# Tensor Field

1. Oct 2, 2007

### joe2317

I have a following problem.

Let D be an operator taking the C^oo functions F to F, and the C^oo vector fields V to V, such that D:F-->F and
D:V-->V, are linear over R(real) and
D(f Y) = f * DY+Df * Y. Here * is a multiplication

Show that D has a unique extension to an operator taking tensor fields of type(k, l) to themselves such that
(1) D is linear over R(real).
(2) D(A $B)= DA$ B+ A $DB. Here$ is a tensor product.
(3) for any contraction C, DC=CD.

If you have Spivak's geometry book. This is a problem 5-15.

Any help would be appreciated.
Thanks.