What is the transformation rule for vector-covector derivatives?

[deadringer]

Homework Statement

We have a vector X^a (n.b ^ indicates superscript) and covector Aa. We need to show that
X^b (d(Aa)/d(x^b) - d(Ab)/d(x^a))

transforms correctly under an artbitrary smooth change of coords. N.b the derivatives are partial.

By using the transformation rules for the vector and covector respectively I get four terms, two of which give us the required transformation rule. I can't get the other two to disappear. I'd appreciate any hints.

 

[Dick]

When I do it I find the other two terms vanish because they contain a term like $$\frac{ \partial^2 f}{\partial x \partial y}-\frac{\partial^2 f}{\partial y \partial x}$$.

 

[deadringer]

I managed to get this. I was differentiating with respect to the wrong coordinate system, which messed up the calculation. I then tried using the chain rule and differentiating with respect to the other coord system and it all fell out.