Partial derivatives of composition

[cantgetright]

Homework Statement

Find the partial derivatives with respect to \(\displaystyle u,v\) of \(\displaystyle \bar{U}(\bar{x}(u,v))\), where \(\displaystyle \bar{U}\) is the unit normal to a surface given by the parametrization \(\displaystyle \bar{x}(u,v)\). (This, of course, is part of a larger problem, but I just am looking for advice with the calculus.)

Homework Equations

Chain rule?

The Attempt at a Solution

I am showing some properties of the shape operator, so everything is in "general terms" (that's why there's no explicit expression for the unit normal or the parametrization). I'm used to explicit expressions--e.g., where I could evaluate what the composition looked like explicitly and take partial derivatives. Here, I am at a loss. I would like to take derivative of U evaluated at x(u,v) with respect to u and then multiply by derivative of x(u,v) with respect to u (and then repeat the process for the partial derivative with respect to v), but in "general terms", how do I express the partial derivative of U?

I am sorry if this is so elementary, but I have not seen calculus in almost 20 years. Be gentle.

 

[mighty2000]

Thank you for your question. It seems like you are on the right track with using the chain rule to find the partial derivatives of \bar{U}(\bar{x}(u,v)). In general terms, the partial derivative of U evaluated at x(u,v) with respect to u (or v) can be expressed as \partial U/\partial x * \partial x/\partial u (or v), where \partial U/\partial x is the gradient of U and \partial x/\partial u (or v) is the partial derivative of x with respect to u (or v).

Another way to think about it is to first find the partial derivative of U with respect to each variable (u and v) separately, and then plug in the values of x(u,v) to get the final result. This might be easier to visualize and work with in "general terms".

I hope this helps. Good luck with your problem!