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76 INTRODUCTION TO TENSORS
The functions x! = X ~ ( X ~ , X ~ , X ~ ) relate the primed Cartesian coordinates to the unprimed Cartesian coordinates. To help keep things straight, notice there is a pattern to these transformation equations. Every time we convert from the unprimed system
to the primed system, whether we are dealing with a basis vector or the components
of some vector, we sum over the second subscript of ai,. In contrast, conversions from the primed system to the unprimed system always sum over thefirst subscript.
4.3.4 Tensor Transformations To understand why the elements of a tensor must change values when expressed in different coordinate systems, consider the conductivity tensor. If in one set of coordinates, current flows more easily in the 1-direction than the 2-direction, then u 11 > a 2 2. Observation of the same physical situation in a new, primed coordinate system where the 1'-direction is equivalent to the original 2-direction and the 2'- direction is the same as the original 1-direction,would show that ui1 < ui2. Clearly the elements of the conductivity tensor must take on different values in the two
systems, even though they are describing the same physical situation. This is also
true of a vector quantity; the same velocity vector will have different components in
different coordinate systems. Tensor transformationsfollow the same pattern as vector transformations.A vector expressed in a primed and unprimed system is still the same vector:
Likewise, using the notation of Equation 4.19, the expressions for a second-rank tensor in the two systems must obey
Herein lies the beauty of this notation. The relationship between the elements, Ti,
and T,$ is built into Equation 4. 47 and is easily obtained by applying two successive
dot products to both sides. The first dot product yields
TRANSFORMATIONS BETWEEN COORDINATE SYSTEMS 77
Applying a second dot product in the same manner yields
Tlm = T:sariasm. (4.49)
To invert Equation 4.49 use the inverted matrix [a]-' twice and remember, for
orthonormal coordinate systems, a[;' = aji. This gives
Ti; = Trsalra,. (4.50)
In general, tensor transformations require one ai, factor for each subscript in the tensor. In other words, an rth rank tensor needs r different aij factors. If the transformation goes from the unprimed to the primed system, all the aij's are summed over their second subscript. For the inverse transformation, going from the primed to the unprimed system, the sums are over the first subscript. Tensor transformations, for arbitrary rank tensors, can be summarized as follows:
where the elements of the mamx [a] are given by Equation 4.35.
There is another important feature in the tensor notation of Equation 4.19. Unlike in a matrix equation where all terms must be in the same basis, the tensor/vector notation allows equations to have mixed bases. Imagine the elements of Ohm's law expressed in both a primed and unprimed coordinate system:
Ohm's law reads
and any combination of the representations in Equations 4.51 can be used in the evaluation. For example:
J1ef = (Ujkei@L) ' (El&[) = UjkEI$I(@L ' &) = U:kE@;Ukl. (4.53)
The fact that elements of from the primed frame are combined with components of E from the unprimed frame presents no problem. The dot product of the basis vectors takes care of the mixed representations, as long as the order of the basis vectors is preserved. This is accomplished in Equation 4.53 by the fact that 2 :. 21 # &. This type of an operation could not be performed using a matrix representation without explicitly converting everything to the same basis. The value of expressing a tensor in the form of 4.19 should now be clear. In addition to handling the same algebraic manipulations as a matrix array, it also contains all
