PROBLEM 1. Let w = w_i dx^i be a 1- form (or covariant vector field) expressed in terms of the coordinate system x = (x^1, ... , x^n). Determine the (covariant) transformation law for the components w_i of w expressed in a new coordinate system y = (y^1, ... , y^n ). Describe the relationship between the contravariant and covariant transformation laws.
PROBLEM 2. Let T be a (0, 2)-tensor such that, in a particular coordinate system x = (x^1, ... , x^n ), we have
T_ij = 1, when i = j and T_ij = 0 otherwise.
Show that the form of T is not preserved by a general coordinate transformation.
PROBLEM 3. Suppose that A is a (2, 0) tensor with components A = A^ij, and suppose that A^ij = -A^ji for all indices i and j . Show that this anti-symmetry property is preserved under an arbitrary coordinate transformation.
These problems illustrate how the components of contravariant and covariant vector fields and rank 2 tensors transform under a change of coordinate system. The style of presentation is similar to that in Lovelock and Rund, "Tensors, Differential Forms, and Variation Principles," a classical treatise on the subject.