co-ordinate systems

Co-ordinate Systems

By,

S.T.Suganthi,

HOD/EEE,

S.Veerasamy Chettiar college of Engineering and

Technology,

Puliangudi-627855

Vector Calculus - Addition

1-2

Associative Law:

Distributive Law:

Scalar and Vector Fields

A scalar field is a function that gives us a single value of some variable for every point in space.

◦ Examples: voltage, current, energy, temperature

A vector is a quantity which has both a magnitude and a direction in space.

◦ Examples: velocity, momentum, acceleration and force

Examples of Vector Fields

Examples of Vector Fields

Examples of Vector Fields

Vector Field

1-7

We are accustomed to thinking of a specific vector:

A vector field is a function defined in space that has magnitude

and direction at all points:

where r = (x,y,z)

The Dot Product

1-8

Commutative Law:

Vector Projections Using the Dot Product

B • a gives the component of B

in the horizontal direction

(B • a) a gives the vector component

of B in the horizontal direction

Projection of a vector on another vector

Operational Use of the Dot Product

Given

Find

where we have used:

Note also:

Cross Product

1-12

Operational Definition of the Cross Product in Rectangular Coordinates

Therefore:

Or…

Begin with:

where

Orthogonal Vector Components

1-14

Orthogonal Unit Vectors

1-15

Vector Representation in Terms of Orthogonal Rectangular Components

1-16

Co-ordinate System

To describe a vector accurately such as lengths, angles, and projections etc…

Three types of Co-ordinate system Rectangular (or) Cartesian Co-ordinates

Cylindrical Co-ordinates

Spherical Co-ordinates

1-17

Rectangular Coordinate SystemCo-ordinates are (x,y,z)

1-18

Rectangular Coordinate System

A Point Locations in Rectangular Coordinates –Intersection of 3 orthogonal planes ( X-constant plane, Y- constant Plane, Z-constant plane)

1-19

Differential Volume Element

1-20

Vector Expressions in Rectangular Coordinates

1-21

General Vector, B:

Magnitude of B:

Unit Vector in the

Direction of B:

Example

1-22

Cylindrical Coordinate Systems

1-23

Cylindrical Coordinate Systems

1-24

Cylindrical Coordinate Systems

1-25

Cylindrical Coordinate Systems

1-26

Differential Volume in Cylindrical Coordinates

1-27

dV = dddz

Point Transformations in Cylindrical Coordinates

1-28

Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

1-29

Example

1-30

Transform the vector, into cylindrical coordinates:

Start with:

Then:

Finally:

Example: cont.

1-32

Spherical Coordinates

1-33

Spherical Coordinates

1-34

Spherical Coordinates

1-35

Spherical Coordinates

1-36

Spherical Coordinates

Spherical Coordinates

1-37

Point P has coordinates

Specified by P(r)

Differential Volume in Spherical Coordinates

1-38

dV = r2sindrdd

Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

1-39

Example: Vector Component Transformation

1-40

Transform the field, , into spherical coordinates and components

Constant coordinate surfaces-Cartesian system

1-41

If we keep one of the coordinate variables

constant and allow the other two to vary,

constant coordinate surfaces are generated in

rectangular, cylindrical and spherical

coordinate systems.

We can have infinite planes:

X=constant,

Y=constant,

Z=constant

These surfaces are perpendicular to x, y and z axes respectively.

Constant coordinate surfaces-cylindrical system

1-42

Orthogonal surfaces in cylindrical

coordinate system can be generated as

ρ=constnt

Φ=constant

z=constant

ρ=constant is a circular cylinder,

Φ=constant is a semi infinite plane with its

edge along z axis

z=constant is an infinite plane as in the

rectangular system.

Constant coordinate surfaces-Spherical system

1-43

Orthogonal surfaces in spherical

coordinate system can be generated

as

r=constant

θ=constant

Φ=constant

θ =constant is a circular cone with z axis as its axis and origin at

the vertex,

Φ =constant is a semi infinite plane as in the cylindrical system.

r=constant is a sphere with its centre at the origin,

Differential elements in rectangularcoordinate systems

1-44

Differential elements in Cylindricalcoordinate systems

1-45

Differential elements in Sphericalcoordinate systems

1-46

1-47

Line integrals

Line integral is defined as any integral that is to be evaluated

along a line. A line indicates a path along a curve in space.

Surface integrals

1-48

Volume integrals

1-49

DEL Operator

1-50

DEL Operator in cylindrical coordinates:

DEL Operator in spherical coordinates:

Gradient of a scalar field

1-51

The gradient of a scalar field V is a vector that represents the

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

For Cartesian Coordinates

For Cylindrical Coordinates

For Spherical Coordinates

Divergence of a vector

1-52

In Cartesian Coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

Gauss’s Divergence theorem

1-53

Curl of a vector

1-54

Curl of a vector

1-55

In Cartesian Coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

Stoke’s theorem

1-57

Laplacian of a scalar

1-58

Laplacian of a scalar

1-59