Consider a two-dimensional spatial coordinate system S' whose coordinates (u,v) are defined by
x = u + v
y = u - v
in terms of the coordinates of a Cartesian coordinate system S. Suppose you are given a vector in S whose contravariant components are Am = (2,8). Determine the contravariant components of this vector in S'.
You'll find different notations for tensors in the literature. The so-called kernel-index notation makes it particularly easy to remember the transformation rules. In this notation you denote the components of a tensor in a transformed coordinate system (S') by putting a prime on the index, instead of using a different name for the tensor itself. You also do this for the coordinates.
A detailed solution is given.