Two More Problems on Vector and Tensor Fields
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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.
(a) Do the partial derivatives w_ij of A_j with resoect to x^i transform as a (0, 2) tensor under an arbitrary change of coordinates?
(b) Now consider the same question, but with w_ij in Part (a) replaced by
F_ij = w_ij - w_ji.
PROBLEM 5. Show that, under a change of coordinates y^i = f^i(x), the components w_i of a 1-form w are transformed linearly by the Jacobian matrix of the inverse transformation y --> x.
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Solution Summary
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.
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