Chow T.L. Mathematical Methods for Physicists: A Concise Introduction
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derivatives exist in the open region, then we can expand the function f x; y about
x x
0
; y y
0
in the form
f x
0
h; y
0
kf x
0
; y
0
h
@
@x
k
@
@y
f x
0
; y
0
1
2!
h
@
@x
k
@
@y
2
f x
0
; y
0
1
n!
h
@
@x
k
@
@y
n
f x
0
; y
0
R
n
;
where h x x ÿ x
0
; k y y ÿ y
0
; R
n
, the remainder after n terms, is given
by
R
n
1
n 1!
h
@
@x
k
@
@y
n1
f x
0
h; y
0
k; 0 <<1;
and where we use the operator notation
h
@
@x
k
@
@y
f x
0
; y
0
hf
x
x
0
; y
0
kf
y
x
0
; y
0
;
h
@
@x
k
@
@y
2
f x
0
; y
0
h
2
@
2
@x
2
2hk
@
2
@x@y
k
2
@
2
@y
2
ý!
f x
0
; y
0
;
etc., when we expand
h
@
@x
k
@
@y
n
formally by the binomial theorem.
When lim
n!1
R
n
0 for all x; y) in a region, the in®nite series expansion is
called a Taylor series in two variables. Extensions can be made to three or more
variables.
Lagrange multiplier
For functions of one variable such as f x to have a stationary value (maximum
or minimum) at x a, we have f
0
a0. If f
n
a < 0 it is a relative maximum
while if f a > 0 it is a relative minimum.
Similarly f x; y has a relative maximum or minimum at x a; y b if
f
x
a; b0, f
y
a; b0. Thus possible points at which f x; y has a relative max-
imum or minimum are obtained by solving simultaneously the equations
@f =@x 0;@f =@y 0:
536
APPENDIX 1 PRELIMINARIES
Sometimes we wish to ®nd the relative maxima or minima of f x; y0 subject to
some constraint condition x; y0. To do this we ®rst form the function
gx; yf x; yf x; y and then set
@g=@x 0;@g=@y 0:
The constant is called a Lagrange multiplier and the method is known as the
method of undetermined multipliers.
537
LAGRANGE MULTIPLIER
Appendix 2
Determinants
The determinant is a tool used in many branches of mathematics, science, and
engineering. The reader is assumed to be familiar with this subject. However, for
those who are in need of review, we prepared this appendix, in which the deter-
minant is de®ned and its properties developed. In Chapters 1 and 3, the reader will
see the determinant's use in proving certain properties of vector and matrix
operations.
The concept of a determinant is already familiar to us from elementary algebra,
where, in solving systems of simultaneous linear equation, we ®nd it convenient to
use determinants. For example, consider the system of two simultaneous linear
equations
a
11
x
1
a
12
x
2
b
1
;
a
21
x
1
a
22
x
2
b
2
;
)
A2:1
in two unknowns x
1
; x
2
where a
ij
i; j 1; 2 are constants. These two equations
represent two lines in the x
1
x
2
plane. To solve the system (A2.1), multiplying the
®rst equation by a
22
, the second by ÿa
12
and then adding, we ®nd
x
1
b
1
a
22
ÿ b
2
a
12
a
11
a
22
ÿ a
21
a
12
: A2:2a
Next, by multiplying the ®rst equation by ÿa
21
, the second by a
11
and adding, we
®nd
x
2
b
2
a
11
ÿ b
1
a
21
a
11
a
22
ÿ a
21
a
12
: A2:2b
We may write the solutions (A2.2) of the system (A2.1) in the determinant form
x
1
D
1
D
; x
2
D
2
D
; A2:3
538
where
D
1
b
1
a
12
b
2
a
22
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; D
2
a
11
b
1
a
21
b
2
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; D
a
11
a
12
a
21
a
22
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
A2:4
are called determinants of second order or order 2. The numbers enclosed between
vertical bars are called the elements of the determinant. The elements in a
horizontal line form a row and the elements in a vertical line form a column
of the determinant. It is obv ious that in Eq. (A2.3) D 6 0.
Note that the elements of determinant D are arranged in the same order as they
occur as coecients in Eqs. (A1.1). The numerator D
1
for x
1
is constructed from
D by replacing its ®rst column with the coecients b
1
and b
2
on the right-hand
side of (A2.1). Similarly, the numerator for x
2
is formed by replacing the second
column of D by b
1
; b
2
. This procedure is often called Cramer's rule.
Comparing Eqs. (A2.3) and (A2.4) with Eq. (A2.2) , we see that the determinant
is computed by summing the prod ucts on the rightward arrows and subtracting
the products on the leftward arrows:
a
11
a
12
a
21
a
22
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
a
11
a
22
ÿ a
12
a
21
; etc:
ÿ
This idea is easily extended. For example, consider the system of three linear
equations
a
11
x
1
a
12
x
2
a
13
x
3
b
1
;
a
21
x
1
a
22
x
2
a
23
x
3
b
2
;
a
31
x
1
a
32
x
2
a
33
x
3
b
3
;
9
>
>
=
>
>
;
A2:5
in three unknowns x
1
; x
2
; x
3
. To solve for x
1
, we multiply the equations by
a
22
a
33
ÿ a
32
a
23
; ÿa
12
a
33
ÿ a
32
a
13
; a
12
a
23
ÿ a
22
a
13
;
respectively, and then add, ®nding
x
1
b
1
a
22
a
33
ÿ b
1
a
23
a
32
b
2
a
13
a
32
ÿ b
2
a
12
a
33
b
3
a
12
a
23
ÿ b
3
a
13
a
22
a
11
a
22
a
33
ÿ a
11
a
32
a
23
a
21
a
32
a
13
ÿ a
21
a
12
a
33
a
31
a
12
a
23
ÿ a
31
a
22
a
13
;
which can be written in determinant form
x
1
D
1
=
D; A2:6
539
APPENDIX 2 DETERMINANTS
where
D
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; D
1
b
1
a
12
a
13
b
2
a
22
a
23
b
3
a
32
a
33
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
: A2:7
Again, the elements of D are arranged in the same order as they appear as
coecients in Eqs. (A2.5), and D
1
is obtained by Cramer's rule. In the same
manner we can ®nd solutions for x
2
; x
3
. Moreover, the expansion of a determi-
nant of third order can be obtained by diagon al multiplication by repeating on the
right the ®rst two columns of the determinant and adding the signed products of
the elements on the various diagonals in the resulting array:
This method of writing out determinants is correct only for second- and third-
order determinants.
Problem A2.1
Solve the following system of three linear equations using Cramer's rule:
2x
1
ÿ x
2
2x
3
2;
x
1
10x
2
ÿ 3x
3
5;
ÿx
1
x
2
x
3
ÿ3:
Problem A2.2
Evaluate the following determinants
a
12
43
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; b
518
1536
1042
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; c
cos ÿsin
sin cos
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
:
Determinants, minors, and cofactors
We are now in a position to de®ne an nth-order determinant. A determinant of
order n is a square array of n
2
quantities enclosed between vertical bars,
540
APPENDIX 2 DETERMINANTS
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
a
11
a
21
a
31
a
12
a
22
a
32
ÿ ÿ ÿ
D
a
11
a
12
a
1n
a
21
a
22
a
2n
.
.
.
.
.
.
.
.
.
a
n1
a
n2
a
nn
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
: A2:8
By deleting the ith row and the kth column from the determinant D we obtain
an (n ÿ 1)st order determinant (a square array of n ÿ 1 rows and n ÿ 1 columns
between vertical bars), which is called the minor of the element a
ik
(which belongs
to the deleted row and column) and is denoted by M
ik
. The minor M
ik
multiplied
by ÿ
ik
is called the cofact or of a
ik
and is denoted by C
ik
:
C
ik
ÿ1
ik
M
ik
: A2:9
For example, in the determinant
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
;
we have
C
11
ÿ1
11
M
11
a
22
a
23
a
32
a
33
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; C
32
ÿ1
32
M
32
ÿ
a
11
a
13
a
21
a
23
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
; etc:
It is very convenient to get the proper sign (plus or minus) for the cofactor
ÿ1
ik
by thinking of a checkerboard of plus and minus signs like this
ÿÿ
ÿÿ
ÿÿetc:
ÿÿ
etc:
.
.
.
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
thus, for the element a
23
we can see that the checkerboard sign is minus.
Expansion of determinants
Now we can see how to ®nd the value of a determinant: multiply each of one row
(or one column) by its cofactor and then add the results, that is,
541
EXPANSION OF DETERMINANTS
D a
i1
C
i1
a
i2
C
i2
a
in
C
in
X
n
k1
a
ik
C
ik
i 1; 2; ...; or nA2:10a
cofactor expansion along the ith row
or
D a
1k
C
1k
a
2k
C
2k
a
nk
C
nk
X
n
i1
a
ik
C
ik
k 1; 2; ...; or n: A2:10b
cofactor expansion along the kth column
We see that D is de®ned in terms of n determinants of order n ÿ 1, each of
which, in turn, is de®ned in terms of n ÿ 1 determinants of order n ÿ 2, and so on;
we ®nally arrive at second-order determinants, in which the cofactors of the
elements are single elements of D. The method of evaluating a determinant just
described is one form of Lapl ace's development of a determinant.
Problem A2.3
For a second-order determinant
D
a
11
a
12
a
21
a
22
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
show that the Laplace's development yields the same value of D no matter which
row or column we choose.
Problem A2.4
Let
D
130
264
ÿ102
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
:
Evaluate D, ®rst by the ®rst-row expansion, then by the ®rst-column expansion.
Do you get the same value of D?
Properties of determinants
In this section we develop some of the fundamental properties of the determinant
function. In most cases, the proofs are brief.
542
APPENDIX 2 DETERMINANTS
(1) If all elements of a row (or a column) of a determinant are zero, the value of
the determinant is zero.
Proof: Let the elements of the kth row of the determinant D be zero. If we
expand D in terms of the ith row, then
D a
i1
C
i1
a
i2
C
i2
a
in
C
in
:
Since the elements a
i1
; a
i2
; ...; a
in
are zero, D 0. Similarly, if all the elements in
one column are zero, expanding in terms of that column shows that the determi-
nant is zero.
(2) If all the elements of one row (or one column) of a determinant are multi-
plied by the same factor k, the value of the new determinant is k times the value of
the original determinant. That is, if a determinant B is obtained from determinant
D by multiplying the elements of a row (or a column) of D by the same factor k,
then B kD.
Proof: Suppose B is obtained from D by multiplying its ith row by k. Hence the
ith row of B is ka
ij
, where j 1; 2; ...; n, and all other elements of B are the same
as the corresponding elements of A. Now expand B in terms of the ith row:
B ka
i1
C
i1
ka
i2
C
i2
ka
in
C
in
ka
i1
C
i1
a
i2
C
i2
a
in
C
in
kD:
The proof for columns is similar.
Note that property (1) can be considered as a special case of property (2) with
k 0.
Example A2.1
If
D
123
011
4 ÿ10
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
and B
163
031
4 ÿ30
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
;
then we see that the second column of B is three times the second column of D.
Evaluating the determinants, we ®nd that the value of D is ÿ3, and the value of B
is ÿ9 which is three times the value of D, illustrating property (2).
Property (2) can be used for simplifying a given determinant, as shown in the
following example.
543
PROPERTIES OF DETERMINANTS
Example A2.2
130
264
ÿ102
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2
130
132
ÿ102
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2 3
110
112
ÿ102
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2 3 2
110
111
ÿ101
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ12:
(3) The value of a determinant is not altered if its rows are written as columns,
in the same order.
Proof: Since the same value is obtained whether we expand a determinant by
any row or any column, thus we have property (3). The following example will
illustrate this property.
Example A2.3
D
102
ÿ110
2 ÿ13
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
1
10
ÿ13
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ 0
ÿ10
23
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2
ÿ11
2 ÿ1
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
1:
Now interchanging the rows and the columns, then evaluating the value of the
resulting determinant, we ®nd
1 ÿ12
01ÿ1
203
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
1
1 ÿ1
03
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿÿ1
0 ÿ1
23
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2
01
20
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
1;
illustrating property (3).
(4) If any two rows (or two columns) of a determinant are interchanged, the
resulting determinant is the negative of the original determinant.
Proof: The proof is by induction. It is easy to see that it holds for 2 2 deter-
minants. Assuming the result holds for n n determinants, we shall show that it
also holds for n 1n 1 determinants, thereby proving by induction that it
holds in general.
Let B be an n 1 n 1 determinant obtained from D by interchanging
two rows. Expanding B in terms of a row that is not one of those interchanged,
such as the kth row, we have
B
X
n
j1
ÿ1
jk
b
kj
M
0
kj
;
where M
0
kj
is the minor of b
kj
. Each b
kj
is identical to the corresponding a
kj
(the
elements of D). Each M
0
kj
is obtained from the corresponding M
kj
(of a
kj
)by
544
APPENDIX 2 DETERMINANTS
interchanging two rows. Thus b
kj
a
kj
, and M
0
kj
ÿM
kj
. Hence
B ÿ
X
n
j1
ÿ1
jk
b
kj
M
kj
ÿD:
The proof for columns is similar.
Example A2.4
Consider
D
102
ÿ110
2 ÿ13
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
1:
Now interchanging the ®rst two rows, we have
B
ÿ110
102
2 ÿ13
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ1
illustrating property (4).
(5) If corresponding elements of two rows (or two columns) of a determinant
are proportional, the value of the determinant is zero.
Proof: Let the elements of the ith and jth rows of D be proportional, say,
a
ik
ca
jk
; k 1; 2; ...; n.Ifc 0, then D 0. For c 6 0, then by property (2),
D cB, where the ith and jth rows of B are identical. Interchanging these two
rows, B goes over to ÿB (by property (4)). But the rows are identical, the new
determinant is still B. Thus B ÿB; B 0, and D 0.
Example A2.5
B
112
ÿ1 ÿ10
228
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
0; D
36ÿ4
1 ÿ 13
ÿ6 ÿ12 8
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
0:
In B the ®rst and second columns are identical, and in D the ®rst and the third
rows are proportional.
(6) If each element of a row of a determinant is a binomial, then the determi-
nant can be written as the sum of two determinants, for example,
4x 232
x 43
3x ÿ 121
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
4x 32
x 43
3x 21
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
232
043
ÿ121
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
:
545
PROPERTIES OF DETERMINANTS