I am trying to figure out the exact meaning of the concepts of 4-vector and relativistic tensor in the Minkowski spacetime. In my understanding, a tensor is a map that assigns an array of numbers to each basis in such a way that certain transformation rules apply. A vector can be viewed as a...
It's been three years but I will answer this question anyway for future reference, as I feel it has not been answered satisfactorily.
What you need is a Riemannian volume form. Area does not make sense on a general manifold because the length of a tangent vector is undefined. However, if you...