Page 83 - Intro to Tensor Calculus

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Solution: We multiply the equation (1.3.18) by g im (inner product) and use equation (1.3.17) to simplify
j
m
j
m
the results. This produces the equation g im A i = g im g ij A = δ A = A . Changing indices produces the
j
result given in equation (1.3.19). Conversely, if we start with equation (1.3.19) and multiply by g km (inner
jk
j
k
product) we obtain g km A = g km g A j = δ A j = A m which is another form of the equation (1.3.18) with
m
the indices changed.
Notice the consequences of what the equations (1.3.18) and (1.3.19) imply when we are in an orthogonal
Cartesian coordinate system where
   
1 0 0 1 0 0
g ij =  0 1 0  and g ij =  0 1 0  .
0 0 1 0 0 1
In this special case, we have
1
2
3
A 1 = g 11 A + g 12 A + g 13 A = A 1
1
3
2
A 2 = g 21 A + g 22 A + g 23 A = A 2
3
1
2
3
A 3 = g 31 A + g 32 A + g 33 A = A .
These equations tell us that in a Cartesian coordinate system the contravariant and covariant components
are identically the same.
EXAMPLE 1.3-6. We have previously shown that if A i is a covariant tensor of rank 1 its components in
a barred system of coordinates are
∂x j
A i = A j i . (1.3.20)
∂x
Solve for the A j in terms of the A j . (i.e. find the inverse transformation).
i
∂x
Solution: Multiply equation (1.3.20) by ∂x m (inner product) and obtain
j
∂x i ∂x ∂x i
A i m = A j i m . (1.3.21)
∂x ∂x ∂x
j
∂x ∂x i ∂x j j j m
In the above product we have i = = δ m since x and x are assumed to be independent
∂x ∂x m ∂x m
coordinates. This reduces equation (1.3.21) to the form
∂x i j
A i = A j δ m = A m (1.3.22)
∂x m
which is the desired inverse transformation.
This result can be obtained in another way. Examine the transformation equation (1.3.20) and ask the
question, “When we have two coordinate systems, say a barred and an unbarred system, does it matter which
system we call the barred system?” With some thought it should be obvious that it doesn’t matter which
system you label as the barred system. Therefore, we can interchange the barred and unbarred symbols in
∂x j
equation (1.3.20) and obtain the result A i = A j i which is the same form as equation (1.3.22), but with
∂x
a different set of indices.