Vector area is unit normal times scalar area
============================================

**Vector area**, or **oriented area**, is a vector quantity equal to the surface integral of the
surface normal. If the unit normal is constant at all points of the surface, integral can be
reduced to a product of the unit normal to the scalar area of the surface.

**Notes:**

#. If the normal changes direction across the surface, you can divide the surface into parts of
   constant unit normal and sum up the vector areas of all parts:

   .. math::

       \vec A = \sum_i \vec A_i = \sum_i \vec n_i A_i

   Alternatively, use the surface integral that sums up infinitesimal vector areas across the
   surface:

   .. math::

       \vec A = \iint \limits_S d \vec A = \iint \limits_S \vec n (\vec r) dA

**Conditions:**

#. The surface is bounded (i.e. finite).

#. The surface normal is the same thoughout the given region, which can be achieved by choosing a
   small enough (or infinitesimal) surface.

**Links:**

#. `Wikipedia — Vector area <https://en.wikipedia.org/wiki/Vector_area#Definition>`__.

.. py:currentmodule:: symplyphysics.definitions.vector.vector_area_is_unit_normal_times_scalar_area

.. py:data:: vector_area

    Vector :attr:`~symplyphysics.symbols.classical_mechanics.area` pertaining to the given region.

Symbol:
    :code:`A`

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

Dimension:
    :code:`area`


.. py:data:: unit_normal

    Unit vector `normal <https://en.wikipedia.org/wiki/Normal_(geometry)#>`__ to the surface.

    **Notes:**

    #. :math:`\left \Vert \vec n \right \Vert = 1`, i.e. it has unit magnitude.

Symbol:
    :code:`n`

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

Dimension:
    :code:`dimensionless`


.. py:data:: scalar_area

    Scalar :attr:`~symplyphysics.symbols.classical_mechanics.area` of the given region.

Symbol:
    :code:`A`

Latex:
    :math:`A`

Dimension:
    :code:`area`


.. py:data:: law

    :code:`A = n * A`


    Latex:
        .. math::
            {\vec A} = {\vec n} A