What is the mathematical statement of Gauss's Divergence Theorem relating surface and volume integrals?

Conditions

• $S$ is a piecewise smooth closed surface bounding volume $V$
• $\vec{n}$ is the outward unit normal vector on $S$
• $\vec{F}$ is continuously differentiable in $V$

Reasoning, step by step

1. Identify the closed surface $S$ and the enclosed volume $V$.
2. Compute the dot product of the vector field $\vec{F}$ and the outward normal $\vec{n}$ on the surface.
3. Integrate this product over the surface area $dS$ to find total outward flux.
4. Alternatively, compute the divergence $\nabla \cdot \vec{F}$ at each point inside the volume.
5. Integrate the divergence over the volume element $dV$.
6. Equate the two results as per the theorem.

Example

The video displays the explicit expression: $\iint_S \vec{F} \cdot \vec{n} dS = \iiint_V (\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}) dV$.

Common misconceptions

• Believing the theorem applies to open surfaces without closing boundaries.
• Confusing the direction of the normal vector; it must be outward for the standard formulation.

Watch the explanation

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