Divergence Theorem

The Essentials

The Divergence Theorem:

\[ \iint_S\vec{F}\cdot d\vec{S}=\iiint_E\text{div}\vec{F}dV \]

It can be used to turn the vector surface integral of a function over a closed surface into the triple integral of the divergence of the function and vice versa.

Example

Use the divergence theorem to solve the surface integral where \( \vec{F}=\langle\frac{x^3}{3}-e^{yz-z^3},x^3\sin(z)+y^3, z^3+x^3+xy^2+2zx^2-xy+3x+e^2\rangle \) and S is the sphere with radius 3 centered on the origin:

\[ \iint_S\vec{F}\cdot d\vec{S} \]

Using the divergence theorem we can convert the surface integral to a triple integral of the divergence:

\[ \iiint_E (x^2+3y^2+3z^2+2x^2)dV \]
\[ 3\int_0^3\int_0^{2\pi}\int_0^\pi \rho^4\sin(\phi)d\rho d\theta d\phi \]
\[ 3\int_0^3\rho^4d\rho \int_0^{2\pi}d\theta \int_0^\pi\sin(\phi) d\phi \]
\[ 3\cdot\frac{243}{5}\cdot2\pi\cdot2 \]
\[ \frac{2916\pi}{5} \]

Practice

Use the divergence theorem to solve the surface integral where \( \vec{F}=\langle xz,3x^2y,z^2+3y^2z\rangle \) and S is the cylinder \( x^2+y^2=4 \) from \( 0\leq z\leq x+2 \):

\[ \iint_S\vec{F}\cdot d\vec{S} \]

Solution:

\( 78\pi \)

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