Specific parametrization of a sphere

Question

I would like to parametrize the region \begin{equation} M=\{(x,y,z):x^2+y^2+z^2 \leq 1 \text{ with } z \leq x+y \} \end{equation} and calculate the flux $\iint_S F\cdot n \ dS$ where $S=\partial M$. The problem is'nt the flux but this parametrization. With spherical coordinates i know how to write the sphere, but the inequality \begin{equation} z\leq x+y \end{equation} stays \begin{equation} \cos(\varphi)\leq\cos(\theta)\sin(\varphi)+\sin(\theta)\sin(\varphi) \end{equation} and im stucked at this part. My geometric intuition tells me that $\theta$ is dependent of $\varphi$, but how write this? Probably in the flux part we are going to use divergence theorem, because the surface is an sphere with lid $z=x+y$, so we need a parametrization here in order to write $\iiint_M div(F) \ dV$, right?

Answer 1

Having an insight: my vector field is $F=(-2xy,y^2,3z)$, so $div(F)=3$. Divergence theorem yields \begin{equation} \iint_S F \cdot n \ dS=\iiint_M div(F)\ dV=3\iiint_M dV. \end{equation} But the plane $z=x+y$ divides the sphere in two equal parts, hence \begin{equation} 3\iiint_M dV=3\dfrac{Vol(B_1(0))}{2}=2\pi. \end{equation}

The result is ok, but the main problem of parametrization remains. I remember seeing somewhere the question of whether or not there is a parameterization for a given surface, does anyone know anything about it?

Answer 2

Your parametrization is fine. The bounds would be

$$\int_0^{2\pi}\int_{\cot^{-1}\left(\sin\theta+\cos\theta\right)}^{\pi}F\cdot n d\varphi d\theta$$

Doing it the other way would split the integral in two.