# Transform a boundary condition into polar coordinates

Suppose I have the boundary condition $u_y(x,0) = 0$. How can I transform it into polar coordinates?

$$x= r\cos(\theta), y = r\sin(\theta),$$ so $$\frac{\partial u}{\partial y} = ?$$

I'm a little confused since I don't think I can use the chain rule here.

• Real my answer to your previous question. Feb 5, 2015 at 16:34

By the chain rule you have $$\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r} \cdot \frac{dr}{dy} + \frac{\partial u}{\partial \theta} \cdot \frac{d\theta}{dy}$$
• so $\frac{\partial r}{\partial y} = \left(\frac{\partial y}{\partial r}\right)^{-1} = \frac{1}{\sin\theta}$? Feb 5, 2015 at 16:33
• Yes. In the end, you'll have an equation of the form $A \frac{\partial u}{\partial r} + B \frac{\partial u}{\partial \theta} = 0$. This equation holding on the boundary of the problem (which you can no longer simply describe as the line $y = 0$) is equivalent to the original boundary condition. Feb 5, 2015 at 22:02