1. Introduction Coordinate Systems L1-2

Vector calculus

and

different coordinate systems

UNIT I-INTRODUCTION

S.Krishnaveni AP/EEE 1

Objective

Broadly, the study of Physics improves one‟s ability to

think logically about the problems of science and

technology and obtain their solutions. The present course

is aimed to offer a broad aspect of those areas of Physics

which are specifically required as an essential background

to all engineering students for their studies in higher

semesters.

Learning outcomes

At the end of the course, the students will have sufficient

scientific understanding of electromagnetic fields and

waves.

Three-Dimensional

Coordinate Systems

In this section, we will learn about:

Aspects of three-dimensional coordinate systems.

VECTORS AND THE GEOMETRY OF SPACE


COORDINATE SYSTEMS

• RECTANGULAR or Cartesian

• CYLINDRICAL

• SPHERICAL

Choice is based on

symmetry of problem

Examples: Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

To understand the Electromagnetics, we must know basic vector algebra and

coordinate systems. So let us start the coordinate systems.


Cylindrical Symmetry Spherical Symmetry

Visualization (Animation) S.Krishnaveni AP/EEE 5

Orthogonal Coordinate Systems:

3. Spherical Coordinates

2. Cylindrical Coordinates

1. Cartesian Coordinates

P (x, y, z)

P (r, θ, Φ)

P (r, Φ, z)

x

y

z P(x,y,z)

Φ

z

r x

y

z

P(r, Φ, z)

θ

Φ

r

z

y x

P(r, θ, Φ)

Rectangular Coordinates

Or

X=r cos Φ,

Y=r sin Φ,

Z=z

X=r sin θ cos Φ,

Y=r sin θ sin Φ,

Z=z cos θ S.Krishnaveni AP/EEE 6

//upload.wikimedia.org/wikipedia/commons/b/b2/Cylindrical_coordinate_surfaces.png

//upload.wikimedia.org/wikipedia/commons/a/ad/Spherical_coordinate_surfaces.png

Cartesian Coordinates

P(x, y, z)

Spherical Coordinates

P(r, θ, Φ)

Cylindrical Coordinates

P(r, Φ, z)

x

y

z P(x,y,z)

Φ

z

r x

y

z

P(r, Φ, z)

θ

Φ

r

z

y x

P(r, θ, Φ)


•The three lines are called the coordinate

axes.

•They are labeled:

– x-axis

– y-axis

– z-axis

COORDINATE AXES


•Curl the fingers of your right hand

around the z-axis in the direction of a 90° counterclockwise rotation

from the positive

x-axis to the positive y-axis.

– Then, your thumb

points in the positive

direction of the z-axis.

COORDINATE AXES


•The three coordinate axes determine

the three coordinate planes.

– The xy-plane contains

the x- and y-axes.

– The yz-plane contains

the y- and z-axes.

– The xz-plane contains

the x- and z-axes.

COORDINATE PLANES


•Look at any bottom corner of a room and call the corner the

origin.

3-D COORDINATE SYSTEMS


•The wall on your left is in the xz-plane.

•The wall on your right is in the yz-plane.

•The floor is in the xy-plane.


•The x-axis runs along the intersection of the floor and the left wall.

•The y-axis runs along that of the floor and the right wall.

•The z-axis runs up from the floor toward the ceiling along the intersection of the two

walls.


•Now, if P is any point in space,

let:

– a be the (directed) distance from the yz-plane to P.

– b be the distance from the xz-plane to P.

– c be the distance from the xy-plane to P.

3-D COORDINATE


•We represent the point P by the ordered

triple of real numbers (a, b, c).

•We call a, b, and c the coordinates of P.

– a is the x-coordinate.

– b is the y-coordinate.

– c is the z-coordinate.



•Thus, to locate the point (a, b, c), we can start at the origin O and proceed

as follows:

– First, move a units along the x-axis.

– Then, move b units

parallel to the y-axis.

– Finally, move c units

parallel to the z-axis.



• The point P(a, b, c) determines a rectangular box.

• If we drop a perpendicular from P to the xy-plane, we get a point Q with

coordinates (a, b, 0).

– This is called

the projection of P

on the xy-plane.

•Similarly, R(0, b, c) and S(a, 0, c)

are the projections of P on the

yz-plane and xz-plane, respectively.

3-D COORDINATE S


•As numerical illustrations, the points

(–4, 3, –5) and (3, –2, –6) are plotted here.



• we construct a rectangular box as shown,

where:

– P1 and P2 are

opposite vertices.

– The faces of the box

are parallel to the

coordinate planes.



•If A(x2, y1, z1) and B(x2, y2, z1) are the vertices of the box, then

|P1A| = |x2 – x1|, |AB| = |y2 – y1|, |BP2| = |z2 – z1|


Combining those equations, we get:

|P1P2|2 = |P1A|2 + |AB|2 + |BP2|

2

= |x2 – x1|2 + |y2 – y1|

2 + |z2 – z1|2

= (x2 – x1)2 + (y2 – y1)

2 + (z2 – z1)2

2 2 2

1 2 2 1 2 1 2 1( ) ( ) ( )PP x x y y z z


Cartesian coordinate system

• dx, dy, dz are infinitesimal displacements along X,Y,Z.

• Volume element is given by

dv = dx dy dz

• Area element is

da = dx dy or dy dz or dxdz

• Line element is

dx or dy or dz

Ex: Show that volume of a cube of edge a is a3.

P(x,y,z)

X

Y

Z

3

000

adzdydxdvVaa

v

a

dx

dy

dz


Cartesian Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv


AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant


Cylindrical Coordinate System F.ppt

Cylindrical coordinate system

(r,φ,z)

X

Y

Z

r

φ

Z


Spherical polar coordinate system

• dr is infinitesimal displacement

along r, r dφ is along φ and

dz is along z direction.


dv = dr r dφ dz

• Limits of integration of r, θ, φ

are

0<r<∞ , 0<z <∞ , o<φ <2π

φ is azimuth angle

Cylindrical coordinate system

(r,φ,z)

X

Y

Z

r φ

r dφ

dz

dr

r dφ

dr

dφ


Volume of a Cylinder of radius „R‟

and Height „H‟

HR

dzdrdr

dzddrrdvV

R H

v

2

0

2

0 0

Try yourself:

1) Surface Area of Cylinder = 2πRH .

2) Base Area of Cylinder (Disc)=πR2.


Differential quantities:

Length element:

Area element:

Volume element:

dzardadrald zrˆˆˆ

rdrdasd

drdzasd

dzrdasd

zz

rr

ˆ

ˆ

ˆ

dzddrrdv

Limits of integration of r, θ, φ are 0<r<∞ , 0<z <∞ , o<φ <2π

Cylindrical Coordinates: Visualization of Volume element


Spherically Symmetric problem

(r,θ,φ)

X

Y

Z

r

φ

θ


Spherical polar coordinate system (r,θ,φ)

• dr is infinitesimal displacement along r, r dθ is along θ and r sinθ dφ is along φ direction.


dv = dr r dθ r sinθ dφ

• Limits of integration of r, θ, φ are

0<r<∞ , 0<θ <π , o<φ <2π

P(r, θ, φ)

X

Y

Z

r

φ

θ

dr P

r dθ

r sinθ dφ

θ is zenith angle( starts from +Z reaches up to –Z) ,

φ is azimuth angle (starts from +X direction and lies in x-y plane only)

r cos θ

r sinθ


Volume of a sphere of radius „R‟

33

0 0

2

0

2

2

3

42.2.

3

sin

sin

RR

dddrr

dddrrdvV

R

v

Try Yourself:

1)Surface area of the sphere= 4πR2 . S.Krishnaveni AP/EEE 29

Spherical Coordinates: Volume element in space


Points to remember

System Coordinates dl1 dl2 dl3

Cartesian x,y,z dx dy dz

Cylindrical r, φ,z dr rdφ dz

Spherical r,θ, φ dr rdθ r sinθdφ

• Volume element : dv = dl1 dl2 dl3

• If Volume charge density ‘ρ’ depends only on ‘r’:

Ex: For Circular plate: NOTE

Area element da=r dr dφ in both the

coordinate systems (because θ=900)

drrdvQv l

24


Quiz: Determine

a) Areas S1, S2 and S3.

b) Volume covered by these surfaces.

Radius is r,

Height is h,

X

Y

Z

r

dφ

S1 S2

S3

21

hr

dzrddrVb

rrddrSiii

rhdzdrSii

rhdzrdSia

Solution

h r

r

r h

h

)(2

..)

)(2

.3)

2)

)(1))

:

12

2

0 0

12

2

0

0 0

12

0

2

1

2

1

2

1


Coordinate Transformation

Cylindrical coordinate—Cartesian coordinate



Change of variables

Cylindrical coordinate—Cartesian coordinate



Change of variables

Spherical coordinate—Cartesian coordinate



Spherical coordinate—Cartesian coordinate


Metric Coefficients


Basic Orthogonal Coordinate Systems


Vector Analysis

• What about A.B=?, AxB=? and AB=?

• Scalar and Vector product:

A.B=ABcosθ Scalar or

(Axi+Ayj+Azk).(Bxi+Byj+Bzk)=AxBx+AyBy+AzBz

AxB=ABSinθ n Vector (Result of cross product is always

perpendicular(normal) to the plane

of A and B A

B

n


An orthogonal system is one in which te coordinates are

mutually perpendicular.

Examples of orthogonal coordinate systems include the

Cartesian, cylindrical and spherical coordinates.

There must be three independent variables. e.g: u1 , u2

and u3.

, and are unit vectors for each surface and the

direction normal to their surfaces.

ORTHOGONAL COORDINATE SYSTEM

1u

2u

3u


The cross product between the unit vector is:

While the dot product is:

•

2ˆ

1ˆ

3ˆ ,

1ˆ

3ˆ

2ˆ ,

3ˆ

2ˆ

1ˆ uuuuuuuuu

13ˆ

3ˆ

2ˆ

2ˆ

1ˆ

1ˆ

01ˆ

3ˆ

3ˆ

2ˆ

2ˆ

1ˆ

uuuuuu

uuuuuu


Vector Addition and

Subtraction


Product of Vector

• Multiplication by a scalar

• Dot (scalar) product

• Cross (vector)


Product of Vector

Multiplication by a scalar

Dot (scalar) product


Product of Vector

• Cross (vector) product

• A new product perpendicular to the plane containing the

two Vectors.


Gradient, Divergence and Curl

• Gradient of a scalar function is a

vector quantity.

• Divergence of a vector is a scalar

quantity.

• Curl of a vector is a vector

quantity.

f Vector

xA

A

.

The Del Operator


Gradient:

gradT: points the direction of maximum increase of the

function T.

Divergence:

Curl:

Operator in Cartesian Coordinate System

kz

Tj

y

Ti

x

TT ˆˆˆ

y zxV VV

Vx y z

ky

V

x

Vj

x

V

z

Vi

z

V

y

VV xyzxyz ˆˆˆ

kVjViVV zyxˆˆˆ

where

as

Fundamental theorem for

divergence and curl

• Gauss divergence

theorem:

• Stokes curl theorem

v s

daVdvV .).(

s l

dlVdaVx .).(

Conversion of volume integral to surface integral and vice verse.

Conversion of surface integral to line integral and vice verse. S.Krishnaveni AP/EEE 48

Operator in Cylindrical Coordinate System

Volume Element:

Gradient:

Divergence:

Curl:

dzrdrddv

zz

TˆT

rr

r

TT

1

1 1 z

r

V VV rV

r r r z

zV

rVrr

ˆr

V

z

Vr

z

VV

rV rzrz

11

zVVrVV zr ˆˆˆ


Operator In Spherical Coordinate System

Gradient :

Divergence:

Curl:

ˆT

sinrˆT

rr

r

TT

11

2

2

sin1 1 1

sin sin

rr V VVV

r r r r

ˆVrV

rr

ˆrVr

V

sinrr

VVsin

sinrV

r

r

1

111

ˆˆˆ VVrVV r S.Krishnaveni AP/EEE 50

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

Basic Vector Calculus

2

( )

0, 0

( ) ( )

F G G F F G

F

F F F

Divergence or Gauss’ Theorem

SV

SdFdVF


dSnSd

Oriented boundary L

n

Stokes’ Theorem

S L

ldFSdF

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


The following operations involving operator :


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1. Introduction Coordinate Systems L1-2

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