Equations
=========

Virtual Casing Principle
------------------------

Let ``Gamma`` be a closed toroidal surface with outward unit normal ``n``.
Let ``B`` be the total magnetic field on ``Gamma``, decomposed into
interior and exterior contributions:

.. math::

   \mathbf{B} = \mathbf{B}_{\mathrm{int}} + \mathbf{B}_{\mathrm{ext}}.

The virtual casing principle provides an explicit expression for
``B_ext`` on the surface in terms of singular layer potentials:

.. math::

   \mathbf{B}_{\mathrm{ext}}(\mathbf{r}) = \frac{1}{2}\mathbf{B}(\mathbf{r})
   + \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r})
   + \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}),

where ``G`` denotes the Laplace single-layer potential:

.. math::

   G[\sigma](\mathbf{r}) = \frac{1}{4\pi}\int_{\Gamma}
   \frac{\sigma(\mathbf{r}')}{\lVert \mathbf{r} - \mathbf{r}' \rVert}
   \, d a(\mathbf{r}').

The interior field is obtained by reversing the signs:

.. math::

   \mathbf{B}_{\mathrm{int}}(\mathbf{r}) = \frac{1}{2}\mathbf{B}(\mathbf{r})
   - \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r})
   - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).

Internal vs External Off-Surface
--------------------------------

For off-surface targets, the jump term is absent. The external and
internal fields are:

.. math::

   \mathbf{B}_{\mathrm{ext}}(\mathbf{r}) =
   \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r})
   - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}),

.. math::

   \mathbf{B}_{\mathrm{int}}(\mathbf{r}) =
   -\nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r})
   + \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).

The off-surface gradients are obtained by differentiating these
expressions and using the second-derivative Laplace kernel.

The two vector layer potentials used are:

- A Laplace single-layer gradient (``FxdU`` in the reference code)
- A Biot-Savart single-layer (``FxU``)

The kernels are (up to scaling) derivatives of ``1 / |r|``.

Kernel Formulas
---------------

Let ``r = x - x'`` be the displacement from a source to a target point.
The kernels implemented follow the BIEST convention with a ``1/(4*pi)`` factor.

Laplace single-layer:

.. math::

   G(r) = \\frac{1}{4\\pi \\lVert r \\rVert}

Gradient of single-layer:

.. math::

   \\nabla G(r) = -\\frac{r}{4\\pi \\lVert r \\rVert^3}

Second derivatives:

.. math::

   \\partial_i\\partial_j G(r) = \\frac{1}{4\\pi}\\left(-\\delta_{ij}\\lVert r \\rVert^{-3} + 3 r_i r_j \\lVert r \\rVert^{-5}\\right)

Biot-Savart (single-layer):

.. math::

   \\mathbf{K}(r) = \\frac{1}{4\\pi} \\frac{\\mathbf{f} \\times r}{\\lVert r \\rVert^3}

The derivative ``FxdU`` for Biot-Savart is implemented explicitly to match the
reference code in ``biest/kernel.hpp``.

Off-Surface Evaluation
----------------------

For off-surface targets ``r``:

.. math::

   \mathbf{B}_{\mathrm{ext}}(\mathbf{r}) = \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r})
   - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).

The singular ``+1/2`` jump term is absent off the surface.

Field-Period Symmetry
---------------------

The surface and field may be defined on a half field period using
stellarator symmetry. The toroidal grid is shifted by half a grid point
for the symmetric case.

A full description of the grid conventions is in the reference code
and reproduced in the :doc:`numerics` section.