To motivate many of our later developments, we start with the simplest case of a rotation: a rotation about a fixed axis through an angle . This example should be familiar to you from many different venues. To describe this rotation, we consider the action of this rotation on the set of fixed, orthonormal, right-handed basis vectors . As depicted in Figure 1, we suppose that these vectors are transformed to the set by the rotation.
A matrix representation of the rotation
Using a matrix notation, we can represent the transformation from the set of basis vectors to as
It is easy to see that the matrix in (1) has a unit determinant, and its inverse is its transpose: . That is, the matrix is proper-orthogonal. By differentiating (1) with respect to time, we find that
Using the fact that , we can easily replace in (2) with to obtain
Notice that this is equivalent to the familiar results and from cylindrical polar coordinates. It should also be clear from (3) that is a skew-symmetric matrix. A vector can be introduced that has the useful property that
You should notice how the vector can be inferred from the components of .
A tensor representation of the rotation
where is the tensor
It can be shown that this representation of the rotation is equivalent to (1). Furthermore, because
we can express the tensor entirely in terms of the axis of rotation and the angle of rotation :
As we shall see later, these equivalent representations naturally lead to the general form of a tensor that represents a rotation about an arbitrary axis through an arbitrary angle of rotation. As with the matrix representation of this simple rotation, the rotation tensor is a proper-orthogonal tensor because it has a determinant of one and its inverse is its transpose: and . Differentiating (5), we find that
where we have used the fact that are constant vectors. Some straightforward computations involving (6) reveal that
With the help of these results, we conclude from (10) that
- For those unfamiliar with tensor notation, a review of tensors is available in the background section on linear algebra.