Transformations are maps of a set into itself. The symmetries of a mathematical structure are given by those transformations that do not change the structure. We can even define a mathematical structure by starting with the transformations, and finding the mathematical structure compatible with these transformations. Thus we can define Euclidean geometry as the study of those mathematical structures invariant under orthogonal transformations. Similarly, the mathematics of special relativity can be defined as the study of all structures compatible with Lorentz transformations. This approach is quite abstract and far removed from the world of experiment. Still, it is an interesting approach.

We can also use symmetry transformations to solve problems. Sometimes a problem can be transformed into a simpler problem. If the structure of the problem is invariant under the transformation, then the solution of the simpler problem gives us a solution of the original problem. At one time it was hoped that all problems in mechanics could be solved by finding transformations that reduced them to trivial problems. As might have been expected, such transformations were even harder to find than solutions of the original problem.

The prime tool in the study of transformations is the infinitesimal transformation, described by a vector field. The operation of this vector field on geometric objects is given by the Lie derivative, one of the most important ideas in this book.