Maxwell's equations can be written either as integral laws over curves and surfaces or as differential equations that describe the fields locally at each point in space and time.

The bridge between the two descriptions is provided by the divergence theorem and Stokes' theorem.

Gauss's law for electricity

From

$$\oint_S\mathbf E\cdot d\mathbf A=\frac{Q_{\mathrm{enc}}}{\varepsilon_0},$$

write the enclosed charge as the volume integral of charge density $\rho$ and apply the divergence theorem. The local form is

$$\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}.$$

Electric charge density is therefore a local source or sink of electric field.

Gauss's law for magnetism

Applying the same argument to magnetic flux gives

$$\nabla\cdot\mathbf B=0.$$

Magnetic field has zero local divergence: there is no classical magnetic-charge density acting as a source or sink.

Faraday's law

Stokes' theorem converts circulation of electric field around a closed curve into the flux of its curl through a spanning surface. Faraday's law becomes

$$\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.$$

A changing magnetic field therefore produces local circulation of electric field.

Ampère–Maxwell law

Likewise,

$$\nabla\times\mathbf B =\mu_0\mathbf J+\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t},$$

where $\mathbf J$ is the electric current-density vector.

Current density and changing electric field both contribute to local circulation of magnetic field.

Integral and differential descriptions

The integral equations describe accumulated fluxes and circulations over finite regions. The differential equations describe the same field laws point by point.

Under the regularity assumptions required by the vector-calculus theorems, neither formulation contains different physics: they are two mathematical representations of the same electromagnetic theory.

The differential form is especially useful for deriving field equations, studying spatially varying systems and analyzing electromagnetic-wave propagation.