Theorem
If  {jatex options:inline}\mathbf{H}{/jatex}  be the magnetic field intensity due a current density  {jatex options:inline}\mathbf{j}{/jatex}  in a region of space. We can draw any surface  {jatex options:inline}S{/jatex}  in the space, with boundary  {jatex options:inline}C{/jatex}.
Then  {jatex options:inline}\oint \mathbf{H} \cdot d \mathbf{r} = I{/jatex}
Proof
Use Stoke's Theorem  {jatex options:inline}\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS{/jatex}
One of Maxwell's Laws states  {jatex options:inline}\mathbf{\nabla} \times \mathbf{H} = I{/jatex}  where  {jatex options:inline}H{/jatex}  is the magnetic field strength and  {jatex options:inline}I{/jatex}  is the magnetic field strength, so let  {jatex options:inline}\mathbf{F} = \mathbf{H}{/jatex}  to give
{jatex options:inline}\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{/jatex}