Find the flux across a part of the surface $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$

Question

Consider the vector field $$F(x, y, z)= \frac{(x{\rm i} + y{\rm j} + z{\rm k})} {(x ^2+ y ^2 + z ^2)^\frac{3}{2}},$$ and let $S$ be the part of the surface $$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$$ in the first octant bounded by the planes $y = 0, y =x\sqrt{3}$ and $z = 0$, oriented upwards. Find the flux of $F$ across the surface $S$.

Answer 1

The vector field $\displaystyle\vec{F}=\frac{\vec r}{r^3}$ has i) radial direction and ii) zero divergence, which implies (why?) that its flux across $S$ coincides with the flux across any surface $S'$ bounded by the same planes.

Now if we take $S'$ to be the corresponding part of the sphere $x^2+y^2+z^2=R^2$ of any radius, then $\vec{F}\cdot \vec{n}=\frac1{R^2}$ (i.e. constant on $S'$). This in turn implies that
$$\text{flux}=\frac{\text{area}(S')}{R^2}.$$ I leave to you the computation of the appropriate area (hint: use spherical coordinates) and the check that the result is independent of $R$.