PROBLEM 4. Let A_i(x) denote the components of a rank (0, 1) tensor field on a smooth n-dimensional manifold M , and let x = (x^1 , · · · , x^n ) denote a coordinate chart on an open subset U of M.
(b) Now consider the same question, but with w_ij in Part (a) replaced by
F_ij = w_ij - w_ji.
Two problems related to vector and tensor fields are considered here. The first problem shows that the "skew-symmetrized" derivative (i.e., the exterior derivative) of a covariant vector field, or one form, is a covariant 2-tensor. The second problem discusses how the components of a one form transform under a change of coordinates. The solution is provided in a detailed, step-by-step process and accompanied by a verbal explanation.