Page 42 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

32 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
3.1.2 Orthonormal Sets of Vectors
The vector n = u n /||u n || has magnitude unity, and if u 1 and u 2 are orthogonal then 1 and
2 are orthonormal and their inner product is

( n , m ) = δ nm = 0, m = n (3.6)
= 1, m = n

where δ nm is called the Kronecker delta.
If 1 , 2 ,and 3 are three vectors that are mutually orthogonal to each other then every
vector in three-dimensional space can be written as a linear combination of 1 , 2 ,and 3 ;
that is,

f(r) = c 1 1 + c 2 2 + c 3 3 (3.7)
Note that due to the fact that the vectors n form an orthonormal set,

(f, 1 ) = c 1 , (f, 2 ) = c 2 , (f, 3 ) = c 3 (3.8)
Simply put, suppose the vector f is

f = 2 1 + 4 2 + 3 . (3.9)
Taking the inner product of f with 1 we find that

(f, 1 ) = 2( 1 , 1 ) + 4( 1 , 2 ) + ( 1 , 3 ) (3.10)

and according to Eq. (3.8) c 1 = 2. Similarly, c 2 = 4and c 3 = 1.

3.2 FUNCTIONS
3.2.1 Orthonormal Sets of Functions and Fourier Series
Suppose there is a set of orthonormal functions n (x) defined on an interval a < x < b
√
( 2sin(nπx) on the interval 0 < x < 1 is an example). A set of orthonormal functions is
b
defined as one whose inner product, defined as n (x) m (x)dx,is
x=a
b

( n , m ) = n m dx = δ nm (3.11)
x=a
Suppose we can express a function as an infinite series of these orthonormal functions,
∞

f (x) = c n n on a < x < b (3.12)
n=0
Equation (3.12) is called a Fourier series of f (x) in terms of the orthonormal function set n (x).