Stokes' theorem is used in the conversion of a line integral of H into a surface integral.

Explanation:
Stokes' theorem relates the line integral of a vector field over a closed curve to the surface integral of the curl of the vector field over the surface bounded by the curve. It is a fundamental theorem in vector calculus and is named after the Irish mathematician George Gabriel Stokes.

Statement of Stokes' theorem:
Stokes' theorem states that for a smooth, oriented surface S with boundary curve C, and a vector field F continuously differentiable on an open region containing S and C, the line integral of F around C is equal to the surface integral of the curl of F over S. Mathematically, the theorem can be stated as:

∮C F · dr = ∬S (curl F) · dS

where ∮C represents the line integral around the boundary curve C, F is the vector field, dr is the differential length element along the curve C, ∬S represents the surface integral over the surface S, curl F is the curl of the vector field F, and dS is the differential surface area element on the surface S.

Application to converting line integral of H into surface integral:
When converting a line integral of a vector field H into a surface integral, we can use Stokes' theorem. The line integral of H around a closed curve C can be converted to the surface integral of the curl of H over the surface bounded by the curve.

In this context, the vector field H represents the magnetic field in electromagnetism. The line integral of H around a closed curve represents the circulation of the magnetic field around the curve. By applying Stokes' theorem, we can convert this line integral into the surface integral of the curl of H over the surface bounded by the curve.

This conversion is particularly useful in electromagnetic theory, as it allows us to relate the circulation of the magnetic field around a closed loop to the distribution of the magnetic field over the surface bounded by the loop. This relationship is crucial in understanding and calculating electromagnetic phenomena, such as electromagnetic induction and magnetic flux.

Overall, Stokes' theorem provides a powerful tool for converting line integrals into surface integrals, allowing us to relate the circulation of a vector field around a closed curve to the distribution of the field over the surface bounded by the curve.