Dissemination of IT for the Promotion of Materials Science (DoITPoMS)


Transformation of axes

As with a vector, every tensor is described with respect to a basis, and if we choose a different basis or different orientation from which to look at the problem, the physical meaning is the same but the components of the tensor will change.

Some orientations are easier to work in than others, due to the geometry of the problem or properties of the physical situation. We must learn how to move our problem from one frame into another. The transformation matrices which we require are pure rotations and are therefore given the symbol R.

Transforming the basis

Let us consider the 2-dimensional simplification first:

. Diagram of transforming the basis with basis vectors

We are rotating from the basis with basis vectors x and y into a new basis with basis vectors x' and y'. The new basis can be written in terms of the old basis by resolving the vectors: x' = xcosθ + ysinθ and y' = −xsinθ + ycosθ, which can be written as a matrix as:

cosθ sinθ
-sinθ cosθ

The components of this rotation matrix, R, are the cosines of the angles involved (known as direction cosines).

The component rij is the cosine of the angle between xj (old basis) and xi' (new basis) i.e. the component of xj resolved along xi'. For example, in general we can say:

x1' = r11x1 + r12x2 + r13x3

Since for unit vectors the scalar product is just the cosine of the angle between the two vectors, we can write: rij = xi'.xj

Written in full we get:

R = 
x1'.x1 x1'.x2 x1'.x3
x2'.x1 x2'.x2 x2'.x3
x3'.x1 x3'.x2 x3'.x3

This is the transformation matrix to go from old to new basis. To go from new to old basis, it is easily seen that the matrix is the transpose of the one above. So rotating and then rotating back gives: RRT = I i.e. the original result.

Therefore the inverse of the rotation matrix is its transpose.

Transforming a vector

Consider the vector: a = a1x1 +  a2x2 +  a3x3 =  a1'x1' +  a2'x2' +  a3'x3'

We know that the component of a resolved onto the new 1 axis is: a1' = a.x1' = a1r11 + a2r12 + a3r13
The other components can be similarly resolved, giving the above result.

Therefore, to rotate a vector we use the following equation: a' Ra

Transforming a second rank tensor

To derive the transformation law for a second rank tensors, let us consider the general tensor equation: p Tq (in the old basis), and p' T'q' (in the new basis). To transform the vectors we use: p' Rp and q' Rq

The above knowledge allows us to make some simple substitutions to see that: p' Rp RTq RTR-1q' RTRTq', and also that p' T'q'

Hence: T' = RTRT

In suffix notation, this can be written as the final transformation law: Tij' = rimrjnTmn, (or conversely) Tij = rmirnjTmn'

Note again that these both represent 9 equations, one for each component of the tensor.

Transforming an nth rank tensor

For an nth rank tensor the transformation law is as follows Tijk...'   = rimrjnrko...Tmno... where there are n transformation matrices. The transformation laws are useful as we can then give a mathematical definition of a tensor as 'an object whose coefficients transform according to the rules above.'

This results in an object that retains its physical meaning whatever basis is used to describe it. This is an important concept, because transforming to a well-known basis usually simplifies the mathematics of a problem, as we will see in the next section.