ELEC 3600 TUTORIAL 2

VECTOR CALCULUS

Alwin Tam

amwtam@ust.hk

Rm. 3121A

WHAT HAVE WE LEARNT SO FAR?

• Classification of vector & scalar fields• Differential length, area and volume• Line, surface and volume integrals• Del operator• Gradient of a scalar• Divergence of a vector

– Divergence theorem• Curl of a vector

– Stokes’ theorem• Laplacian of a scalar

SCALAR AND VECTOR FIELD

What is scalar field? Quantities that can be completely described

from its magnitude and phase. i.e. weight, distance, speed, voltage, impedance, current, energy

What is a vector field? Quantities that can be completely described

from its magnitude, phase and LOCATION. i.e. force, displacement, velocity, electric field, magnetic field

Need some sense of direction i.e. up, down right and left to specify

SCALAR AND VECTOR FIELD (CONT.)

Is temperature a scalar quantity?A. YesB. NoAnswer: A, because it can be completely described by a number when someone ask how hot is today.

Is acceleration a scalar quantity?A. YesB. NoAnswer: B, because it requires both magnitude and some sense of direction to describe i.e. is it accelerating upward, downwards, left or right etc.

VECTOR CALCULUS

What is vector calculus? Concern with vector differentiation and line, surface

and volume integral

So why do we need vector calculus?? To understand how the vector quantities i.e. electric

field, changes in space (vector differential) To determine the energy require for an object to

travel from one place to another through a complicated path under a field that could be spatially varying (line integral) i.e.

To pass ELEC 3600!! (vector differential and line integral)

W =

DIFFERENTIAL LENGTH, VOLUME AND SURFACE (CARTESIAN COORDINATE)

Differential length A vector whose magnitude is

close to zero i.e. dx, dy and dz → 0

Differential volume An object whose volume

approaches zero i.e. dv = dxdydz → 0 (scalar)

Differential surface A vector whose direction is

pointing normal to its surface area

Its surface area |dS| approach zero i.e. shaded area ~ 0

Calculated by cross product of two differential vector component

Differential is infinitely small difference between 2 quantities

DIFFERENTIAL LENGTH, VOLUME AND SURFACE (CYLINDRICAL

COORDINATE)All vector components MUST

have spatial units i.e. meters, cm, inch etc.

DIFFERENTIAL LENGTH, VOLUME AND SURFACE (SPHERICAL COORDINATE)

z

x

y

All vector components MUST have spatial units i.e. meters,

cm, inch etc.

LINE INTEGRAL Line integral: Integral of the tangential component

of vector field A along curve L. 2 vectors are involve inside the integral Result from line integral is a scalar

Line integral Definite integral

Diagram

Maths description

Result

Area under the curve

A measure of the total effect of a given field along a given path

Information required

1. Vector field expression A

2. Path expression

1. Function f(x)2. Integral limits

Integral limits depends on path

SURFACE & VOLUME INTEGRAL

Surface integral: Integral of the normal component of vector field A along curve L. Two vectors involve inside the integral Result of surface integral is a scalar

Volume integral: Integral of a function f i.e. inside a given volume V. Two scalars involve inside the integral Result of volume integral is a scalar

SURFACE & VOLUME INTEGRAL (CONT.)

Surface integral Volume integral

Diagram

Maths description

Result

A measure of the total effect of a scalar function i.e. temperature, inside a given volume

A measure of the total flux from vector field passing through a given surface

Information required

1. Vector field expression A

2. Surface expression

1. Scalar Function rv

2. Volume expression

Integral limits depends on surface

Integral limits depends on volume

PROBLEM 1

Given that , calculate the circulation of F around the (closed) path shown in the following figure.

zyx2 aaaF 2yxzx

Solution:

DEL OPERATOR

Vector differential operator

Must operate on a quantity (i.e. function or vector) to have a meaning

Mathematical form:

Cartesian Cylindrical Spherical

SUMMARY OF GRAD, DIV & CURL

Gradient Divergence Curl

must operate on Scalar f(x,y) Vector A Vector A

Expression (Cartesian)

Expression (Cylindrical)

Expression (Spherical)

Result Vector Scalar Vector

SUMMARY OF GRAD, DIV & CURL

Gradient Divergence Curl

Physical meaning

A vector that gives direction of the maximum rate of change of a quantity i.e. temp

A scalar that measures the magnitude of a source or sink at a given point

Sink Source

A vector operator that describes the rotation/ununiformity of a vector field

RHC rotation

LHC rotation

Irrotational

i.e. Flux out < flux in

i.e. Flux out > flux in

IncompressibleFlux out = flux in

DIVERGENCE THEOREM

Divergence theorem: Total outward flux of a vector field A through a

closed surface S is the same as the volume integral of divA. i.e. Transformation of volume integral involving divA to surface integral involving A

Equation:

Physical meaning: The total flux from field A passing through a volume V is equivalent to summing all the flux at the surface of V.

PROBLEM 2 (MIDTERM EXAM 2013)

Verify the divergence theorem for the vector r2ar within the semisphere.

STOKE’S THEOREM

Stoke’s Theorem: The line integral of field A at the boundary of a

closed surface S is the same as the total rotation of field A at the surface. i.e. Transformation of surface integral involving curlA to line integral of A

Equation:

Physical meaning: The total effect of field A along a closed path is equivalent to summing all the rotational component of the field inside the surface of which the path enclose.

LAPLACIAN OF A SCALAR FUNCTION

U is a scalar function of x, y, z (i.e. temperature)

Laplacian of a scalar = Divergence of a Gradient of scalar function.

Important operator when working with MAXWELL’S EQUATION!!

2 2 2

22 2 2

U U UU U

x y z

2 2 22

2 2 2, ,

x y z

PROBLEM 3

Given that , find(a) Where L is shown in the following figure

(b) Where S is the area bounded by L

(c) Is Stokes’s theorem satisfied?

yx aaF yyx 2

LdlF

S

dSF

1 2

3