VECTOR REPRESENTATATION OF SURFACES

Consider a plane surface S which is either an open surface or part of a closed

surface. The vector surface is defined as:

ˆnS Sa

Where S is the area of the given surface, and ˆna , is the unit vector

perpendicular to the surface.

If S is a part of a closed surface, positive ˆna is taken to be an outward

normal. If S is an open surface, first its periphery is oriented and then ˆna is

defined with the right-hand rule. Positive ˆna is thus selected arbitrarily.

ORTHOGONAL COORDINATE SYSTEMS

In a three dimensional space a point can be located as the intersection of

three surfaces. When the surfaces intersect perpendicularly we have an

orthogonal coordinate system.

Cartesian Coordinates:

Here, the constant surfaces are .x const , .y const and .z const

Point 1 1 1 1( , , )P x y z is located at the intersection of three surfaces.

Constant Surfaces

Fig. 2-1 Cartesian coordinates constant surfaces

.x const Planar Surface

.y const Planar Surface

.z const Planar Surface

Unit Vectors

ˆxa x const. Surface

ˆya y const. Surface

ˆza z const. Surface

Differential Length Element

ˆ ˆ ˆx y zdl a dx a dy a dz

Fig. 2-2 Differential Length Elements in the Cartesian coordinate system

Differential Surface Elements:

ˆ

ˆ

ˆ

x x

y y

z z

ds dydz a

ds dxdz a

ds dz dya

Fig.2-3 Differential Surface Elements in the Cartesian coordinate system

Differential Volume Element

dV dxdydz

Cylindrical Coordinates

The constant surfaces are: .const , .const , .z const

Point 1 1 1( , , )P z is located at the intersection of three surfaces.

Fig. 2-4 Cylindrical Coordinates, constant surfaces.

Constant Surfaces

.,const Circular cylinders

.,const Planes

.,z const Planes

Ranges of variables:

0

0 2

z

Unit Vectors

Fig. 2-5 Cylindrical Coordinates, unit vectors.

ˆ , .a const Surface

ˆ , .a const Surface

ˆ , .za z const Surface

Differential Length Elements

ˆ ˆ ˆzd a d a d a dz

Fig. 2-6 Differential Length Elements in the Cylindrical Coordinate system.

Differential Volume Element

dV d d dz

Differential Surface Elements

ˆds a d dz

ˆds a d dz

ˆz zds a d d

Fig. 2-7 Differential surface elements in the cylindrical coordinate system.

The unit vectors obey the following right-hand cyclic relations:

ˆ ˆ ˆza Xa a

ˆ ˆ ˆza Xa a

ˆ ˆ ˆza Xa a

Also, like the other vectors:

ˆ ˆ. 1a a

ˆ ˆ. 1a a

ˆ ˆ. 1z za a

ˆ ˆ. 0a a

ˆ ˆ. 0za a

ˆ ˆ. 0za a

Cylindrical Unit Vectors in Terms of Cartesian Unit Vectors

Fig. 2-8 Cylindrical Unit Vectors in Terms of Cartesian Unit Vectors

ˆ ˆ ˆcos sinx ya a a

ˆ ˆ ˆsin sx ya a a co

ˆ ˆz za a

Similarly;

ˆ ˆ ˆcos sinxa a a

ˆ ˆ ˆsin sya a a co

ˆ ˆz za a

Relationship between ( , , )x y z and ( , , )z :

Fig.2-9 Relationship between ( , , )x y z and ( , , )z

2 2 1

cos , sin ,

, tan ,

x y z z

xx y z z

y

Spherical Coordinates

The constant surfaces are: .r const , .const , .const

Point 1 1 1( , , )P r is located at the intersection of three surfaces.

Fig. 2-10 Unit vectors in spherical coordinates.

Unit Vectors

ˆ , .ra r const Surface

ˆ , .a const Surface

ˆ , .a const Surface

Ranges of variables:

0

0

0 2

r

Constant Surfaces

.,r const Spherical Surfaces

.,const Conical Surfaces

.,const Planes

Fig. 2-11 Constant surfaces in spherical coordinates.

Differential Length Elements

ˆ ˆ ˆ sinrd a dr a rd a r d

Fig. 2-11 Differential length elements for spherical coordinates.

Differential Volume Element

2 sindV r drd d

Differential Surface Elements

2ˆ sinr rds a r d d

ˆ sinds a r drd

ˆds a rdrd

Fig. 2-12 Differential surface elements for spherical coordinates.

The unit vectors obey the following right-hand cyclic relations:

ˆ ˆ ˆra Xa a

ˆ ˆ ˆra Xa a

ˆ ˆ ˆra Xa a

Also, like the other vectors:

ˆ ˆ. 1r ra a

ˆ ˆ. 1a a

ˆ ˆ. 1a a

ˆ ˆ. 0ra a

ˆ ˆ. 0a a

ˆ ˆ. 0ra a

Spherical Unit Vectors in Terms of Cartesian Unit Vectors

Fig. 2-13 Unit vector relations between Cartesian and spherical coordinates.

ˆ ˆ ˆ ˆsin cos cos cos sin

ˆ ˆ ˆ ˆsin sin cos sin s

ˆ ˆ ˆcos sin

x r

y r

z r

a a a a

a a a co a

a a a

ˆ ˆ ˆ ˆsin cos sin sin cos

ˆ ˆ ˆ ˆcos cos cos sin sin

ˆ ˆ ˆsin s

r x y z

x y z

x y

a a a a

a a a a

a a co a

Relationship between ( , , )x y z and ( , , )r :

Fig. 2-14 Relationship between Cartesian coordinates and spherical

coordinates.

2 2

2 2 2 1 1, tan , tan

sin cos , sin sin , cos

x y yr x y z

z x

x r y r z r