Use the Divergence Theorem to evaluate ∬s 𝐅 ·d 𝐒, where 𝐅(x,...

Use the Divergence Theorem to evaluate $\iint_{s} \mathbf{F} \cdot d \mathbf{S}$, where $$ \mathbf{F}(x, y, z)=z^{2} x \mathbf{i}+\left(\frac{1}{3} y^{3}+\tan ^{-1} z\right) \mathbf{j}+\left(x^{2} z+y^{2}\right) \mathbf{k} $$ and $S$ is the top half of the sphere $x^{2}+y^{2}+z^{2}=1$. [Hint: Note that $S$ is not a closed surface. First compute integrals over $S_{1}$ and $S_{2}$, where $S_{1}$ is the disk $x^{2}+y^{2} \leq 1$, oriented downward, and $S_{2}=S \cup S_{1 .}$ ]

Use the Divergence Theorem to evaluate $\iint_{s} \mathbf{F} \cdot d \mathbf{S}$, where
$$
\mathbf{F}(x, y, z)=z^{2} x \mathbf{i}+\left(\frac{1}{3} y^{3}+\tan ^{-1} z\right) \mathbf{j}+\left(x^{2} z+y^{2}\right) \mathbf{k}
$$
and $S$ is the top half of the sphere $x^{2}+y^{2}+z^{2}=1$. [Hint: Note that $S$ is not a closed surface. First compute integrals over $S_{1}$ and $S_{2}$, where $S_{1}$ is the disk $x^{2}+y^{2} \leq 1$, oriented downward, and $S_{2}=S \cup S_{1 .}$ ]

Key Concepts

Divergence Theorem

The Divergence Theorem is a fundamental result in vector calculus that relates the flux of a vector field through a closed surface to the integral of the divergence of that field over the volume enclosed by the surface. It allows for the conversion of a surface integral into a typically simpler volume integral, which can greatly simplify the problem of evaluating the net flow of a field.

Surface Integral (Flux)

A surface integral, particularly in the context of flux, measures how much of a vector field passes through a given surface. Flux integrals account for both the magnitude and the direction of the field relative to the surface's normal vector, making them useful in physical applications such as fluid flow and electromagnetism.

Computing the Divergence

Computing the divergence of a vector field involves taking the sum of the partial derivatives of its components with respect to their corresponding variables. This scalar quantity represents the net rate of 'outflow' from an infinitesimal volume and is central to applying the Divergence Theorem.

Closed Surface Extension

When presented with an open surface, it is often beneficial to extend the surface to a closed one so that the Divergence Theorem can be applied. This involves adding additional surfaces that, together with the given surface, form a closed boundary, allowing the volume integral of the divergence to be computed and then adjusted for the extra surfaces.

Surface Orientation

Surface orientation refers to the choice of the normal vector's direction when calculating surface integrals. Correctly orienting surfaces is crucial since the sign of the flux integral depends on it. When adding auxiliary surfaces to a non-closed surface, careful attention must be paid to the orientation of these surfaces to ensure consistency with the theorem's requirements.

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