Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
will be introduced later. Consider the vector A expressed in terms of the unit
coordinate vectors …^e 1 ; ^e 2 ; ^e 3 †:
A ˆ A 1^e 1 ‡ A 2^e 2 ‡ A^e 3 ˆ X3
iˆ1
A i^e i :
Relative to a new system …^e 0 1; ^e 0 2; ^e 0 3† that has a di€erent orientation from that of
the old system …^e 1 ; ^e 2 ; ^e 3 †, vector A is expressed as
A ˆ A 0 1^e 0 1 ‡ A 0 2^e 0 2 ‡ A 0^e 0 3 ˆ X3
iˆ1
A 0
i ^e 0
i :
Note that the dot product A ^e 1 0 is equal to A1, 0 the projection of A on the direction
of ^e 1; 0 A ^e 2 0 is equal to A2, 0 and A ^e 3 0 is equal to A3. 0 Thus we may write
A1 0 ˆ…^e 1 ^e 1†A 0 1 ‡…^e 2 ^e 1†A 0 2 ‡…^e 3 ^e 1†A 0 9
3 ; >=
A2 0 ˆ…^e 1 ^e 2†A 0 1 ‡…^e 2 ^e 2†A 0 2 ‡…^e 3 ^e 2†A 0 3 ;
…1:23†
>;
A3 0 ˆ…^e 1 ^e 3†A 0 1 ‡…^e 2 ^e 3†A 0 2 ‡…^e 3 ^e 3†A 0 3 :
The dot products …^e i ^e j 0 † are the direction cosines of the axes of the new coordinate
system relative to the old system: ^e i 0 ^e j ˆ cos…xi 0 ; x j †; they are often called the
coecients of transformation. In matrix notation, we can write the above system
of equations as
0
A1
0 1 0
^e 1 ^e 1 0 ^e 2 ^e 1 0 ^e 3 ^e 0 10
1
1 A 1
B
@ A2
0 C B
A ˆ ^e 1 ^e 2 0 ^e 2 ^e 2 0 ^e 3 ^e 2
0 CB
C
@
A@
A 2 A:
A3
0 ^e 1 ^e 3 0 ^e 2 ^e 3 0 ^e 3 ^e 3
0
The 3 3 matrix in the above equation is called the rotation (or transformation)
matrix, and is an orthogonal matrix. One advantage of using a matrix is that
successive transformations can be handled easily by means of matrix multiplication.
Let us digress for a quick review of some basic matrix algebra. A full account
of matrix method is given in Chapter 3.
A matrix is an ordered array of scalars that obeys prescribed rules of addition
and multiplication. A particular matrix element is speci®ed by its row number
followed by its column number. Thus a ij is the matrix element in the ith row and
jth column. Alternative ways of representing matrix ~A are [a ij ] or the entire array
0
1
a 11 a 12 ::: a 1n
a 21 a 22 ::: a 2n
~A ˆ
B ::: ::: ::: ::: C
@
A :
a m1 a m2 ::: a mn
A 3
12