Flux is integral along flat curve
=================================

The flux of a vector field exiting a boundary flat curve is defined as the line integral of the
component of the field normal to the curve along the curve.

**Conditions:**

#. The normal to the curve is outward (see right-hand rule).
#. The vector field is continuously differentiable everywhere within the region of the surface.
#. The curve is closed and flat.

.. py:currentmodule:: symplyphysics.laws.fields.flux_is_integral_along_flat_curve

.. py:data:: flux

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

Symbol:
    :code:`H`

Latex:
    :math:`H`

Dimension:
    :code:`any_dimension`


.. py:data:: field

    Vector field, i.e. a vector-valued function of the position vector.

Symbol:
    :code:`F`

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

Dimension:
    :code:`any_dimension`


.. py:data:: unit_normal

    Unit normal vector to the curve.

Symbol:
    :code:`n`

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

Dimension:
    :code:`any_dimension`


.. py:data:: curve

    Curve which is the boundary of the surface along which :attr:`~flux` is calculated.

    Symbol:
        :code:`C`

    Latex:
        :math:`C`


.. py:data:: initial_parameter

    Initial value of the curve parameter.

Symbol:
    :code:`u_1`

Latex:
    :math:`u_{1}`

Dimension:
    :code:`dimensionless`


.. py:data:: final_parameter

    Final value of the curve parameter.

Symbol:
    :code:`u_2`

Latex:
    :math:`u_{2}`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`H = LineIntegral(dot(F, n) * norm(dr), C)`


    Latex:
        .. math::
            H = \int \limits_{C} \left( {\vec F}, {\vec n} \right) \left \Vert d \vec r \right \Vert