Tensors are another abstract mathematical tool at our disposal for solving problems from rotating rigid bodies to the structure of the Universe. Not only does the use of tensor notation clean up complex equations, but also allows us to embody the invariance of physical quantities within tensors. This is worth repeating: we can use tensors to write physical equations independent of the choice of coordinate systems.
In the most general way we define a tensor based on how it behaves under coordinate transformations
where the tensor rank is , the contravariant rank is and the covariant rank is . Note that rank is also referred to as the tensor order.
Although Eq. (1) can be intimidating at first, one can familiarize themselves by working simple examples and pulling from experience with scalars, vectors and matrices. A scalar quantity such as temperature or density is invariant under coordinate transformation and is labeled as a tensor of rank zero. The next step is a tensor of rank 1. If and , then and we have a contravariant vector
If and , then and we have a covariant vector
It is important to realize the differences between Eq. (2) and Eq. (3), i.e. Eq. (2) can represent the transformation of familiar vectors, while Eq. (3) can represent the transformation between basis vectors. This can best be illustrated with the simple examples of the transformation between cartesian coordinates and polar coordinates and the transformation between cartesian basis vectors and polar basis vectors.