Page 9 - Intro to Tensor Calculus

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EXAMPLE 1.1-2. For y i = a ij x j ,i, j =1, 2, 3and x i = b ij z j ,i, j =1, 2, 3solve for the y variables in
terms of the z variables.
Solution: In matrix form the given equations can be expressed:

           
y 1 a 11 a 12 a 13 x 1 x 1 b 11 b 12 b 13 z 1
 y 2  =  a 21 a 22 a 23   x 2  and  x 2  =  b 21 b 22 b 23   z 2  .
y 3 a 31 a 32 a 33 x 3 x 3 b 31 b 32 b 33 z 3
Now solve for the y variables in terms of the z variables and obtain

       
y 1 a 11 a 12 a 13 b 11 b 12 b 13 z 1
 y 2  =  a 21 a 22 a 23   b 21 b 22 b 23   z 2  .
y 3 a 31 a 32 a 33 b 31 b 32 b 33 z 3

The index notation employs indices that are dummy indices and so we can write

y n = a nm x m , n, m =1, 2, 3and x m = b mj z j , m, j =1, 2, 3.

Here we have purposely changed the indices so that when we substitute for x m , from one equation into the
other, a summation index does not repeat itself more than twice. Substituting we find the indicial form of
the above matrix equation as
y n = a nm b mj z j , m,n,j =1, 2, 3

where n is the free index and m, j are the dummy summation indices. It is left as an exercise to expand
both the matrix equation and the indicial equation and verify that they are different ways of representing
the same thing.

EXAMPLE 1.1-3. The dot product of two vectors A q ,q =1, 2, 3and B j ,j =1, 2, 3 can be represented
~
~
with the index notation by the product A i B i = AB cos θ i =1, 2, 3, A = |A|, B = |B|. Since the
subscript i is repeated it is understood to represent a summation index. Summing on i over the range
specified, there results
A 1 B 1 + A 2 B 2 + A 3 B 3 = AB cos θ.
Observe that the index notation employs dummy indices. At times these indices are altered in order to
conform to the above summation rules, without attention being brought to the change. As in this example,
the indices q and j are dummy indices and can be changed to other letters if one desires. Also, in the future,
if the range of the indices is not stated it is assumed that the range is over the integer values 1, 2and 3.

To systems containing subscripts and superscripts one can apply certain algebraic operations. We
present in an informal way the operations of addition, multiplication and contraction.