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Using Faraday's Law of Electromagnetic Induction to Prove One of Maxwell's Laws

Faraday's Law of Electromagnetic Induction states that  

\[\oint_C \mathbf{E} d \mathbf{r} =- \frac{1}{c} \frac{\partial}{\partial t} \int \int \mathbf{H} \cdot \mathbf{n} dS\]

From Stoke's Theorem  
\[\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS\]
  for a vector field  
\[\mathbf{F}\]
  with continuous derivatives.
If  
\[\mathbf{H}\]
  varies smoothly with time then we can write  
\[\frac{1}{c} \frac{\partial}{\partial t} \int \int \mathbf{H} \cdot \mathbf{n} dS = \frac{1}{c} \int \int \frac{\partial \mathbf{H}}{\partial t} \cdot \mathbf{n} dS\]

Hence  
\[ \int \int_S (\mathbf{\nabla} \times \mathbf{E}) \cdot \mathbf{n} dS= \int \int_S - \frac{1}{c} \frac{\partial \mathbf{H}}{\partial t} \cdot \mathbf{n} dS \]

Equating the integrands gives
\[\mathbf{\nabla} \times \mathbf{E} \cdot \mathbf{n}= - \frac{1}{c} \frac{\partial \mathbf{H}}{\partial t} \cdot \mathbf{n} \]

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