Let (mathcal{S}) be the ellipsoid (left(frac{x}{4}ight)^{2}+left(frac{y}{3}ight)^{2}+left(frac{z}{2}ight)^{2}=1). Calculate the flux of (mathbf{F}=z mathbf{i}) over the portion of (mathcal{S})

Question:

Let $\mathcal{S}$ be the ellipsoid $\left(\frac{x}{4}ight)^{2}+\left(\frac{y}{3}ight)^{2}+\left(\frac{z}{2}ight)^{2}=1$. Calculate the flux of $\mathbf{F}=z \mathbf{i}$ over the portion of $\mathcal{S}$ where $x, y, z \leq 0$ with upward-pointing normal. Hint: Parametrize $\mathcal{S}$ using a modified form of spherical coordinates $(\theta, \phi)$.

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