Thread: chain rule question

1. chain rule question

see if you can answer this one

Let f = f(u,v) and u = x + y, v = x - y

i) assume f to be twice differentiable and compute f_{xx} and f_{yy} in terms of f_{u}, f_{v}, f_{uu}, f_{uv} and f_{vv}.

ii)express the wave equation:

((∂²f)/(∂x²))-((∂²f)/(∂y²))=0

in terms of partial derivatives of f with respect to u and v.

2. Originally Posted by dexza666
see if you can answer this one

Let f = f(u,v) and u = x + y, v = x - y

i) assume f to be twice differentiable and compute f_{xx} and f_{yy} in terms of f_{u}, f_{v}, f_{uu}, f_{uv} and f_{vv}.

ii)express the wave equation:

((∂²f)/(∂x²))-((∂²f)/(∂y²))=0

in terms of partial derivatives of f with respect to u and v.
Well lets get started

$\displaystyle \frac{\partial f}{\partial x}=\frac{\partial f}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}=f_u \cdot 1+f_v \cdot (1) = f_u + f_v$

Now we need the 2nd partial with respect to x

$\displaystyle f_{xx}=\frac{\partial}{\partial x}\frac{\partial f}{\partial x}=\frac{\partial }{\partial x} \left( f_u +f_v \right) = \frac{\partial f_{u}}{\partial x}+ \frac{\partial f_v}{\partial x} =$

$\displaystyle \frac{\partial f_u}{\partial x}=\frac{\partial f_u}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f_u}{\partial v} \frac{\partial v}{\partial x}=f_{uu} \cdot 1+f_{vu} \cdot (1) = f_{uu} + f_{vu}$

$\displaystyle \frac{\partial f_v}{\partial x}=\frac{\partial f_v}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f_v}{\partial v} \frac{\partial v}{\partial x}=f_{uv} \cdot 1+f_{vv} \cdot (1) = f_{uv} + f_{vv}$

so finally we have

$\displaystyle f_{xx}=(f_{uu}+f_{uv})+(f_{uv}+f_{vv})=f_{uu}+2f_{ uv}+f_{vv}$

The last step is by clairaut's theorem.

You will need to do the same thing for partials with respect to y.

Good luck.

3. im a bit confused im assuming this is the answer to part i or is it a conjugation of part i and ii. if not what is part ii

4. Originally Posted by dexza666
im a bit confused im assuming this is the answer to part i or is it a conjugation of part i and ii. if not what is part ii
This half of number i.

you need to find $\displaystyle f_{yy}$

Then you plug both $\displaystyle f_{xx},f_{yy}$
into the wave equaiton for part ii.