Circulation is integral of curl over surface
============================================

The circulation of a vector field, defined as :ref:`such <Circulation is integral along curve>`,
can be re-written as a double integral of the flux of that field over any surface whose boundary
is the curve along which the circulation is calculated.

**Notes:**

#. This law is also known as the `Stokes' theorem <https://en.wikipedia.org/wiki/Stokes%27_theorem>`__.

**Conditions:**

#. The surface is smooth.
#. The orientation of the surface and the orientation of the curve are related by the right-hand
   rule.

.. py:currentmodule:: symplyphysics.laws.fields.circulation_is_integral_of_curl_over_surface

.. py:data:: circulation

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

Symbol:
    :code:`G`

Latex:
    :math:`G`

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:`G = SurfaceIntegral(dot(curl(F), dS), S, ((u_1, u_2), (v_1, v_2)))`


    ..
        FIXME Unimplemented latex printer for the surface integral

    Latex:

        .. math::

            G = \iint \limits_S \left( \nabla \times \vec F \right) \cdot d \vec S