- #1
Felafel
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Homework Statement
let u=f(x,y) , x=x(s,t), y=y(s,t) and u,x,y##\in C^2##
find:
##\frac{\partial^2u}{\partial s^2}, \frac{\partial^2u}{\partial t^2}, \frac{\partial^2u}{\partial t \partial s}## as a function of the partial derivatives of f.
i'm not sure I'm using the chain rules correctly, my ideas are a bit confused..
The Attempt at a Solution
1- write f as f(x,y)=f(x(s,t),y(s,t))
##\frac{\partial u}{\partial s} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}=\frac{\partial f}{\partial x}(x'(s)*x'(s,t))+\frac{\partial f}{\partial y}(y'(s)*y'(s,t))##
Thus, the second derivative:
##\frac{\partial^2 u}{\partial s^2} = \frac{\partial^2 f}{\partial x^2}(x''(s)x'(s,t)+x'(s)x''(s,t))+\frac{\partial^2 f}{\partial y^2}(y''(s)y'(s,t)+y'(s)y''(s,t))##
and then i apply the same for the other two second derivatives.
am i using the right formula?
thank you