Partial derivatives, equation help

[tweety1234]

Homework Statement

Heat is being conducted radially through a cylindrical pipe. The temperature at a radius r is T(r). In Cartesian co-ordinates, $$r = \sqrt{(x^{2}+ y^{2}})$$

show that $$\frac{\partial T}{\partial x} = \frac{x}{r} \frac{dT}{dr}$$

 

[madah12]

cant you just say $$\frac{\partial T}{\partial x} = \frac{\partial T}{\partial r} * \frac{\partial r}{\partial x}$$
but since T is function of r you can write $$\frac{\partial T}{\partial r}$$ as dT/dr I am not sure this is correct though.

 

[tweety1234]

cant you just say $$\frac{\partial T}{\partial x} = \frac{\partial T}{\partial r} * \frac{\partial r}{\partial x}$$
but since T is function of r you can write $$\frac{\partial T}{\partial r}$$ as dT/dr I am not sure this is correct though.

oh I thought we might have to make use of the equation relating x and r given in the question ?

 

[madah12]

yes to get the partial of r with respect to x you need the equation right?

 

[tweety1234]

But the expression wants x/r?

 

[madah12]

I think that the $$\frac{\partial T}{\partial r}$$ will give you the x/r part