Question

consider \(g(s,t)=f(s^2-t^2,t^2-s^2)\) where \(f\) is a differentiable function of two variables. Use the multivariable chain rule to show that \(satisfies\)\[t\frac{\partial g}{\partial s}+s\frac{\partial g}{\partial t}=0\]

Answer 1

oops the last sentence should read: Use the multivariable chain rule to show that \(g\) satisfies

Answer 2

\[\frac{\partial g}{\partial s}=\frac{\partial f}{\partial s},~\frac{\partial g}{\partial t}=\frac{\partial f}{\partial t}\]If you are confused on how to start, this is how: calculate the partial derivatives of \(f\) with respect to \(s\) and \(t\), because they are the same as those of \(g\).

Answer 3

Notationally, this problem could get sort of confusing. I would recommend writing \(u=s^2-t^2\) and \(v=t^2-s^2\). You'll see why this is significant once you get to that point.

Answer 4

Do you need help with the chain rule itself?

Answer 5

nope writing the u and the v was all it took. I couldn't keep my work organized before. thanks a lot!