EM-07 · Electrostatics toolkitTool equip

The Divergence Theorem

The net flux through a closed surface equals the divergence at every interior point summed over the volume. Subdivide the region to see interior faces cancel and only the outer boundary remain, and pull out the differential form of Gauss.

Linking the surface and the interior

Subdivide the region into small cells. On a face shared by two neighbors, what one sends out the other takes in, so they cancel. What is left over, never cancelled?

Subdivisions N (per side)

Drag to orbit. Buttons set the subdivisions N.

The divergence theorem

∮E·dA = ∫(∇·E)dV · surface integral = volume integral of the divergence

Even split into 8 cells, interior faces cancel and only the outer boundary remains.

Neighbors cancel

A bridge between surface and interior

The Gauss law speaks of the flux through a closed surface (the outside). The divergence theorem bridges this to the sum, over the whole volume, of the divergence at every interior point: ∮E·dA = ∫(∇·E)dV. It says what you measure on the surface equals what you measure throughout the volume.

Observe∮E·dA = ∫(∇·E)dV

The surface integral equals the volume integral of the divergence.

Why it holds — neighbors cancel

Split the region into small cells and add up each cell’s net flux. On an interior face shared by two cells, the flow leaving one is the flow entering the other, with opposite sign. So interior faces cancel in pairs. The only faces left unpaired are those on the outer boundary. Adding the divergence of every cell therefore leaves just the flux through the outer surface.

Choose∇·E = ?

Divergence is the net flux per unit volume.

Divergence, the source per unit volume

Shrink a cell to nothing and its net flux divided by volume becomes the divergence ∇·E at a point — the amount of field welling up there per unit volume. Where the divergence is positive the point is a source (positive charge); where negative, a sink (negative charge). The divergence theorem says that gathering this pointwise welling over the volume gives the surface flux.

The differential form of Gauss

Put the divergence theorem into the Gauss law. The surface flux ∮E·dA is Q/ε₀, and the interior is ∫(∇·E)dV. The enclosed charge Q is the density integrated over volume, ∫ρ dV. Then ∫(∇·E)dV = ∫(ρ/ε₀)dV holds for any volume, so the integrands must match: ∇·E = ρ/ε₀. The one-line integral form has become a pointwise differential form — the first of Maxwell’s equations.

Fill in∫(∇·E)dV = ?

By the Gauss law the volume integral becomes the enclosed charge.

On your own∇·E = ?

Equal for every volume means the integrands match: ∇·E = ρ/ε₀.

Back to the first screen

On the first screen, the finer you split the region, the denser the shared interior faces became — yet they all cancelled in pairs. The only ones left were the faces on the outer boundary. So the volume integral of every cell’s divergence exactly equals the surface integral: ∮E·dA = ∫(∇·E)dV. Lay this bridge onto the Gauss law and out comes the differential form ∇·E = ρ/ε₀.

The divergence theorem is the identity that the net flux through a closed surface equals the volume integral of the divergence at every interior point: ∮E·dA = ∫(∇·E)dV. It holds because the flow that interior cells exchange with their neighbors cancels, leaving only the outer boundary. Combined with the Gauss law it yields the differential form ∇·E = ρ/ε₀.

Tool equipOnce you equip this tool

The divergence theorem is the tool for moving freely between surface and volume integrals. Once equipped, the differential form of the Gauss law, ∇·E = ρ/ε₀ (Maxwell 1, EM-22), is in your hands, paired with its loop-going counterpart, the Stokes theorem (EM-16). The integral form serves symmetric problems; the differential form, the local structure of the field and Maxwell’s equations.