, 3).

Solution (a)

(b)

Direction of maximum increase

Vector Field

Electric field E E(x, y, z),

Magnetic field H H(x, y, z)

Every point in a region of space is assigned a

vector value obtained from a vector function A(x,

y, z), then a vector field A(x, y, z) is defined

in the region.

R(t1, t2) acos t1i asin t1j t2k

Divergence of a Vector Field

Representing field variations graphically by

directed field lines - flux lines

Divergence of a Vector Field

The divergence of a vector field A at a point is

defined as the net outward flux of A per unit

volume as the volume about the point tends to

zero

It indicates the presence of a source (or sink)!

? term the source as flow source. And div A is a

measure of the strength of the flow source.

Divergence of a Vector Field

In rectangular coordinate, the divergence of A

can be calculated as

For instance, if A 3xzi 2xyj yz2k, then

div A 3z 2x 2yz

At (1, 2, 2), div A 0 at (1, 1, 2), div A 4,

there is a source at (1, 3, 1), div A -1,

there is a sink.

Curl of a Vector Field

The curl of a vector field A is a vector whose

magnitude is the maximum net circulation of A per

unit area as the area tends to zero and whose

direction is the normal direction of the area.

It is an indication of a vortex source, which

causes a circulation of a vector field around it.

Water whirling down a sink drain is an example of

a vortex sink causing a circulation of fluid

velocity.

If A is electric field intensity, then the

circulation will be an electromotive force around

the closed path.

Curl of a Vector Field

In rectangular coordinate, curl A can be

calculated as

Curl of a Vector Field

Example

If A yzi 3zxj zk, then

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