and their differential and integral forms are related by the special 3-dimensional (vector calculus) case of Stokes' theorem. Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below (see Differentiation under the integral sign): The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in SI units. In other systems of units, such as CGS or Gaussian units, the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms: respectively, where is the speed of light in vacuum. Divergence theorem Likewise, the divergence theorem is a special case if we identify a vector field with the -form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case where is an arbitrary constant vector. Working out the divergence of the product gives Since this holds for all we find Volume integral of gradient of scalar field Let be a scalar field. Then where is the normal vector to the surface at a given point. Proof: Let be a vector. Then Since this holds for any (in particular, for every basis vector), the result follows. See also Chandrasekhar–Wentzel lemma Footnotes References Further reading External links Proof of the Divergence Theorem and Stokes' Theorem Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation Differential topology Differential forms Duality theories Integration on manifolds Theorems in calculus Theorems in differential geometry