1. ## Change of Coordinates

If f(x,y) becomes g(r,t) in a transformation from Cartesian to Polar cooardinates, show that

i) $\displaystyle \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2= \left(\frac{\partial g}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial g}{\partial t}\right)^2$

ii) $\displaystyle \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 g}{\partial r^2}+\frac{1}{r}\frac{\partial^2 g}{\partial r} + \frac{1}{r^2}\frac{\partial^2 g}{\partial t^2}$

I know the partials of x and y with respect to r and t and vice-versa; I just can't get my head around how to approach the question.

I know how to find $\displaystyle \frac{\partial f}{\partial r}$ and $\displaystyle \frac{\partial f}{\partial t}$ with the chain rule, and same for g, but not the other way round. Help!

2. Originally Posted by harbottle
If f(x,y) becomes g(r,t) in a transformation from Cartesian to Polar cooardinates, show that

i) $\displaystyle \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2= \left(\frac{\partial g}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial g}{\partial t}\right)^2$

ii) $\displaystyle \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 g}{\partial r^2}+\frac{1}{r}\frac{\partial^2 g}{\partial r} + \frac{1}{r^2}\frac{\partial^2 g}{\partial t^2}$

I know the partials of x and y with respect to r and t and vice-versa; I just can't get my head around how to approach the question.

I know how to find $\displaystyle \frac{\partial f}{\partial r}$ and $\displaystyle \frac{\partial f}{\partial t}$ with the chain rule, and same for g, but not the other way round. Help!

So x = r cos t, y = r sin t and then g(r,w) = f(x,y){under change) = f(r cos t, r sin t), so:

dg/dr = (df/dx)(dx/dr) + (df/dy)(dy/dr)= (df/dx)cos t + (df/dy)sin t

dg/dt = (df/dx)(-r sin t) + (df/dy)(rcos t)

Now just square above, add both lines and make some order in that mess and you'll get what you want, both (i) and (ii)

Tonio

3. So we just use the fact that g=f.. simple.

thank you!