Gauss’ divergence theorem

The divergence of a vector field $\vec{F}$ at a point $M$ is defined as the limit of the flux per unit volume as the enclosing volume shrinks to zero around $M$. The formula is $\lim_{V \to 0} \frac{1}{V} \iint_S \vec{F} \cdot \vec{n} dS = \text{div } \vec{F}(M)$.

Conditions: The volume $V$ shrinks to the point $M$ (radius tends to zero); The vector field $\vec{F}$ is smooth in the neighborhood of $M$

Divergence represents the intensity of dispersion (or net outward flow) of the vector field at that specific point. It quantifies how much the field acts as a source or sink locally.

Conditions: Applicable to any continuous vector field; Evaluated at a specific point $M$

It is defined by dividing the total outward flux across a closed surface $S$ by the volume $V$ enclosed by that surface. The formula is $\frac{1}{V} \iint_S \vec{F} \cdot \vec{n} dS$, which represents the average net outflow of the vector field per unit volume within the region.

Conditions: The surface $S$ encloses a finite non-zero volume $V$; $\vec{F}$ is defined throughout the interior of $S$

The introduction establishes a 3D Cartesian coordinate system with a mesh-like sphere at the origin. Yellow arrows represent the outward normal vector $\vec{n}$ on the surface, and red arrows illustrate the distributed vector field $\vec{F}$.

Conditions: Visualization of a sphere centered at the origin; Clear distinction between surface normals and field vectors

Gauss's Divergence Theorem states that the flux of a vector field $\vec{F}$ through a closed surface $S$ equals the triple integral of the divergence of $\vec{F}$ over the volume $V$ enclosed by $S$. Mathematically, it is expressed as $\iint_S \vec{F} \cdot \vec{n} dS = \iiint_V (\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}) dV$.

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 $V$