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.

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## A matrix representation of the rotation

Using a matrix notation, we can represent the transformation from the set of basis vectors to as

(1)

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

(2)

Using the fact that , we can easily replace in (2) with to obtain

(3)

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

(4)

You should notice how the vector can be inferred from the components of .

## A tensor representation of the rotation

It is convenient to use a tensor notation^{1} to describe the rotation we have been discussing. In particular, we can write (1) in the form

(5)

where is the tensor

(6)

It can be shown that this representation of the rotation is equivalent to (1). Furthermore, because

(7)

we can express the tensor entirely in terms of the axis of rotation and the angle of rotation :

(8)

or, equivalently,

(9)

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

(10)

where we have used the fact that are constant vectors. Some straightforward computations involving (6) reveal that

(11)

and

(12)

With the help of these results, we conclude from (10) that

(13)

As expected, this result is in agreement with (4). Expression (13) implies that the vector is the axial vector of the skew-symmetric tensor .

## Notes

- For those unfamiliar with tensor notation, a review of tensors is available in the background section on linear algebra.