Applied Electromagnetics Using QuickField and MATLAB

By James R. Claycomb

Chapter 1 - Mathematical Preliminaries: Vector Anyalysis

In This Chapter

Vector Analysis
Vector Derivatives
The Scalar Laplacian
The Vector Laplacian
Integral Theorems
Integral Transforms
The Dirac Delta Function
Special Functions

A scalar physical quantity is described by a numerical factor with units. In general, the numerical factor will depend on the system of units we choose. Examples of scalars include mass, temperature, and time. Two scalars may be added or subtracted provided they have the same units. For example, it would be meaningless to add scalars of differing units such as temperature and time. Scalars of differing units may be multiplied or divided to form a new scalar, however, such as the average density obtained as the ratio of an object’s mass to its volume.

A vector physical quantity is described by a numerical factor with units and a specified direction. Examples of vectors include electric and magnetic fields and forces that have a specified direction in space. Vectors are usually designated with boldfaced symbols such as A in printed work or with an arrow in handwritten work, such as on a chalkboard or on notebook paper.

Vector Addition

A three-dimensional vector A may be expressed in Cartesian coordinates as

where i, j, and k are unit vectors in the x-, y-, and z-directions, respectively. A_x, A_y, and A_z are the components of A along each coordinate axis. Given the vector B = B_xi + B_y j + B_zk, the vector sum or difference A ± B may be written as

Vector addition is commutative,

In MATLAB vectors are expressed as matrices

x=[1 2 3]	% specifies the row vector x
y=[2 3 4]	% specifies the row vector y
x´	% gives the column vector with x entries above
y´	% gives the column vector with y entries above
x+y	% gives the sum of vectors x and y specified above

The magnitude or length of the vector A is given by the square root of the sum of squares of each vector component

Dot Product

The dot product, also known as the scalar product, between two vectors A and B results in a scalar quantity that depends on both the magnitude and angle θ between each vector or

The dot product is zero between orthogonal vectors and is maximal between parallel vectors and is written in terms of the unit vectors

Since i · i = j · j = k · k =1 and i · j = i · k = j · k = 0 we have that

The magnitude of a vector may also be expressed as the dot product with itself square rooted

The dot product is commutative, that is,

The dot product may also be distributed across a sum

The dot product may be evaluated in MATLAB as follows

x=[1 2 3]	% specifies the row vector x
y=[2 3 4]	% specifies the row vector y
z=[3 4 5]	% specifies the row vector z
sqrt(x*x´)	% gives the magnitude of the vector x
x*y´	% gives the dot product between vectors x and y
dot(x,y)	% also gives the dot product between x and y
x*(y+z)´	% gives the dot product between x and (y+z)
xy´+xz´	% also gives the dot product between x and (y+z)
dot(x,y+z)	% also gives the dot product between x and (y+z)

Cross Product

The cross product between two vectors, A and B, also known as the vector product, results in a third vector that depends on the magnitude and direction of A and B. The magnitude of the cross product is given by

The cross product is zero between parallel vectors and is maximal between orthogonal vectors. In terms of the unit vectors, the cross product may be expressed as a determinant

expanded across the top row. Note that the vector formed by A x B is perpendicular to both A and B so that A • (A x B) = B • (Α × Β) = 0. Interchanging the last two rows in the determinate we can see that A x B = −B x A.

The cross product may be calculated in MATLAB as

x=[1 2 3]	% specifies the row vector x
y=[2 3 4]	% specifies the row vector y
cross(x,y)	% gives the cross product between x and y
dot(x,cross(x,y))	% gives the dot product between x and cross(x,y)

Triple Vector Product

The vector formed by the cross product of three vectors A, B, and C is frequently encountered in vector analysis

A mnemonic for remembering the triple cross identity is the “BAC CAB” rule, where the brackets go on the back.

x=[1 2 3]	% specifies the row vector x
y=[2 3 4]	% specifies the row vector y
z=[3 4 5]	% specifies the row vector z
cross(x,cross(y,z))	% gives the triple cross product % between x, y and z

Next Excerpt >

Purchase This Book

TABLE OF CONTENTS

Chapter 1 - Mathematical Preliminaries: Vector Anyalysis

Contact Preferences

This is embarrasing...

Customize Your GlobalSpec Experience

Select Your Free Newsletters

Industry Newsletters

Select Your Free Product Alerts

This is embarrasing...