• Various equations in physics contain partial derivatives of components of tensors. A possible combination of such derivatives corresponds to an important geometrical object known as the Lie derivative of a tensor field.

If we speak about a derivative of a tensor field, we should compare (subtract) its values at infinitesimally close points. However, two tensors at different points (no matter how close they are to one another) represent elements of completely different linear spaces and therefore it is not possible to perform their subtraction (linear combination) straight from the definition (if no tricks are used). A general way to validate the required comparison should consist in some kind of transport of the tensor from one point to another. Making use of the concept of transport, comparison may be defined as follows: given two tensors sitting at two nearby points, one of them is to be transported to the point where the other resides. In this way two tensors are now available at the same point. If the two tensors happen to coincide, we may infer that their values at the original points “are equal” (in the sense of the particular rule of transport) and, consequently, that the derivative of the tensor (field) in the direction given by the two points vanishes. If the two tensors do not coincide, we get a non-zero derivative.

In this chapter we thrash out the question of how to carry out this simple idea in the case where Lie transport is used in the above-mentioned scheme. A highly important and useful way of differentiating tensor fields emerges from these considerations, namely the Lie derivative.