I have to write down the Neumann boundary value problem for the Poisson equation on the unit disk $x^2 + y^2 ≤ 1$ that is satisfied by $u(x,y)=x^3 + xy^2$.
We have to do some self study on the Neumann boundary conditions before the first lecture. The only problem is that I tried to answer this question without any success. So any help would be grateful.
First method. We perform all operations in the standard cartesian coordinate system $(x,y)$. Let $B_1=\{(x,y)\in\Bbb R^2:x^2+y^2\le1\}$ be the unit disk and remember that $$ \left.\frac{\partial u}{\partial \nu}\right|_{\partial B_1}=\left\langle\nabla u,\nu\ \right\rangle|_{\partial B_1}=\nabla u\cdot\nu|_{\partial B_1} $$ where $\nu$ is the outer normal to $B_1$ and by ${\langle.,.\!\rangle}= .\!\cdot .$ we mean the scalar product. Now we have $$ \nu|_{\partial B_1}=\frac{(x,y)}{\Vert (x,y)\Vert}=\frac{(x,y)}{\sqrt{x^2+y^2}}=(x,y)\quad \forall(x,y)\in{\partial B_1}, $$ and $$ \nabla u=\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right)=(3x^2+y^2,2xy). $$ Then $$ \left.\frac{\partial u}{\partial \nu}\right|_{\partial B_1}=(3x^2+y^2)x+2xy^2=3x(x^2+y^2)=3x. $$ Second method. We change the coordinates from cartesian $(x,y)$ to polar $(r,\theta)$ and perform all operations in the latter coordinate system, as suggested by Mattos. Then, the first thing to note is that we have that $$ \left.\frac{\partial u}{\partial \nu}\right|_{\partial B_1}=\left.\frac{\partial u}{\partial r} \right|_{\partial B_1}. $$ Then, by using the coordinate change, we get $$ u(x,y)\equiv u(r,\theta)=r^3\cos\theta\:\:\text{ and }\:\:u(r,\theta)|_{\partial B_1}=\cos\theta $$ and $$ \left.\frac{\partial u}{\partial \nu}\right|_{\partial B_1}=\left.\frac{\partial u}{\partial r} \right|_{\partial B_1}=3\cos \theta\equiv 3x\quad \forall(x,y)\in \partial B_1. $$ As you can see, proceeding in this way is faster than by using the standard coordinates, due to the fact that the analytic formula of the unit normal $\nu$.