1-1

Engineering Electromagnetics

Chapter 1:

Vector Analysis

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Vector Addition

Associative Law:

Distributive Law:

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Rectangular Coordinate System

1-4

Point Locations in Rectangular Coordinates

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Differential Volume Element

1-6

Summary

1-7

Orthogonal Vector Components

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Orthogonal Unit Vectors

1-9

Vector Representation in Terms of Orthogonal Rectangular Components

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Summary

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Vector Expressions in Rectangular Coordinates

General Vector, B:

Magnitude of B:

Unit Vector in the Direction of B:

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Example

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Vector Field

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)

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The Dot Product

Commutative Law:

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Vector Projections Using the Dot Product

B • a gives the component of Bin the horizontal direction

(B • a)a gives the vector component of B in the horizontal direction

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Operational Use of the Dot Product

Given

Find

where we have used:

Note also:

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Cross Product

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Operational Definition of the Cross Product in Rectangular Coordinates

Therefore:

Or…

Begin with:

where

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Circular Cylindrical Coordinates

Point P has coordinatesSpecified by P(z)

z

x

y

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Orthogonal Unit Vectors in Cylindrical Coordinates

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Differential Volume in Cylindrical Coordinates

dV = dddz

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Summary

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Point Transformations in Cylindrical Coordinates

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Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

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Transform the vector,

into cylindrical coordinates:

Example

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Transform the vector,

into cylindrical coordinates:

Start with:

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Transform the vector,

into cylindrical coordinates:

Then:

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Transform the vector,

into cylindrical coordinates:

Finally:

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Spherical Coordinates

Point P has coordinatesSpecified by P(r)

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Constant Coordinate Surfaces in Spherical Coordinates

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Unit Vector Components in Spherical Coordinates

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Differential Volume in Spherical Coordinates

dV = r2sindrdd

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

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Example: Vector Component Transformation

Transform the field, , into spherical coordinates and components

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Summary Illustrations