Flux is integral of divergence over surface
===========================================

The flux of a vector field exiting a closed flat curve can be re-written as a surface integral of
the divergence of the given field over the surface whose boundary is the given curve.

**Notes:**

#. This is also known as the `Green's theorem <https://en.wikipedia.org/wiki/Green%27s_theorem>`__.

**Conditions:**

#. The vector field is continuously differentiable everywhere within the region of the surface.
#. The curve is closed and flat.
#. The curve and the surface are oriented according to the right-hand rule.

.. py:currentmodule:: symplyphysics.laws.fields.flux_is_integral_of_divergence_over_surface

.. py:data:: flux

    Flux of the vector :attr:`~field`.

Symbol:
    :code:`H`

Latex:
    :math:`H`

Dimension:
    :code:`any_dimension`


.. py:data:: field

    Any vector field, i.e. a vector-valued function that depends on the position vector.

Symbol:
    :code:`F`

Latex:
    :math:`{\vec F}`

Dimension:
    :code:`any_dimension`


.. py:data:: surface

    Surface along which the integral is evaluated.

    Symbol:
        :code:`S`

    Latex:
        :math:`S`


.. py:data:: initial_first_parameter

    Initial value of the first surface parameter.

Symbol:
    :code:`u_1`

Latex:
    :math:`u_{1}`

Dimension:
    :code:`dimensionless`


.. py:data:: final_first_parameter

    Final value of the first surface parameter.

Symbol:
    :code:`u_2`

Latex:
    :math:`u_{2}`

Dimension:
    :code:`dimensionless`


.. py:data:: initial_second_parameter

    Initial value of the second surface parameter.

Symbol:
    :code:`v_1`

Latex:
    :math:`v_{1}`

Dimension:
    :code:`dimensionless`


.. py:data:: final_second_parameter

    Final value of the second surface parameter.

Symbol:
    :code:`v_2`

Latex:
    :math:`v_{2}`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`H = SurfaceIntegral(div(F) * norm(dS), S, ((u_1, u_2), (v_1, v_2)))`


    ..
        FIXME Implement the surface integral in the latex printer

    Latex:

        .. math::

            H = \iint \limits_S \text{div} \vec F \, dS