:orphan:

.. admonition:: Work in Progress
   :class: warning

   This document is still under development and may change frequently.


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Tangential Magnetic Field Boundary Condition
============================================

.. toctree::
   :maxdepth: 1
   :caption: Tangential Magnetic Field Boundary Condition


As boundary conditions, we support the *tangential magnetic field* given by

.. math::

   \left. \mathbf{H} \times \mathbf{n} \right|_{\Gamma_0} = \mathbf{H}_{\text{ext}} \quad,

where :math:`\mathbf{H}_{\text{ext}}` is the *user-defined* external magnetic field imposed on the boundary :math:`\Gamma_0`.

This condition is implemented as a inhomogeneous Neumann condition through a linear form given by

.. math::
    
   f_{\Gamma_0} = \left( \mathbf{n} \times \mathbf{H}_{\text{ext}}, \mathbf{A}' \right)_{\Gamma_0} \quad,

where :math:`\mathbf{A}'` is a test function in the function space.