Proof That Integral of Magnetic Field Strength Around a Closed Loop Equals Current Through the Loop

Theorem
If  
\[\mathbf{H}\]
  be the magnetic field intensity due a current density  
\[\mathbf{j}\]
  in a region of space.We can draw any surface  
\[S\]
  in the space, with boundary  
\[C\]
.
Then  
\[\oint \mathbf{H} \cdot d \mathbf{r} = I\]

Proof
Use Stoke's Theorem  
\[\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS\]

One of Maxwell's Laws states  
\[\mathbf{\nabla} \times \mathbf{H} = I\]
  where  
\[H\]
  is the magnetic field strength and  
\[I\]
  is the magnetic field strength, so let  
\[\mathbf{F} = \mathbf{H}\]
  to give
\[\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{H}) \cdot \mathbf{n} dS = \int \int_S \mathbf{j} \cdot \mathbf{n} = I\]