We now substitute these expressions into Equation 1 and obtain
where
Since and as and since and
are continuous at , we see that and as .
Therefore is differentiable at .
M
共a, b兲f
共⌬x, ⌬y兲 l 共0, 0兲
2
l 0
1
l 0共a, b兲
f
y
f
x
共⌬x, ⌬y兲 l 共0, 0兲共a, v兲 l 共a, b兲共u, b ⫹⌬y兲 l 共a, b兲
2
苷 f
y
共a, v兲 ⫺ f
y
共a, b兲
1
苷 f
x
共u, b ⫹⌬y兲 ⫺ f
x
共a, b兲
苷 f
x
共a, b兲 ⌬x ⫹ f
y
共a, b兲 ⌬y ⫹
1
⌬x ⫹
2
⌬y
苷 ⫹ 关 f
y
共a, v兲 ⫺ f
y
共a, b兲兴 ⌬y
苷 f
x
共a, b兲 ⌬x ⫹ 关 f
x
共u, b ⫹⌬y兲 ⫺ f
x
共a, b兲兴 ⌬x ⫹ f
y
共a, b兲 ⌬y
⌬z 苷 f
x
共u, b ⫹⌬y兲 ⌬x ⫹ f
y
共a, v兲 ⌬y
A48
||||
APPENDIX F PROOFS OF THEOREMS
COMPLEX NUMBERS
A complex number can be represented by an expression of the form , where and
are real numbers and is a symbol with the property that . The complex num-
ber can also be represented by the ordered pair and plotted as a point in a
plane (called the Argand plane) as in Figure 1. Thus the complex number is
identified with the point .
The real part of the complex number is the real number and the imaginary
part is the real number . Thus the real part of is and the imaginary part is .
Two complex numbers and are equal if and ; that is, their real
parts are equal and their imaginary parts are equal. In the Argand plane the horizontal axis
is called the real axis and the vertical axis is called the imaginary axis.
The sum and difference of two complex numbers are defined by adding or subtracting
their real parts and their imaginary parts:
For instance,
The product of complex numbers is defined so that the usual commutative and distributive
laws hold:
Since , this becomes
EXAMPLE 1
M 苷 ⫺2 ⫹ 5i ⫹ 6i ⫺ 15共⫺1兲 苷 13 ⫹ 11i
共⫺1 ⫹ 3i 兲共2 ⫺ 5i兲 苷 共⫺1兲共2 ⫺ 5i兲 ⫹ 3i共2 ⫺ 5i兲
共a ⫹ bi 兲共c ⫹ di兲 苷 共ac ⫺ bd兲 ⫹ 共ad ⫹ bc兲i
i
2
苷 ⫺1
苷 ac ⫹ adi ⫹ bci ⫹ bdi
2
共a ⫹ bi 兲共c ⫹ di兲 苷 a共c ⫹ di兲 ⫹ 共bi 兲共c ⫹ di兲
共1 ⫺ i 兲 ⫹ 共4 ⫹ 7i 兲 苷 共1 ⫹ 4兲 ⫹ 共⫺1 ⫹ 7兲i 苷 5 ⫹ 6i
共a ⫹ bi 兲 ⫺ 共c ⫹ di 兲 苷 共a ⫺ c兲 ⫹ 共b ⫺ d 兲i
共a ⫹ bi 兲 ⫹ 共c ⫹ di 兲 苷 共a ⫹ c兲 ⫹ 共b ⫹ d 兲i
b 苷 da 苷 cc ⫹ dia ⫹ bi
⫺344 ⫺ 3ib
aa ⫹ bi
共0, 1兲
i 苷 0 ⫹ 1 ⴢ i
共a, b兲a ⫹ bi
i
2
苷 ⫺1ib
aa ⫹ bi
G