vector representatation of surface1

8/11/2019 Vector Representatation of Surface1

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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:

nS Sa

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

perpendicular to the surface.

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

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

defined with the right-hand rule. Positive na 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 anorthogonal 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.


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Constant Surfaces

Fig. 2-1 Cartesian coordinates constant surfaces

.x const Planar Surface

.y const Planar Surface.z const Planar Surface

Unit Vectors

xa x const. Surface

ya y const. Surface

za z const. Surface


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Differential Length Element

x y zdl a dx a dy a dz

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

Differential Surface Elements:

x x

y y

z z

ds dy dz a

ds dx dz a

ds dz dy a


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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.


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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.


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, .a const Surface

, .a const Surface

, .z

a z const Surface

Differential Length Elements

zd a d a d a dz

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


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dV d d dz

Differential Surface Elements

ds a d dz

ds a d dz

z zds a d d

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


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

za Xa a

z

a Xa a

za Xa a

Also, like the other vectors:

. 1a a

. 1a a

. 1z z

a a

. 0a a

. 0z

a a

. 0za a


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Cylindrical Unit Vectors in Terms of Cartesian Unit Vectors

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

cos sinx ya a a

sin sx ya a a co

z z

a a

Similarly;

cos sinxa a a

sin sya a a co

z z

a a


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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


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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

, .ra r const Surface

, .a const Surface


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, .a const Surface

Ranges of variables:

0

0

0 2

r

Constant Surfaces

.,r const Spherical Surfaces.,const Conical Surfaces.,const Planes


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Fig. 2-11 Constant surfaces in spherical coordinates.

Differential Length Elements

sinr

d a dr a rd a r d


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


2 sindV r drd d

Differential Surface Elements

2 sinr rds a r d d

sinds a r drd

ds a rdrd


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

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

ra Xa a

r

a Xa a

ra Xa a

Also, like the other vectors:

. 1r ra a

. 1a a

. 1a a

. 0ra a

. 0a a

. 0r

a a


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Spherical Unit Vectors in Terms of Cartesian Unit Vectors

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

sin cos cos cos sin

sin sin cos sin s

cos sin

x r

y r

z r

a a a a

a a a co a

a a a

sin cos sin sin cos

cos cos cos sin sin

sin s

r x y z

x y z

x y

a a a a

a a a a

a a co a


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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

vector representatation of surface1

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