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Appendix A
Review of
Vectors and Matrices
A.1. VECTORS
A.1.1 Definition
For
the purposes of this text, a vector is an object which has magnitude and
direction. Examples include forces,
electric fields, and the normal to a surface.Â
A vector is often represented pictorially as an arrow and symbolically
by an underlined letter   or using bold type . Its magnitude is denoted  or . There are two special cases of vectors: the
unit vector  has ;
and the null vector  has .
A.1.2 Vector Operations
 Â Addition
Let  and  be vectors.Â
Then  is also a vector. The vector  may be shown diagramatically by placing
arrows representing  and  head to tail, as shown in the figure.
 Multiplication
1. Multiplication by a scalar. Let  be a vector, and  a scalar.Â
Then  is a vector.Â
The direction of  is parallel to  and its magnitude is given by .
Note that you can form a unit
vector n which is parallel to a by setting .
2. Dot
Product (also called the scalar
product). Let a and b be two vectors. The dot product of a and b is a scalar
denoted by ,
and is defined by
,
where
 is the angle subtended by a and b. Note that ,
and . If  and  then  if and only if ;
i.e. a and b are perpendicular.
3. Cross
Product (also called the vector
product). Let a and b be two
vectors. The cross product of a and b is a vector denoted by . The direction of c is perpendicular to a
and b, and is chosen so that (a,b,c) form a right handed triad,
Fig. 3. The magnitude of c is given by
Note
that  and .
 Some useful vector identities
A.1.3 Cartesian components of vectors
Let
 be three mutually perpendicular unit vectors
which form a right handed triad, Fig. 4.Â
Then  are said to form and orthonormal basis. The
vectors satisfy
We
may express any vector a as a
suitable combination of the unit vectors ,
 and . For example, we may write
where
 are scalars, called the components of a in the
basis .  The components of a have a simple physical interpretation. For example, if we evaluate the dot product
 we find that
in view of the properties
of the three vectors ,
 and . Recall that
Then, noting that ,
we have
Thus,
 represents the projected length of the
vector a in the direction of ,
as illustrated in the figure.Â
Similarly,  and  may be shown to represent the projection of  in the directions  and ,
respectively.
The
advantage of representing vectors in a Cartesian basis is that vector
addition and multiplication can be expressed as simple operations on the
components of the vectors. For
example, let a, b and c be vectors, with components ,
 and ,
respectively. Then, it is
straightforward to show that
A.1.4 Change of basis
Let
a be a vector, and let  be a Cartesian basis. Suppose that the components of a in the basis  are known to be . Now, suppose that we wish to compute the
components of a in a second
Cartesian basis, . This means we wish to find components ,
such that
To do so, note that
This transformation is
conveniently written as a matrix operation
,
where
 is a matrix consisting of the components of a in the basis ,
 is a matrix consisting of the components of a in the basis ,
and  is a `rotation matrix’ as follows
Note
that the elements of  have a simple physical interpretation. For example, ,
where  is the angle between the  and  axes.Â
Similarly  where  is the angle between the  and  axes.Â
In practice, we usually know the angles between the axes that make up
the two bases, so it is simplest to assemble the elements of  by putting the cosines of the known angles
in the appropriate places.
Index notation provides
another convenient way to write this transformation:
You
don’t need to know index notation in detail to understand this  all you need to know is that
The same approach may be
used to find an expression for  in terms of . If you work through the details, you will
find that
Comparing this result
with the formula for  in terms of ,
we see that
where
the superscript T denotes the
transpose (rows and columns interchanged). The transformation matrix  is therefore orthogonal, and satisfies
where [I] is the identity matrix.
A.1.5 Useful vector operations
 Calculating areas
The
area of a triangle bounded by vectors a,
b¸and b-a is
The
area of the parallelogram shown in the picture is 2A.
 Calculating
angles
The angle between two vectors a and b is
 Calculating
the normal to a surface.
If two vectors a and b can be found
which are known to lie in the surface, then the unit normal to the surface is
If the surface is specified by a
parametric equation of the form ,
where s and t are
two parameters and r is the
position vector of a point on the surface, then two vectors which lie in the
plane may be computed from
 Calculating
Volumes
The volume of the parallelopiped defined by three vectors a, b, c is
The volume of the tetrahedron shown outlined in red is V/6.
A.2. VECTOR FIELDS AND VECTOR CALCULUS
A.2.1. Scalar field.
Let  be a Cartesian basis with origin O in three
dimensional space. Â Let
denote
the position vector of a point in space.Â
A scalar field is a scalar
valued function of position in space.Â
A scalar field is a function of the components of the position vector,
and so may be expressed as .
The value of  at a particular point in space must be
independent of the choice of basis vectors.Â
A scalar field may be a function of time (and possibly other
parameters) as well as position in space.
A.2.2. Vector field
Let  be a Cartesian basis with origin O in three
dimensional space. Let
denote
the position vector of a point in space.Â
A vector field is a vector
valued function of position in space.Â
A vector field is a function of the components of the position vector,
and so may be expressed as . The vector may also be expressed as
components in the basis
The
magnitude and direction of  at a particular point in space is
independent of the choice of basis vectors. Â
A vector field may be a function of time (and possibly other
parameters) as well as position in space.
A.2.3. Change of basis for scalar fields.
Let
 be a Cartesian basis with origin O in three
dimensional space. Express the position vector of a point relative to O in  as
and let  be a scalar field.
Let
 be a second Cartesian basis, with origin
P. Let  denote the position vector of P relative to O. Express the position
vector of a point relative to P in  as
To
find ,
use the following procedure. First,
express p as components in the basis ,
using the procedure outlined in Section 1.4:
where
or, using index notation
where the transformation
matrix  is defined in Sect 1.4.
Now, express c as components in ,
and note that
so that
A.2.4. Change of basis for vector fields.
Let
 be a Cartesian basis with origin O in three
dimensional space. Express the position vector of a point relative to O in  as
and let  be a vectorÂ
field, with components
Let
 be a second Cartesian basis, with origin
P. Let  denote the position vector of P relative to O. Express the position
vector of a point relative to P in  as
To
express the vector field as components in  and as a function of the components of p, use the following procedure. First, express  in terms of  using the procedure outlined for scalar
fields in the preceding section
for
k=1,2,3. Now, find the components of v
in  using the procedure outlined in Section
1.4. Using index notation, the result
is
A.2.5. Time derivatives of vectors
Let
a(t) be a vector whose
magnitude and direction vary with time, t. Suppose that  is a fixed basis, i.e. independent of
time. We may express a(t)
in terms of components  in the basis  as
.
The time derivative of a is defined using the usual rules of
calculus
,
or in component form as
The
definition of the time derivative of a vector may be used to show the
following rules
A.2.6. Using a rotating basis
Â
It
is often convenient to express position vectors as components in a basis
which rotates with time. To write
equations of motion one must evaluate time derivatives of rotating vectors.
Let  be a basis which rotates with instantaneous
angular velocity . Then,
A.2.7. Gradient of a scalar field.
Let
 be a scalar field in three dimensional
space. The gradient of  is a vector field denoted by  or ,
and is defined so that
for every position r in space and for every vector a.
Let  be a Cartesian basis with origin O in three
dimensional space. Let
denote
the position vector of a point in space.Â
Express  as a function of the components of r . The gradient of  in this basis is then given by
A.2.8. Gradient of a vector field
Let v be a vector field in three dimensional space. The gradient of v is a tensor field denoted by  or ,
and is defined so that
for every position r in space and for every vector a.
Let  be a Cartesian basis with origin O in three
dimensional space. Let
denote
the position vector of a point in space.Â
Express v as a function of
the components of r, so that . The gradient of v
in this basis is then given by
Alternatively, in index
notation
A.2.9. Divergence of a vector field
Let
v be a vector field in three
dimensional space. The divergence of v is a scalar field denoted by  or . Formally, it is defined as  (the trace of a tensor is the sum of its
diagonal terms).Â
Let  be a Cartesian basis with origin O in three
dimensional space. Let
denote
the position vector of a point in space.Â
Express v as a function of
the components of r: .
The divergence of v is then
A.2.10. Curl of a vector field.
Let
v be a vector field in three
dimensional space. The curl of v is a vector field denoted by  or . It is best defined in terms of its components
in a given basis, although its magnitude and direction are not dependent on
the choice of basis.
Let  be a Cartesian basis with origin O in three
dimensional space. Let
denote
the position vector of a point in space.Â
Express v as a function of
the components of r .
The curl of v in this basis is then given by
Using index notation,
this may be expressed as
A.2.11 The Divergence Theorem.
Let
V be a closed region in three
dimensional space, bounded by an orientable surface S. Let n  denote the unit vector normal to S, taken so that n points out of V. Let u be a vector field which is
continuous and has continuous first partial derivatives in some domain
containing T. Then
alternatively, expressed
in index notation
For
a proof of this extremely useful theorem consult e.g. Kreyzig, Advanced Engineering Mathematics, Wiley, New York, (1998).
A.3. MATRICES
A.3.1 Definition
An  matrix  is a set of numbers, arranged in m rows and n columns
 A square
matrix has equal numbers of rows and columns
 A diagonal
matrix is a square matrix with elements such that  for
 The identity
matrix  is a diagonal matrix for which all diagonal
elements
 A symmetric
matrix is a square matrix with elements such that
 A skew
symmetric matrix is a square matrix with elements such that
A.3.2 Matrix operations
 Addition Let  and  be two matrices of order  with elements  and . Then
 Multiplication
by a scalar. Let  be a matrix with elements ,
and let k be a scalar. Then
 Multiplication by a matrix. Let  be a matrix of order  with elements ,
and let  be a matrix of order  with elements . The product  is defined only if n=p, and is an  matrix such that
Note
that multiplication is distributive and associative, but not commutative,
i.e.
The multiplication of a vector by a matrix is a
particularly important operation. Let b and c be two vectors with n components, which we think of as  matrices.Â
Let  be an  matrix.Â
Thus
Now,
i.e.
 Transpose.
Let  be a matrix of order  with elements . The transpose of  is denoted . If  is an  matrix such that ,
then ,
i.e.
Note that
 Determinant The determinant is defined only for a
square matrix. Let  be a  matrix with components . The determinant of  is denoted by  or  and is given by
Now, let  be an  matrix.Â
Define the minors  of  as the determinant formed by omitting the ith row and jth column of . For example, the minors  and  for a  matrix are computed as follows.  Let
Then
Define the cofactors  of  as
Then,
the determinant of the  matrix  is computed as follows
The result is the same whichever row i is chosen for the expansion.Â
For the particular case of a  matrix
 The determinant may also be evaluated by
summing over rows, i.e.
and as
before the result is the same for each choice of column j. Finally, note that
 Inversion. Let  be an  matrix.Â
The inverse of  is denoted by  and is defined such that
The inverse of  exists if and only if . A matrix which has no inverse is said to be
singular. The inverse of a matrix may be computed
explicitly, by forming the cofactor
matrix  with components  as defined in the preceding section. Then
In practice, it is faster to compute the inverse of a
matrix using methods such as Gaussian elimination.Â
Note
that
For a diagonal matrix, the inverse is
For a  matrix, the inverse is
 Eigenvalues and eigenvectors. Let  be an  matrix, with coefficients . Consider the vector equation
                                              Â
(1)
where x is a vector with n
components, and  is a scalar (which may be complex). The n
nonzero vectors x and
corresponding scalars  which satisfy this equation are the eigenvectors and eigenvalues of .
Formally, eighenvalues and eigenvectors may be computed as
follows. Rearrange the preceding
equation to
                                    (2)
This has nontrivial solutions for x only if the determinant
of the matrix  vanishes.Â
The equation
is
an nth order polynomial which may
be solved for . In general the polynomial will have n roots, which may be complex. The eigenvectors may then be computed using
equation (2). For example, a  matrix generally has two eigenvectors, which
satisfy
Solve
the quadratic equation to see that
The two
corresponding eigenvectors may be computed from (2), which shows that
so that, multiplying out the first row of the matrix (you
can use the second row too, if you wish  since we chose  to make the determinant of the matrix
vanish, the two equations have the same solutions. In fact, if ,
you will need to do this, because the first equation will simply give 0=0
when trying to solve for one of the eigenvectors)
which
are satisfied by any vector of the form
where p and q are arbitrary real numbers.
It is often convenient to normalize eigenvectors so that they have unit `length’. For this purpose, choose p
and q so that . (For vectors of dimension n, the generalized dot product is
defined such that  )
One
may calculate explicit expressions for eigenvalues and eigenvectors for any
matrix up to order ,
but the results are so cumbersome that, except for the  results, they are virtually useless. In practice, numerical values may be
computed using several iterative techniques.Â
Packages like Mathematica, Maple or Matlab make calculations like this
easy.
The
eigenvalues of a real symmetric matrix are always real, and its eigenvectors
are orthogonal, i.e. the ith and jth eigenvectors (with  ) satisfy .
The eigenvalues of a skew
symmetric matrix are pure imaginary.
 Spectral
and singular value decomposition.Â
Let  be a real symmetric  matrix. Denote the n (real) eigenvalues of  by ,
and let  be the corresponding normalized eigenvectors, such that . Then, for any arbitrary vector b,
Let  be a diagonal matrix which contains the n
eigenvalues of  as elements of the diagonal, and let  be a matrix consisting of the n eigenvectors as columns, i.e.
Then
Note that this gives
another (generally quite useless) way to invert
where  is easy to compute since  is diagonal.
 Square
root of a matrix.  Let  be a real symmetric  matrix.Â
Denote the singular value decomposition of  by  as defined above. Suppose that  denotes the square root of ,
defined so that
One way
to compute  is through the singular value decomposition
of
where
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