Divergence and Gauss's Theorem

The Divergence Operator

For a vector field $\mathbf{F} = (F_x, F_y, F_z)$:

$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$

The result is a scalar field.

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Physical Interpretation

$\nabla \cdot \mathbf{F}$ measures expansion: the rate at which flow “spreads out” from each point.

• $\nabla \cdot \mathbf{F} > 0$: Source (flow diverges)
• $\nabla \cdot \mathbf{F} < 0$: Sink (flow converges)
• $\nabla \cdot \mathbf{F} = 0$: Incompressible (volume preserved)

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The Laplacian

Applying divergence to gradient of a scalar field:

$\nabla \cdot (\nabla \phi) = \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$

The Laplacian $\nabla^2$ is fundamental in PDEs:

PDE	Equation
Laplace’s equation	$\nabla^2\phi = 0$
Heat equation	$\nabla^2\phi = \frac{1}{D}\frac{\partial\phi}{\partial t}$
Wave equation	$\nabla^2\phi = \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}$

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Example

For $\mathbf{F} = (x^2, y^2, z^2)$:

$\nabla \cdot \mathbf{F} = 2x + 2y + 2z = 2(x + y + z)$

At origin: $\nabla \cdot \mathbf{F} = 0$.

On axis $(x, 0, 0)$: $\nabla \cdot \mathbf{F} = 2x$, increases with distance.

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Gauss’s (Divergence) Theorem

For a closed surface $S$ enclosing volume $V$: $\int_S \mathbf{F} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{F})\,dV$

The flux through a closed surface equals the volume integral of divergence inside.

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Proof Sketch (Cuboid)

Consider a cuboid $V$: $0 \leq x \leq a$, $0 \leq y \leq b$, $0 \leq z \leq c$.

Faces perpendicular to $x$:

• $x = 0$: flux $= -\int F_x(0, y, z)\,dy\,dz$ (normal points inward)
• $x = a$: flux $= \int F_x(a, y, z)\,dy\,dz$

Combined: $\int [F_x(a, y, z) - F_x(0, y, z)]\,dy\,dz$

By fundamental theorem of calculus: $= \int_0^a \frac{\partial F_x}{\partial x} dx\,dy\,dz$

Sum over all six faces:

$\Phi = \int_V \left(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right)dV$

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Proof Extension

To generalise from cuboid to any shape:

1. Subdivide volume into small cuboids
2. Apply theorem to each cuboid
3. Sum: internal surfaces cancel (opposite normals)
4. Result: theorem holds for whole volume

Requirement: Surface is smooth enough for subdivision.

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Applications

Flux Calculation

Often easier to compute volume integral than surface integral.

Example: Flux of $\mathbf{F} = (x^2, y^2, z^2)$ through unit sphere.

$\nabla \cdot \mathbf{F} = 2(x + y + z)$

$\Phi = \int_V 2(x + y + z)\,dV$

By symmetry: $\int x\,dV = \int y\,dV = \int z\,dV = 0$.

So $\Phi = 0$.

Vector Area of Closed Surface

$\oint_S d\mathbf{S} = \mathbf{0}$

(Apply divergence theorem to constant vector field.)

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Solenoidal Fields

A vector field with $\nabla \cdot \mathbf{F} = 0$ everywhere is solenoidal.

From divergence theorem: flux through any closed surface is zero.

Examples:

• Magnetic field $\mathbf{B}$: $\nabla \cdot \mathbf{B} = 0$ (no magnetic monopoles)
• Incompressible fluid velocity

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Why These Theorems Matter

Gauss’s theorem converts between:

• Surface integrals (flux)
• Volume integrals (divergence)

This is powerful because:

• Choose whichever is easier to evaluate
• Derive conservation laws in fluid mechanics
• Apply to Maxwell’s equations in electromagnetism