The representation surface
The representation surface (or representation quadric) is a geometrical representation of a second rank tensor and is useful for giving us a visual image of the tensor as well as being useful for example in calculating magnitudes of material properties described by second rank tensors.
Let us consider the equation Tijxixj = 1. Here Tij represents a second rank tensor and xi and xj are coordinates.
This can be written in full as:
T11x12 + T12x1x2 + T13x1x3 + T21x2x1 + T22x22 + T23x2x3 + T31x3x1 + T32x3x2 + T33x32 = 1
which can be plotted to obtain a 3-dimensional graph. This graph is in fact a surface that is a complete description of T.
If we want to transform the surface to a new basis, making the substitutions xi = rkixk' and xj = rljxl', we can write the equation of the representation surface as Tijrkixk'rljxl' = 1, or equivalently as Tkl'xk'xl' = 1.
This means that: Tkl' = rklrijTij
Therefore any transformations of the tensor result in identical transformations of the 3D plot.
For symmetric tensors, the quadric equation for the representation surface simplifies to:
T11x12 + T22x22 + T33x32 + 2T12x1x2 + 2T13x1x3 + 2T23x2x3 = 1, which is the equation of an ellipsoid.
For non-symmetric tensors and those with negative principal values, the representation surfaces are more complex. As with the tensor itself, the representation surface has its simplest form when referred to the principal axes (when the basis vectors are in line with the radii of the ellipsoid), where the equation becomes: T1x12 + T2x22 + T3x32 = 1 (where T1, T2 and T3 are the principal values of the tensor).