Struggle with derivation

Question

I have $\Omega$ the following domain $$ \Omega = \left\{\left(x_1,x_2\right) \in \mathbb{R}^2, \ 1 \leq \sqrt{x_1^2+x_2^2} \leq 2\right\} \text{ and }u\left(x_1,x_2\right)=\ln\left(\sqrt{x_1^2+x_2^2}\right) $$

I'm asked to calculate $\displaystyle \frac{ \partial u }{\partial n}$ on $\Gamma$.

I guess $\Gamma$ is the boundary of $\Omega$ but I dont know what is $n$ and i've no idea how to calculate this. I know how to calculate derivative relatively to $x_1$ or $x_2$

Answer 1

Recall that for a differentiable function the directional derivative is given by

$$\frac{ \partial u }{\partial n}=\nabla u \cdot n$$

where, in that case if we refer to the boundary, $n$ is the unit normal vectors to the circle centered at the origin that is for $P=(x,y)$

$$n=\pm \left( \frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}} \right)$$

and

$$\nabla u=(u_x,u_y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)$$

Answer 2

As you are dealing with an annulus and the function is radially symmetric, switch to polar coordinates.

The normal to your boundary is then just $r/|r|$ as you want to move orthogonally to the circle.