ch2 summary vector algebra
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tableTRANSCRIPT
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Chapter 2:
Vector Algebra
Electromagnetics for Engineers
F.T. Ulaby
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2-1 Basic Laws of Vector Algebra
2-2 Orthogonal Coordinate Systems
2-3 Transformations between Coordinate Systems
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2-1 Basic Laws of Vector Algebra
Magnitude and direction of a vector
Unit and base vectors
Equal and identical vectors
Addition and subtraction of vectors
Position and distance vectors
Vector multiplication: Simple product = scalarvector kA,
Scalar or dot product AB, Vector or cross product A x B (right-hand rule)
Scalar A(BxC), and vector triple products Ax(BxC).
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Magnitude and direction of a vector
Unit and base vectors
Equal and identical vectors
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Addition and subtraction of vectors
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Position vectors
Distance vector
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Vector multiplication
Simple product = scalarvector
Scalar or dot product, AB
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Properties of the vectors dot product
Finding the smaller angle between two specified
vectors:
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Vector or cross product, A x B
Properties of the vectors cross product
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Scalar and vector triple products
Scalar triple product,
Vector triple product,
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2-2 Orthogonal Coordinate Systems
(three perpendicular dimensional coordinates)
1. Cartesian Coordinates
Differential length, area, and volume in Cartesian coordinates
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(cont) Cartesian Coordinates
Differential length:
Differential surface areas:
Differential volume:
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2. Cylindrical Coordinates
(r, , z)
In cylindrical coordinates a vector is expressed as
Point P(r1,1, z1) in cylindrical coordinates:
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(cont) Cylindrical Coordinates
Differential length:
Differential surface areas:
Differential volume:
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3. Spherical Coordinates (R, , )
In spherical coordinates a vector is
expressed as
Point P(R1, 1, 1) in spherical coordinates:
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(cont) Spherical Coordinates
Differential length:
Differential surface areas:
Differential volume:
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2-3 Transformations between Coordinate Systems
A. Cartesian (x, y, z) to Cylindrical (r, , z)Transformations
Both systems share the coordinate z, and the relations between
the other two pairs of coordinates are:
The inverse relations are:
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(cont) Cartesian to Cylindrical Transformations
Unit vectors relations:
The relations in terms of x and y:
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The inverse relations are:
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(cont) Cartesian to Cylindrical Transformations
Example: A vector in Cartesian coordinates can be
transformed into in cylindrical
coordinates as follows:
and conversely:
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B. Cartesian (x, y, z) to Spherical (R, , )Transformations
The inverse relations are:
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(cont) Cartesian to Spherical Transformations
The expressions of ( ) in terms of ( ) are
The inverse relations are:
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(cont) Transformations between Coordinate Systems
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Distance between Two Points in
all Three Orthogonal Coordinates
The distance d between points P1 and P2 is given by
In Cartesian coordinates
In cylindrical coordinates
In spherical coordinates