Fields and Waves

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Fields and Waves
Lesson 2.1
VECTORS and VECTOR CALCULUS
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VECTORS
Todays Class will focus on

• vectors - description in 3 coordinate systems

• vector operations - DOT CROSS PRODUCT

• vector calculus - AREA and VOLUME INTEGRALS

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VECTOR NOTATION
VECTOR NOTATION
Rectangular or Cartesian Coordinate System
Dot Product
(SCALAR)
Cross Product
(VECTOR)
Magnitude of vector
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VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Examples
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
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VECTOR REPRESENTATION CYLINDRICAL COORDINATES
UNIT VECTORS
Cylindrical representation uses r ,f , z
Dot Product
(SCALAR)
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VECTOR REPRESENTATION SPHERICAL COORDINATES
UNIT VECTORS
Spherical representation uses r ,q , f
Dot Product
(SCALAR)
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VECTOR REPRESENTATION UNIT VECTORS
Rectangular Coordinate System
Unit Vector Representation for Rectangular
Coordinate System
The Unit Vectors imply
Points in the direction of increasing x
Points in the direction of increasing y
Points in the direction of increasing z
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VECTOR REPRESENTATION UNIT VECTORS
Cylindrical Coordinate System
The Unit Vectors imply
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing z
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VECTOR REPRESENTATION UNIT VECTORS
Spherical Coordinate System
The Unit Vectors imply
Points in the direction of increasing r
Points in the direction of increasing q
Points in the direction of increasing j
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VECTOR REPRESENTATION UNIT VECTORS Summary
RECTANGULAR Coordinate Systems
CYLINDRICAL Coordinate Systems
SPHERICAL Coordinate Systems
NOTE THE ORDER!
r,f, z
r,q ,f
Note We do not emphasize transformations between
coordinate systems
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METRIC COEFFICIENTS
1. Rectangular Coordinates
When you move a small amount in x-direction, the
distance is dx
In a similar fashion, you generate dy and dz
( dx, dy, dz )
Generate
2. Cylindrical Coordinates
Differential Distances
Distance r df
( dr, rdf, dz )
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METRIC COEFFICIENTS
3. Spherical Coordinates
Differential Distances
Distance r sinq df
( dr, rdq, r sinq df )
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METRIC COEFFICIENTS
Representation of differential length dl in
coordinate systems
rectangular
cylindrical
spherical
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AREA INTEGRALS

• integration over 2 delta distances

Example
AREA
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Note that z constant
In this course, area surface integrals will be
on similar types of surfaces e.g. r constant or
f constant or q constant et c.
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PROBLEM 2
PROBLEM 2A
What is constant?
How is the integration performed? What are the
differentials?
Representation of differential surface element
Vector is NORMAL to surface
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DIFFERENTIALS FOR INTEGRALS
Example of Line differentials
or
or
Example of Surface differentials
or
Example of Volume differentials