The Divergence Theorem

Divergence in three dimensions

Divergence form of Green's theorem $$\oint_C\vec F\cdot \vec n ds=\iint_R\left({\partial M\over \partial x}+{\partial N\over\partial y}\right)dxdy$$We compute the net outward flux of a vector field across a simple closed curve by integrating the divergence of the field over the region enclosed by the curve.

The divergence theorem (in the 3D space) extends Green's theorem (flux density) to three dimensions.

The divergence of a vector field $\vec F=M\vec i+N\vec j+P\vec k$ is the scalar function $$\mbox{div}\vec F = \nabla \cdot \vec F = {\partial M\over \partial x}+{\partial N\over \partial y}+{\partial P\over\partial z}.$$

Physical interpretation:

• If $\vec F$ is the velocity field of a flowing gas, the value of div$\vec F$ at $(x,y,z)$ is the rate at which the gas is compressing or expanding at the point.

• The divergence is the flux per unit volume or flux density at the point. (analogous to circulation density)

Example: Find their divergence and interpret their physical meaning

• Expansion: $\vec F(x, y, z) = x\vec i + y\vec j + z\vec k$ $$\nabla\cdot\vec F=3$$ The gas is undergoing constant uniform expansion at all points.

• Compression: $\vec F(x, y, z) = -x\vec i - y\vec j - z\vec k$ $$\nabla\cdot\vec F=-3$$ The gas is undergoing constant uniform compression at all points.

• Rotation about the $z$-axis: $\vec F(x, y, z) = -y\vec i + x\vec j$ $$\nabla\cdot\vec F=0$$ The gas is neither expanding nor compressing at any point.

• Shearing along parallel horizontal planes: $\vec F(x, y, z) = z\vec j$ $$\nabla\cdot\vec F=0$$ The divergence is zero at all points in the domain of the velocity field, so the gas is neither expanding nor compressing at any point.

Divergence theorem

Theorem (Theorem 8--Divergence theorem) If $\vec F$ is a vector field whose components have continuous first partial derivatives. Let $S$ be a piecewise smooth oriented closed surface. The flux of $\vec F$ across $S$ in the direction of the surface's outward unit normal field $\vec n$ equals the triple integral of the divergence $\nabla\cdot\vec F$ over the region $D$ enclosed by the surface: $$\underbrace{\iint_S\vec F\cdot\vec n d\sigma}_{\mbox{outward flux}}=\underbrace{\iiint_D\nabla\cdot \vec F dV.}_{\mbox{divergence integral}}$$

Corollary The outward flux across a piecewise smooth oriented closed surface $S$ is zero for any vector field $\vec F$ having zero divergence at every point of the region enclosed by the surface.

Examples of zero divergence: the velocity field of a circulating incompressible liquid, constant vector fields, shearing

Example: Verify the divergence theorem for the expanding vector field $\vec F=x\vec i+y\vec j+z\vec k$ over the sphere $x^2+y^2+z^2=a^2$.

Solution: $$\begin{align*} \iiint_D \nabla \cdot \vec F dV=&3\iiint_D dV=4\pi a^3\\ \iint_S\vec F\cdot \vec n d\sigma=& \iint_S{a^2\over a} d\sigma =\iint_Sa d\sigma = 4\pi a^3 \end{align*}$$

Example: Find the flux of $\vec F = xy \vec i + yz \vec j + xz \vec k$ outward through the surface of the cube cut from the first octant by the planes $x = 1$, $y = 1$, and $z = 1$.

Solution: Directly computing the flux requires computing over six sides, so we compute the triple-integral using the divergence theorem instead. $$\begin{align*} \iint \vec F\cdot\vec n d\sigma = \int_0^1\int_0^1\int_0^1 \nabla \cdot \vec F dxdydz=\int_0^1\int_0^1\int_0^1 (x+y+z) dxdydz={3\over 2}. \end{align*}$$

Example: Find the flux of $\vec F = x^2 \vec i + 4xyz \vec j + ze^x \vec k$ out of the box-shaped region $D:0\leq x\leq 3,~0\leq y\leq 2,~0\leq z\leq 1$. Verify the divergence theorem.

Solution: $$\begin{align*} \iiint_D \nabla \cdot \vec F dV=&\int_0^1 \int_0^2 \int_0^3 (2x+4xz+e^x)dxdydz=18+18+2(e^3-1)\\ \iint_S\vec F\cdot \vec n d\sigma=&\int_0^1\int_0^23^2dydz+\int_0^2\int_0^3 e^xdxdy+\int_0^1\int_0^3 8xzdxdz\\ =& 18+2(e^3-1)+18 \end{align*}$$

Divergence and the curl

Theorem If $\vec F = M \vec i + N \vec j + P \vec k$ is a vector field with continuous second partial derivatives, then $$\mbox{div}(\mbox{curl} \vec F)=\nabla\cdot(\nabla \times \vec F)=0.$$

If $\vec{G}=\nabla \times\vec F$, then $\iint_S\vec{G}\cdot \vec n d\sigma =0$ for any closed surface $S$.

Proof of divergence theorem for special regions (no holes/bubbles)

Let $\vec n=(\cos\alpha)\vec i+(\cos\beta)\vec j+(\cos\gamma)\vec k$, then $$\vec F\cdot\vec n=M\cos\alpha+N\cos\beta+P\cos\gamma$$ $$\iint_S{({\color{red}M\cos\alpha}+N\cos\beta+{\color{blue}P\cos\gamma})}d\sigma=\iiint_D\left({{\color{red}{\partial M\over\partial x}}+{\partial N\over\partial y}+{\color{blue}{\partial P\over\partial z}}}\right)dxdydz$$

We prove that $\iint_SP\cos\gamma d\sigma=\iint_{R_{xy}}P(x,y,f_2(x,y))-P(x,y,f_1(x,y))dxdy$

Divergence theorem for other regions

$D$ is the region between two concentric spheres. The surface of the lower half $D_1$ consists of an outer hemisphere, an inner hemisphere, and a plane washer-shaped base.

Example: Find the net outward flux of the field

across the boundary of the region $D: 0< b^2 \leq x^2 + y^2 + z^2 \leq a^2$.

Solution: $$\begin{align*} \mbox{div}\vec F=\nabla\cdot\vec F={0} \end{align*}$$ $$\begin{align*} \vec F\cdot\vec n={1\over \rho^2} \end{align*}$$

What can we learn from this example?

• $\nabla \cdot \vec F=0$.

• $\iint_{S_a}\vec F\cdot\vec n d\sigma=4\pi$.

Gauss's law: one of the four great laws of electromagnetic theory

Let $$\vec E(x,y,z)={q\over 4\pi\epsilon_0}{\vec r\over|\vec r |^3}$$ Then, for any closed surface that encloses the origin, we have $${\color{red}\iint_S\vec E\cdot\vec n d\sigma={q\over\epsilon_0}.}$$

Continuity equation of hydrodynamics

Let $D$ be a region bounded by a closed-oriented surface $S$. If $\vec v(x, y, z)$ is the velocity field of a fluid flowing smoothly through $D$, $\delta = \delta(t, x, y, z)$ is the fluid's density at $(x, y, z)$ at time $t$, and $\vec F = \delta\vec v$, then the continuity equation of hydrodynamics states that

Let $B$ be a solid sphere centered at $Q$: $${\iiint_B\nabla\cdot \vec F dV\over \iiint_B dV}={\iint_S\vec F \cdot \vec n d\sigma\over \iiint_B dV}={{dm\over dt}\over \iiint_B dV}$$

Conservation of mass $$\iiint_B\nabla\cdot \vec F dV=\iint_S\vec F\cdot\vec n d\sigma$$

Unifying

• Fundamental theorem of calculus, normal form of Green's theorem, divergence theorem

• Stokes' theorem, tangential form of Green's theorem