Multivariate Chain Rule

1. Nov 13, 2013

Yagoda

1. The problem statement, all variables and given/known data
Let $h(u,v) = f(a(u,v), b(u,v))$, where $a_u = b_v$ and $a_v = -b_u$.
Show that $h_{uu} + h_{vv} = (f_{xx} + f_{yy}) (a^2_u + a^2_v)$.

2. Relevant equations

3. The attempt at a solution I suppose my first question is where the x's and y's come from. (I thought at first it was a typo in the problem, but this type of setup appears in several other exercises in the book).
To try to make it easier to understand I tried letting the a's and x's and b's be y's so that we get $h(u,v) = f(x(u,v), y(u,v))$, but then I realized that to prove the result we need apparently both a's, b's, x's and y's.

To compute $h_{uu}$ we would begin by getting $h_u$, but I'm having trouble figuring this out since I think all the letters are tripping me up.

2. Nov 13, 2013

Staff: Mentor

x=a(u,v) and y=b(u,v)

First revisit the simple chain rule:

y = g(f(x)) which we can write as y = g(u) with u=f(x)

dy/dx = dg/du * du/dx

and so in your case they want you to compute the second derivative of h

ph/pu = pf/px * px/pu + pf/py * py/pu (think p as the partial derivative operator)