1. The problem statement, all variables and given/known data I'm learning a bit about tensors on my own. I've been given a definition of a tensor as an object which transforms upon a change of coordinates in one of two ways (contravariantly or covariantly) with the usual partial derivatives of the new and old coordinates. (I apologize if I'm screwing up the terminology). I was trying to apply this definition to a scalar field, specifically to the field in Euclidean 2-Space where f = x²+2y² and trying to transform that to the coordinates (g,h) given by g = x+y; h=y-x (a simple rotation). It was easy to solve for f in the new coordinates: f' = (1/4)(3g² + 2gh + 3h²) but I've tried to transform f into f' using the definition and just run around in circles. Is the transformation covariant? contravariant? both? neither? Is this the wrong definition for a rank 0 tensor? If so, what is the correct definition? What is the simplest example of an object which transforms contravariantly and one which transforms covariantly? Is it a rank 1 tensor? (I'm following along with L. Susskind's Stanford video lectures on GR, and he shows how a 2-vector and the grad of it transform, are there any simpler examples or are they it?) 2. Relevant equations f(x,y) = x²+2y²; f'(g,h) = (0.25)(3g²+2gh+3h²) where g = x+y and h=y-x 3. The attempt at a solution I've tried dg = ∂x + ∂y and dh = ∂y - ∂x but then I get (using the definitions of co- and contra- variant tensors): (∂x/∂g)*f + (∂y/∂g)*f and (∂x/∂h)*f + (∂y/∂h)*f and just can't figure how these can be combined to get f'. I must be missing something basic.