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Maxwell's equations

In the physics of electromagnetism, Stokes' theorem provides the justification for the equivalence of the differential form of the Maxwell–Faraday equation and the Maxwell–Ampère equation and the integral form of these equations. For Faraday's law, Stokes' theorem is applied to the electric field, E {\displaystyle \mathbf {E} }: ∮ ∂ Σ E ⋅ d l = ∬ Σ ∇ × E ⋅ d S. {\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{\Sigma }\mathbf {\nabla } \times \mathbf {E} \cdot \mathrm {d} \mathbf {S}.}

For Ampère's law, Stokes' theorem is applied to the magnetic field, B {\displaystyle \mathbf {B} }: ∮ ∂ Σ B ⋅ d l = ∬ Σ ∇ × B ⋅ d S. {\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{\Sigma }\mathbf {\nabla } \times \mathbf {B} \cdot \mathrm {d} \mathbf {S}.}