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Definition

The Maxwell Equations are 4 fundamental equations that describe the relation between the quantities of electromagnetism.

Name Integral Version Differential Version
Gauss's law: $${\mathop{\circ\!\!\!\iint}}_{\partial \Omega} {\boldsymbol E} \cdot {\,\text{d}}A = \frac{1}{\epsilon_0} \iiint_{\Omega} \rho {\,\text{d}}V$$ $${\text{div}}{\boldsymbol D} = \varrho$$
Faraday's law of induction $$\oint_{\partial A} {\boldsymbol E} \cdot {\,\text{d}}{\boldsymbol s} = - \iint_{A} \frac{\partial {\boldsymbol B}}{\partial t} \cdot {\,\text{d}}{\boldsymbol A}$$ $${\text{rot}}{\boldsymbol E} + \frac{\partial {\boldsymbol B}}{\partial t} = 0$$
Gauss's law for magnetism $${\mathop{\circ\!\!\!\iint}}_{\partial \Omega} {\boldsymbol B} \cdot {\,\text{d}}A = 0$$ $${\text{div}}{\boldsymbol B} = 0$$
Ampère's law $$\oint_{\partial A} {\boldsymbol H} \cdot {\,\text{d}}{\boldsymbol s} = \iint_{A} {\boldsymbol j} \cdot {\,\text{d}}{\boldsymbol A} + \iint_{A} \frac{\partial {\boldsymbol D}}{\partial t} \cdot {\,\text{d}}{\boldsymbol A}$$ $${\text{rot}}{\boldsymbol H} = {\boldsymbol j} + \frac{\partial {\boldsymbol D}}{\partial t}$$

Gauss's law

Explanation: The differential form states that if there exists electric charge somewhere, then the divergence of $D$ at that point is nonzero, otherwise it is equal to zero. The integral form states that the amount of charge inside a volume $V$ of enclosed charge is equal to the total amount of Electric Flux $D$ exiting the surface $S$.

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