**************************************
  Geometric Algebra Module for SymPy
**************************************

:Author: Alan Bromborsky

.. |release| replace:: 0.10

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.. topic:: Abstract

   This document describes the implementation of a geometric algebra module in
   python that utilizes the :mod:`sympy` symbolic algebra library.  The python
   module :mod:`GA` has been developed for coordinate free calculations using
   the operations (geometric, outer, and inner products etc.) of geometric algebra.
   The operations can be defined using a completely arbitrary metric defined
   by the inner products of a set of arbitrary vectors or the metric can be
   restricted to enforce orthogonality and signature constraints on the set of
   vectors.  In addition the module includes the geometric, outer (curl) and inner
   (div) derivatives and the ability to define a curvilinear coordinate system.
   The module requires the numpy and the sympy modules.


What is Geometric Algebra?
==========================

Geometric algebra is the Clifford algebra of a real finite dimensional vector
space or the algebra that results when a real finite dimensional vector space
is extended with a product of vectors (geometric product) that is associative,
left and right distributive, and yields a real number for the square (geometric
product) of any vector [Hestenes,Lasenby].  The elements of the geometric
algebra are called multivectors and consist of the linear combination of
scalars, vectros, and the geometric product of two or more vectors. The
additional axioms for the geometric algebra are that for any vectors :math:`a`,
:math:`b`, and :math:`c` in the base vector space:

.. math::
  :nowrap:

  \begin{equation*}
  \begin{array}{c}
  a \left( bc \right) = \left( ab \right) c \\
  a \left( b+c \right) = ab+ac \\
  \left( a + b \right) c = ac+bc \\
  aa = a^{2} \in \Re
  \end{array}
  \end{equation*}


By induction these also apply to any multivectors.

Several software packages for numerical geometric algebra calculations are
available from Doran-Lazenby group and the Dorst group. Symbolic packages for
Clifford algebra using orthongonal bases such as
:math:`e_{i}e_{j}+e_{j}e_{i} = 2\eta_{ij}`, where :math:`\eta_{ij}` is a numeric
array are available in Maple and Mathematica. The symbolic algebra module,
:mod:`GA`, developed for python does not depend on an orthogonal basis
representation, but rather is generated from a set of :math:`n` arbitrary
symbolic vectors,  :math:`a_{1},a_{2},\dots,a_{n}` and a symbolic metric
tensor :math:`g_{ij} = a_{i}\cdot a_{j}`.

In order not to reinvent the wheel all scalar symbolic algebra is handled by the
python module  :mod:`sympy`.

The basic geometic algebra operations will be implemented in python by defining
a multivector class, MV, and overloading the python operators in Table
:ref:`1 <table1>` where ``A`` and ``B``  are any two multivectors (In the case of
``+``, ``-``, ``*``, ``^``, ``|``, ``<<``, and ``>>`` the operation is also defined if ``A`` or
``B`` is a sympy symbol or a sympy real number).

.. _table1:

.. csv-table::
    :header: Operation, Result
    :widths: 10, 40

    ``A+B``,             Sum of multivectors
    ``A-B``,             Difference of multivectors
    ``A*B``,             Geometric product
    ``A^B``,             Outer product of multivectors
    ``A|B``,             Inner product of multivectors
    ``A<B`` or ``A<<B``, Left contraction of multivectors
    ``A>B`` or ``A>>B``, Right contraction of multivectors

Table :ref:`1 <table1>`. Multivector operations for symbolicGA

The option to use ``<`` or ``<<`` for left contraction and ``>`` or ``>>`` is given since the ``<`` and
``>`` operators do not have r-forms (there are no :func:`__rlt__` and :func:`__rgt__` functions to overload) while
``<<`` and ``>>`` do have r-forms so that ``x << A`` and ``x >> A`` are allowed where ``x`` is a scalar (symbol or integer)
and ``A`` is a multivector. With ``<`` and ``>`` we can only have mixed modes (scalars and multivectors) if the
multivector is the first operand.

.. note::

    Except for ``<`` and ``>`` all the multivector operators have r-forms so that as long as one of the
    operands, left or right, is a multivector the other can be a multivector or a scalar (sympy symbol or integer).

.. warning::

    Note that the operator order precedence is determined by python and is not
    neccessarily that used by geometric algebra. It is **absolutely essential** to
    use parenthesis in multivector
    expressions containing ``^``, ``|``, ``<``, ``>``, ``<<`` and/or ``>>``.  As an example let
    ``A`` and ``B`` be any two multivectors. Then ``A + A*B = A +(A*B)``, but
    ``A+A^B = (2*A)^B`` since in python the ``^`` operator has a lower precedence
    than the '+' operator.  In geometric algebra the outer and inner products and
    the left and right contractions have a higher precedence than the geometric
    product and the geometric product has a higher precedence than addition and
    subtraction.  In python the ``^``, ``|``, ``<``, ``>``, ``<<`` and ``>>`` all have a lower
    precedence than ``+`` and ``-`` while ``*`` has a higher precedence than
    ``+`` and ``-``.


.. _vbm:

Vector Basis and Metric
=======================

The two structures that define the :class:`MV` (multivector) class are the
symbolic basis vectors and the symbolic metric.  The symbolic basis
vectors are input as a string with the symbol name separated by spaces.  For
example if we are calculating the geometric algebra of a system with three
vectors that we wish to denote as ``a0``, ``a1``, and ``a2`` we would define the
string variable:

``basis = 'a0 a1 a2'``

that would be input into the multivector setup function.  The next step would be
to define the symbolic metric for the geometric algebra of the basis we
have defined. The default metric is the most general and is the matrix of
the following symbols

.. _eq1:



.. math::
  :label: 1
  :nowrap:

  \begin{equation*}
  g = \left[
  \begin{array}{ccc}
    a0**2 & (a0.a1) & (a0.a2) \\
    (a0.a1) & a1**2 & (a1.a2) \\
    (a0.a2) & (a1.a2) & a2**2 \\
  \end{array}
  \right]
  \end{equation*}


where each of the :math:`g_{ij}` is a symbol representing all of the dot
products of the basis vectors. Note that the symbols are named so that
:math:`g_{ij} = g_{ji}` since for the symbol function
:math:`(a0.a1) \ne (a1.a0)`.

Note that the strings shown in equation :ref:`1 <eq1>` are only used when the values
of :math:`g_{ij}` are output (printed).   In the :mod:`GA` module (library)
the :math:`g_{ij}` symbols are stored in a static member list of the multivector
class :class:`MV` as the double list ``MV.metric`` (:math:`g_{ij}` = ``MV.metric[i][j]``).

The default definition of :math:`g` can be overwritten by specifying a string
that will define :math:`g`. As an example consider a symbolic representation
for conformal geometry. Define for a basis

``basis = 'a0 a1 a2 n nbar'``

and for a metric

``metric = '# # # 0 0, # # # 0 0, # # # 0 0, 0 0 0 0 2, 0 0 0 2 0'``

then calling `MV.setup(basis,metric)` would initialize



.. math::
  :nowrap:

  \begin{equation*}
  g = \left[
  \begin{array}{ccccc}
    a0**2 & (a0.a1)  & (a0.a2) & 0 & 0\\
    (a0.a1) & a1**2  & (a1.a2) & 0 & 0\\
    (a0.a2) & (a1.a2) & a2**2 & 0 & 0 \\
    0 & 0 & 0 & 0 & 2 \\
    0 & 0 & 0 & 2 & 0
  \end{array}
  \right]
  \end{equation*}


Here we have specified that :math:`n` and :math:`nbar` are orthonal to all the
:math:`a`'s, :math:`n**2 = nbar**2 = 0`, and :math:`(n.nbar) = 2`. Using
:math:`\#` in the metric definition string just tells the program to use the
default symbol for that value.

When ``MV.setup`` is called multivector representations of the basis local to
the program are instantiated.  For our first example that means that the
symbolic vectors named ``a0``, ``a1``, and ``a2`` are created and made available
to the programmer for future calculations.

In addition to the basis vectors the :math:`g_{ij}` are also made available to
the programer with the following convention. If ``a0`` and ``a1`` are basis
vectors, then their dot products are denoted by  ``a0sq``, ``a2sq``, and
``a0dota1`` for use as python program varibles. If you print  ``a0sq`` the
output would be ``a0**2`` and the output for ``a0dota1`` would be ``(a0.a1)`` as
shown in equation :ref:`1 <eq1>`.  If the default value are overridden the new
values are output by print.  For examle if :math:`g_{00} = 0` then "``print
a0sq``" would output "0."

More generally, if ``metric`` is not a string, but a list of lists or a two
dimension numpy array, it is assumed that each element of ``metric`` is symbolic
variable so that the :math:`g_{ij}` could be defined as symbolic functions as
well as variables. For example instead of letting :math:`g_{01} = (a0.a1)` we
could have  :math:`g_{01} = cos(theta)` where we use a symbolic :math:`\cos`
function.

.. note::

    Additionally ``MV.setup`` has an option for an othogonal basis where the signature
    of the metric space is defined by a string.  For example if the signature of the
    vector space is :math:`(1,1,1)` (Euclidian 3-space) set

    ``metric = '[1,1,1]'``

    Likewise if the signature is that of spacetime, :math:`(1,-1,-1,-1)` then define

    ``metric = '[1,-1,-1,-1]'``.


Representation and Reduction of Multivector Bases
=================================================

In our symbolic geometric algebra we assume that all multivectors of interest to
us can be obtained from the symbolic basis vectors we have input, via the
different operations available to geometric algebra. The first problem we have
is representing the general multivector in terms of the basis vectors.  To
do this we form the ordered geometric products of the basis vectors and develop
an internal representation of these products in terms of python classes.  The
ordered geometric products are all multivectors of the form
:math:`a_{i_{1}}a_{i_{2}}\dots a_{i_{r}}` where :math:`i_{1}<i_{2}<\dots <i_{r}`
and :math:`r \le n`. We call these multivectors bases and represent them
internally with the list of integers
:math:`\left[ i_{1},i_{2},\dots,i_{r}\right]`. The bases are labeled, for the
purpose of output display, with strings that are concatenations of the strings
representing the basis vectors.  So that in our example  ``[1,2]`` would be
labeled with the string ``'a1a2'`` and represents the geometric product
``a1*a2``. Thus the list ``[0,1,2]`` represents ``a0*a1*a2``.For our example the
complete set of bases and labels are shown in Table :ref:`2 <table2>`

.. note::

    The empty list, ``[]``, represents the scalar 1.

.. _table2:

::

   MV.basislabel = ['1', ['a0', 'a1', 'a2'], ['a0a1', 'a0a2', 'a1a2'],
                    ['a0a1a2']]
   MV.basis      = [[], [[0], [1], [2]], [[0, 1], [0, 2], [1, 2]],
                    [[0, 1, 2]]]

Table :ref:`2 <table2>`. Multivector basis labels and internal basis representation.

Since there are :math:`2^{n}` bases and the number of bases with equal list
lengths is the same as for the grade decomposition of a dimension :math:`n`
geometric algebra we will call the collections of bases of equal length
``psuedogrades``.

The critical operation in setting up the geometric algebra module is reducing
the geomertric product of any two bases to a linear combination of bases so that
we can calculate a multiplication table for the bases.  First we represent the
product as the concatenation of two base lists.  For example ``a1a2*a0a1`` is
represented by the list ``[1,2]+[0,1] = [1,2,0,1]``. The representation of the
product is reduced via two operations, contraction and revision. The state of
the reduction is saved in two lists of equal length.  The first list contains
symbolic scale factors (symbol or numeric types) for the corresponding interger
list representing the product of bases.  If we wish to reduce
:math:`\left[ i_{1},\dots,i_{r}\right]` the starting point is the coefficient list
:math:`C = \left[ 1 \right]` and the bases list
:math:`B = \left[ \left[ i_{1},\dots,i_{r}\right] \right]`.  We now operate on each
element of the lists as follows:

* ``contraction``: Consider a basis list :math:`B` with element
  :math:`B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]` where
  :math:`i_{l} = i_{l+1}`. Then the product of the :math:`l` and :math:`l+1` terms
  result in a scalar and :math:`B[j]` is replaced by the new list representation
  :math:`\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{r}\right]` which is of psuedo
  grade :math:`r-2` and :math:`C[j]` is replaced by the symbol
  :math:`g_{i_{l}i_{l}}C[j]`.

* ``revision``: Consider a basis list :math:`B` with element
  :math:`B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]` where
  :math:`i_{l} > i_{l+1}`. Then the :math:`l` and :math:`l+1` elements must be
  reversed to be put in normal order, but we have :math:`a_{i_{l}}a_{i_{l+1}} = 2g_{i_{l}i_{l+1}}-a_{i_{l+1}}a_{i_{l}}`
  (From the geometric algebra definition of the dot product of two vectors). Thus
  we append the list representing the reduced element,
  :math:`\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{s}\right]`, to the pseudo bases
  list, :math:`B`, and append :math:`2g_{i{l}i_{l+1}}C[j]` to the coefficients
  list, then we replace :math:`B[j]` with
  :math:`\left[ i_{1},\dots,i_{l+1},i_{l},\dots,i_{s}\right]`  and :math:`C[j]` with
  :math:`-C[j]`. Both lists are increased by one element if
  :math:`g_{i_{l}i_{l+1}} \ne 0`.

These processes are repeated untill every basis list in :math:`B` is in normal
(ascending) order with no repeated elements.  Then the coefficents of equivalent
bases are summed and the bases sorted according to psuedograde and ascending
order.  We now have a way of calculating the geometric product of any two bases
as a symbolic linear combination of all the bases with the coefficients
determined by :math:`g`. The base multiplication table for our simple example of
three vectors is given by (the coefficient of each psuedo base is enclosed with
{} for clarity):


.. literalinclude:: multable.dat   


Base Representation of Multivectors
===================================

In terms of the bases defined an arbitrary multivector can be represented as a
list of arrays  (we use the numpy python module to implement arrays).   If we
have :math:`n` basis vectors we initialize the list ``self.mv = [0,0,...,0]``
with :math:`n+1` integers all zero.  Each zero is a placeholder for an array of
python objects (in this case the objects will be sympy symbol objects). If
``self.mv[r] = numpy.array([list of symbol objects])`` each entry in the
``numpy.array`` will be a coefficient of the corresponding psuedo base.
``self.mv[r] = 0`` indicates that the coefficients of every base of psuedo grade
:math:`r` are 0.  The length of the array ``self.mv[r]`` is :math:`n \choose r`
the binomial coefficient. For example the psuedo basis vector ``a1`` would be
represented as a multivector by the list:

``a1.mv = [0,numpy.array([numeric(0),numeric(1),numeric(0)]),0,0]``

and ``a0a1a2`` by:

``a0a1a2.mv = [0,0,0,numpy.array([numeric(1)])]``

The array is stuffed with sympy numeric objects instead of python integers so
that we can perform symbolically manipulate sympy expressions that consist of
scalar algebraic symbols and exact rational numbers which sympy can also
represent.

The ``numpy.array`` is used because operations of addition, substraction, and
multiplication by an object are defined for the array if they are defined for
the objects making up the array, which they are by sympy.  We call this
representation a base type because the ``r`` index is not a grade index since
the bases we are using are not blades. In a blade representation the structure
would be identical, but the bases would be replaced by blades and
``self.mv[r]`` would represent the ``r`` grade components of the multivector.
The first use of the base representation is to store the results of the
multiplication tabel for the bases in the class variable ``MV.mtabel``.  This
variable is a group of nested lists so that the geometric product of the
``igrade`` and  ``ibase`` with the ``jgrade`` and ``jbase`` is
``MV.mtabel[igrade][ibase][jgrade][jbase]``.  We can then use this table to
calculate the geometric product of any two multivectors.


Blade Representation of Multivectors
====================================

Since we can now calculate the symbolic geometric product of any two
multivectors we can also calculate the blades corresponding to the product of
the symbolic basis vectors using the formula

.. math::
  :nowrap:

  \begin{equation*}
    A_{r}\wedge b = \frac{1}{2}\left( A_{r}b-\left( -1 \right)^{r}bA_{r} \right),
  \end{equation*}


where :math:`A_{r}` is a multivector of grade :math:`r` and :math:`b` is a
vector.  For our example basis the result is shown in Table :ref:`3 <table3>`.

.. _table3:

::

   1 = 1
   a0 = a0
   a1 = a1
   a2 = a2
   a0^a1 = {-(a0.a1)}1+a0a1
   a0^a2 = {-(a0.a2)}1+a0a2
   a1^a2 = {-(a1.a2)}1+a1a2
   a0^a1^a2 = {-(a1.a2)}a0+{(a0.a2)}a1+{-(a0.a1)}a2+a0a1a2

Table :ref:`3 <table3>`. Bases blades in terms of bases.

The important thing to notice about Table :ref:`3 <table3>` is that it is a
triagonal (lower triangular) system of equations so that using a simple back
substitution algorithym we can solve for the psuedo bases in terms of the blades
giving Table :ref:`4 <table4>`.

.. _table4:

::

   1 = 1
   a0 = a0
   a1 = a1
   a2 = a2
   a0a1 = {(a0.a1)}1+a0^a1
   a0a2 = {(a0.a2)}1+a0^a2
   a1a2 = {(a1.a2)}1+a1^a2
   a0a1a2 = {(a1.a2)}a0+{-(a0.a2)}a1+{(a0.a1)}a2+a0^a1^a2

Table :ref:`4 <table4>`. Bases in terms of basis blades.

Using Table :ref:`4 <table4>` and simple substitution we can convert from a base
multivector representation to a blade representation.  Likewise, using Table
:ref:`3 <table3>` we can convert from blades to bases.

Using the blade representation it becomes simple to program functions that will
calculate the grade projection, reverse, even, and odd multivector functions.

Note that in the multivector class ``MV`` there is a class variable for each
instantiation, ``self.bladeflg``, that is set to zero for a base representation
and 1 for a blade representation.  One needs to keep track of which
representation is in use since various multivector operations require conversion
from one representation to the other.

.. warning::

    When the geometric product of two multivectors is calculated the module looks to
    see if either multivector is in blade representation.  If either is the result of
    the geometric product is converted to a blade representation.  One result of this
    is that if either of the multivectors is a simple vector (which is automatically a
    blade) the result will be in a blade representation.  If ``a`` and ``b`` are vectors
    then the result ``a*b`` will be ``(a.b)+a^b`` or simply ``a^b`` if ``(a.b) = 0``.


Outer and Inner Products, Left and Right Contractions
=====================================================

In geometric algebra any general multivector :math:`A` can be decomposed into
pure grade multivectors (a linear combination of blades of all the same order)
so that in a :math:`n`-dimensional vector space

.. math::
  :nowrap:

  \begin{equation*}
  A = \sum_{r = 0}^{n}A_{r}
  \end{equation*}


The geometric product of two pure grade multivectors :math:`A_{r}` and
:math:`B_{s}` has the form

.. math::
  :nowrap:

  \begin{equation*}
  A_{r}B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|} + \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|+2} + \cdots + \left\langle A_{r}B_{s} \right\rangle_{r+s}
  \end{equation*}


where :math:`\left\langle  \right\rangle_{t}` projects the :math:`t` grade components of the
multivector argument.  The inner and outer products of :math:`A_{r}` and
:math:`B_{s}` are then defined to be

.. math::
  :nowrap:

  \begin{equation*}
  A_{r}\cdot B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  A_{r}\wedge B_{s} = \left\langle A_{r}B_{s} \right\rangle_{r+s}
  \end{equation*}


and

.. math::
  :nowrap:

  \begin{equation*}
  A\cdot B = \sum_{r,s}A_{r}\cdot B_{s}
  \end{equation*}



.. math::
  :nowrap:

  \begin{equation*}
  A\wedge B = \sum_{r,s}A_{r}\wedge B_{s}
  \end{equation*}


Likewise the right (:math:`\lfloor`) and left (:math:`\rfloor`) contractions are defined as


.. math::
  :nowrap:

  \begin{equation*}
  A_{r}\lfloor B_{s} = \left \{ \begin{array}{cc}
    \left\langle A_{r}B_{s} \right\rangle_{r-s} &  r \ge s \\
                     0                          &  r < s \end{array} \right \}
  \end{equation*}

.. math::
  :nowrap:

  \begin{equation*}
  A_{r}\rfloor B_{s} = \left \{ \begin{array}{cc}
     \left\langle A_{r}B_{s} \right\rangle_{s-r} &  s \ge r \\
                      0                          &  s < r \end{array} \right \}
  \end{equation*}


and

.. math::
  :nowrap:

  \begin{equation*}
  A\lfloor B = \sum_{r,s}A_{r}\lfloor B_{s}
  \end{equation*}


.. math::
  :nowrap:

  \begin{equation*}
  A\rfloor B = \sum_{r,s}A_{r}\rfloor B_{s}
  \end{equation*}


The ``MV`` class function for the outer product of the multivectors ``mv1`` and
``mv2`` is ::

   @staticmethod
   def outer_product(mv1,mv2):
        product = MV()
        product.bladeflg = 1
        mv1.convert_to_blades()
        mv2.convert_to_blades()
        for igrade1 in MV.n1rg:
            if not isint(mv1.mv[igrade1]):
                pg1 = mv1.project(igrade1)
                for igrade2 in MV.n1rg:
                    igrade = igrade1+igrade2
                    if igrade <= MV.n:
                        if not isint(mv2.mv[igrade2]):
                            pg2 = mv2.project(igrade2)
                            pg1pg2 = pg1*pg2
                            product.add_in_place(pg1pg2.project(igrade))
        return(product)


The steps for calculating the outer product are:

#. Convert ``mv1`` and ``mv2`` to blade representation if they are not already
   in that form.

#. Project and loop through each grade ``mv1.mv[i1]`` and ``mv2.mv[i2]``.

#. Calculate the geometric product ``pg1*pg2``.

#. Project the ``i1+i2`` grade from ``pg1*pg2``.

#. Accumulate the results for each pair of grades in the input multivectors.

.. warning::

    In the  ``MV`` class we have overloaded the ``^`` operator to represent the outer
    product so that instead of calling the outer product function we can write ``mv1^ mv2``.
    Due to the precedence rules for python it is **absolutely essential** to enclose outer products
    in parenthesis.

For the inner product of the multivectors ``mv1`` and ``mv2`` the ``MV`` class
function is  ::

    @staticmethod
    def inner_product(mv1,mv2,mode='s'):
        """
        MV.inner_product(mv1,mv2) calculates the inner

        mode = 's' - symmetic (Doran & Lasenby)
        mode = 'l' - left contraction (Dorst)
        mode = 'r' - right contraction (Dorst)
        """
        if isinstance(mv1,MV) and isinstance(mv2,MV):
            product = MV()
            product.bladeflg = 1
            mv1.convert_to_blades()
            mv2.convert_to_blades()
            for igrade1 in range(MV.n1):
                if isinstance(mv1.mv[igrade1],numpy.ndarray):
                    pg1 = mv1.project(igrade1)
                    for igrade2 in range(MV.n1):
                        igrade = igrade1-igrade2
                        if mode == 's':
                            igrade = igrade.__abs__()
                        else:
                            if mode == 'l':
                                igrade = -igrade
                        if igrade >= 0:
                            if isinstance(mv2.mv[igrade2],numpy.ndarray):
                                pg2 = mv2.project(igrade2)
                                pg1pg2 = pg1*pg2
                                product.add_in_place(pg1pg2.project(igrade))
            return(product)
        else:
            if mode == 's':
                if isinstance(mv1,MV):
                    product = mv1.scalar_mul(mv2)
                if isinstance(mv2,MV):
                    product = mv2.scalar_mul(mv1)
            else:
                product = None
        return(product)


The inner product is calculated the same way as the outer product except that in
step 4, ``i1+i2`` is replaced by ``abs(i1-i2)`` or ``i1-i2`` for the right contraction or
``i2-i1`` for the left contraction.  If ``i1-i2`` is less than zero there is no contribution
to the right contraction.  If ``i2-i1`` is less than zero there is no contribution to the
left contraction.

.. warning::

    In the ``MV`` class we have overloaded the ``|`` operator for the inner product,
    ``>`` operator for the right contraction, and ``<`` operator for the left contraction.
    Instead of calling the inner product function we can write ``mv1|mv2``, ``mv1>mv2``, or
    ``mv1<mv2`` respectively for the inner product, right contraction, or left contraction.
    Again, due to the precedence rules for python it is **absolutely essential** to enclose inner
    products and/or contractions in parenthesis.


.. _reverse:

Reverse of Multivector
======================

If :math:`A` is the geometric product of :math:`r` vectors

.. math::
  :nowrap:

  \begin{equation*}
    A = a_{1}\dots a_{r}
  \end{equation*}


then the reverse of :math:`A` designated :math:`A^{\dagger}` is defined by

.. math::
  :nowrap:

  \begin{equation*}
    A^{\dagger} \equiv a_{r}\dots a_{1}.
  \end{equation*}


The reverse is simply the product with the order of terms reversed.  The reverse
of a sum of products is defined as the sum of the reverses so that for a general
multivector A we have

.. math::
  :nowrap:

  \begin{equation*}
    A^{\dagger} = \sum_{i=0}^{N} \left\langle A \right\rangle_{i}^{\dagger}
  \end{equation*}


but

.. _eq14:



.. math::
  :label: 2
  :nowrap:

  \begin{equation*}
    \left\langle A \right\rangle_{i}^{\dagger} = \left( -1\right)^{\frac{i\left( i-1\right)}{2}}\left\langle A \right\rangle_{i}
  \end{equation*}


which is proved by expanding the blade bases in terms of orthogonal vectors and
showing that equation :ref:`2 <eq14>` holds for the geometric product of orthogonal
vectors.

The reverse is important in the theory of rotations in :math:`n`-dimensions.  If
:math:`R` is the product of an even number of vectors and :math:`RR^{\dagger} = 1`
then :math:`RaR^{\dagger}` is a composition of rotations of the vector :math:`a`.
If :math:`R` is the product of two vectors then the plane that :math:`R` defines
is the plane of the rotation.  That is to say that :math:`RaR^{\dagger}` rotates the
component of :math:`a` that is projected into the plane defined by :math:`a` and
:math:`b` where :math:`R=ab`.  :math:`R` may be written
:math:`R = e^{\frac{\theta}{2}U}`, where :math:`\theta` is the angle of rotation
and :math:`u` is a unit blade :math:`\left( u^{2} = \pm 1\right)` that defines the
plane of rotation.


.. _recframe:

Reciprocal Frames
=================

If we have :math:`M` linearly independent vectors (a frame),
:math:`a_{1},\dots,a_{M}`, then the reciprocal frame is
:math:`a^{1},\dots,a^{M}` where :math:`a_{i}\cdot a^{j} = \delta_{i}^{j}`,
:math:`\delta_{i}^{j}` is the Kronecker delta (zero if :math:`i \ne j` and one
if :math:`i = j`). The reciprocal frame is constructed as follows:

.. math::
  :nowrap:

  \begin{equation*}
    E_{M} = a_{1}\wedge\dots\wedge a_{M}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
    E_{M}^{-1} = \displaystyle\frac{E_{M}}{E_{M}^{2}}
  \end{equation*}


Then

.. math::
  :nowrap:

  \begin{equation*}
    a^{i} = \left( -1\right)^{i-1}\left( a_{1}\wedge\dots\wedge \breve{a}_{i} \wedge\dots\wedge a_{M}\right) E_{M}^{-1}
  \end{equation*}


where :math:`\breve{a}_{i}` indicates that :math:`a_{i}` is to be deleted from
the product.  In the standard notation if a vector is denoted with a subscript
the reciprocal vector is denoted with a superscript. The multivector setup
function ``MV.setup(basis,metric,rframe)`` has the argument ``rframe`` with a
default value of ``False``.  If it is set to ``True`` the reciprocal frame of
the basis vectors is calculated. Additionaly there is the function
``reciprocal_frame(vlst,names='')`` external to the ``MV`` class that will
calculate the reciprocal frame of a list, ``vlst``, of vectors.  If the argument
``names`` is set to a space delimited string of names for the vectors the
reciprocal vectors will be given these names.


.. _deriv:

Geometric Derivative
====================

If :math:`F` is a multivector field that is a function of a vector
:math:`x = x^{i}\boldsymbol{\gamma}_{i}` (we are using the summation convention that
pairs of subscripts and superscripts are summed over the dimension of the vector
space) then the geometric derivative :math:`\nabla F` is given by (in this
section the summation convention is used):

.. math::
  :nowrap:

  \begin{equation*}
    \nabla F = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial F}{\partial x^{i}}
  \end{equation*}


If :math:`F_{R}` is a grade-:math:`R` multivector and
:math:`F_{R} = F_{R}^{i_{1}\dots i_{R}}\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}}`
then

.. math::
  :nowrap:

  \begin{equation*}
    \nabla F_{R} = \displaystyle\frac{\partial F_{R}^{i_{1}\dots i_{R}}}{\partial x^{j}}\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge
                   \dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)
  \end{equation*}


Note that
:math:`\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)`
can only contain grades :math:`R-1` and :math:`R+1` so that :math:`\nabla F_{R}`
also can only contain those grades. For a grade-:math:`R` multivector
:math:`F_{R}` the inner (div) and outer (curl) derivatives are defined as


.. math::
  :nowrap:

  \begin{equation*}
  \nabla\cdot F_{R} = \left < \nabla F_{R}\right >_{R-1}
  \end{equation*}


and

.. math::
  :nowrap:

  \begin{equation*}
  \nabla\wedge F_{R} = \left < \nabla F_{R}\right >_{R+1}
  \end{equation*}


For a general multivector function :math:`F` the inner and outer derivatives are
just the sum of the inner and outer dervatives of each grade of the multivector
function.

Curvilinear coordinates are derived from a vector function
:math:`x(\boldsymbol{\theta})` where
:math:`\boldsymbol{\theta} = \left(\theta_{1},\dots,\theta_{N}\right)` where the number of
coordinates is equal to the dimension of the vector space.  In the case of
3-dimensional spherical coordinates :math:`\boldsymbol{\theta} = \left( r,\theta,\phi \right)`
and the coordinate generating function :math:`x(\boldsymbol{\theta})` is

.. math::
  :nowrap:

  \begin{equation*}
  x =  r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+ r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+ r \cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
  \end{equation*}


A coordinate frame is derived from :math:`x` by
:math:`\boldsymbol{e}_{i} = \displaystyle\frac{\partial x}{\partial \theta^{i}}`.  The following show the frame for
spherical coordinates.

.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{e}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{e}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}} - r \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{e}_{{\phi}} = - r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}
  \end{equation*}


The coordinate frame generated in this manner is not necessarily normalized so
define a normalized frame by

.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{\hat{e}}_{i} = \displaystyle\frac{\boldsymbol{e}_{i}}{\sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}} = \displaystyle\frac{\boldsymbol{e}_{i}}{\left| \boldsymbol{e}_{i} \right|}
  \end{equation*}


This works for all :math:`\boldsymbol{e}_{i}^{2} \neq 0` since we have defined
:math:`\left| \boldsymbol{e}_{i} \right| = \sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}`.   For spherical
coordinates the normalized frame vectors are

.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{\hat{e}}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{\hat{e}}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}- \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  \boldsymbol{\hat{e}}_{{\phi}} = - \sin\left({\phi}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}
  \end{equation*}


The geometric derivative in curvilinear coordinates is given by

.. math::
  :nowrap:

  \begin{align*}
    \nabla F_{R} & = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial }{\partial x^{i}}\left( F_{R}^{i_{1}\dots i_{R}}
                     \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
                 & = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}\left( F_{R}^{i_{1}\dots i_{R}}
                     \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
                 & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
                     \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
                     F_{R}^{i_{1}\dots i_{R}}\boldsymbol{e^{j}}
                     \displaystyle\frac{\partial }{\partial \theta^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\
                 & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
                     \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
                     F_{R}^{i_{1}\dots i_{R}}C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\}
  \end{align*}


where

.. math::
  :nowrap:

  \begin{equation*}
  C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\} = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}
                                                                                               \left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)
  \end{equation*}


are the connection multivectors for the curvilinear coordinate system. For a
spherical coordinate system they are

.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_{r}\right\} =  \frac{2}{r}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_{\theta}\right\} = \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
                                                + \frac{1}{r}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\theta}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}{\boldsymbol{\boldsymbol{\hat{e}}_{r}}}\wedge\boldsymbol{\hat{e}}_{{\phi}}+ \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\hat{e}_{r}\wedge\hat{e}_{\theta}\right\} = - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
                                                        \boldsymbol{\hat{e}}_{r}+\frac{1}{r}\boldsymbol{\hat{e}}_{{\theta}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}\boldsymbol{\hat{e}}_{{\phi}}
                                                                            - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{2}{r}\boldsymbol{\hat{e}}_{r}\wedge
                                                                                   \boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  C\left\{\boldsymbol{\hat{e}}_r\wedge\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = 0
  \end{equation*}



Module Components
=================


Initializing Multivector Class
------------------------------

The multivector class is initialized with:


.. function:: MV.setup(basis,metric='',rframe=False,coords=None,debug=False,offset=0)

   The ``basis`` and ``metric`` parameters were described in section :ref:`vbm`. If
   ``rframe=True`` the reciprocal frame of the symbolic bases vectors is calculated.
   If ``debug=True`` the data structure required to initialize the :class:`MV` class
   are printer out. ``coords`` is a list of :class:`sympy` symbols equal in length to
   the number of basis vectors.  These symbols are used as the arguments of a
   multivector field as a function of position and for calculating the derivatives
   of a multivector field (if ``coords`` is defined then ``rframe`` is automatically
   set equal to ``True``). ``offset`` is an integer that is added to the multivector
   coefficient index. For example if one wishes to start labeling vector coefficient
   indexes at one instead of zero then set ``offset=1``.  Additionally, :func:`MV.setup`
   calculates the pseudo scalar, :math:`I` and its inverse, :math:`I^{-1}` and makes
   them available to the programmer as ``MV.I`` and ``MV.Iinv``.

After :func:`MV.setup` is run one can reinialize the :class:`MV` class with
curvilinear coordinates using:


.. function:: MV.rebase(x,coords,base_name,debug=False,debug_level=0)

   A typical usage of ``MV.rebase`` for generating spherical curvilinear coordinate
   is::

      metric = '1 0 0,0 1 0,0 0 1'
      gamma_x,gamma_y,gamma_z = MV.setup('gamma_x gamma_y gamma_z',metric,True)
      coords = r,theta,phi = sympy.symbols('r theta phi')
      x = r*(sympy.cos(theta)*gamma_z+sympy.sin(theta)*\
          (sympy.cos(phi)*gamma_x+sympy.sin(phi)*gamma_y))
      x.set_name('x')
      MV.rebase(x,coords,'e',True)

   The input parameters for ``MV.rebase`` are

*  ``x``: Vector function of coordinates (derivatives define curvilinear basis)

*  ``coords``: List of sympy symbols for curvilinear coordinates

*  ``debug``: If ``True`` printout (LaTeX) all quantities required for derivative calculation

*  ``debug_level``: Set to 0,1,2, or 3 to stop curvilinear calculation before all quatities are
   calculated.  This is done when debugging new curvilinear coordinate systems since simplification
   of expressions is not sufficiently automated to insure success of process of any coordinate system
   defined by vector function ``x``



   To date ``MV.rebase`` works for cylindrical and spherical coordinate systems in
   any number of dimensions  (until the execution time becomes too long).  To make
   it work for these systems required creating some hacks for expression
   simplification since both trigsimp and simplify were not general enough to
   perform the required simplification.


.. function:: MV.set_str_format(str_mode=0)

   If ``str_mode=0`` the string representation of the multivector contains no
   newline characters (prints on one line).   If ``str_mode=1`` the string
   representation of the multivector places a newline after each grade of the
   multivector  (prints one grade per line).  If ``str_mode=2`` the string
   representation of the multivector places a newline after each base of the
   multivector  (prints one base per line). In both cases bases with zero
   coefficients are not printed.

    .. note::

        This function directly affects the way multivectors are printed with the
        print command since it interacts with the :func:`__str__` function for the
        multivector class which is used by the ``print`` command.


        .. csv-table::
            :header: ``str_mode``, Effect on print
            :widths: 10, 50

            0, One multivector per line
            1, One grade per line
            2, One base per line

Instantiating a Multivector
---------------------------

Now that grades and bases have been described we can show all the ways that a
multivector can be instantiated. As an example assume that the multivector space
is initialized with ``e1,e2,e3 = MV.setup('e1 e2 e3')``.

So that multivectors
could be instantiated with statements such as (``a1``, ``a2``, and ``a3`` are
``sympy`` symbols)::

   x = a1*e1+a2*e2+a3*e3
   y = x*e1*e2
   z = x|y
   w = x^y

or with the multivector class constructor:


.. class:: MV(value='',mvtype='',mvname='',fct=False)

   ``mvname`` is a string that defines the name of the multivector for output
   purposes. ``value`` and  ``type`` are defined by the following table and ``fct`` is a
   switch that will convert the symbolic coefficients of a multivector to functions
   if coordinate variables have been defined when :func:`MV.setup` is called:

    .. csv-table::
        :delim: ;
        :header: ``mvtype``, ``value``, Result
        :widths: 10, 30, 30

        default;           default;                       Zero multivector
        'basisvector';     int i;                         :math:`\mbox{i}^{th}` basis vector
        'basisbivector';   int i;                         :math:`\mbox{i}^{th}` basis bivector
        'scalar';          symbol x;                      Symbolic scalar of value x
        ;                  string s;                      Symbolic scalar of value Symbol(s)
        ;                  int i;                         SymPy integer of value i
        'grade';           [int r, 1-D symbol array A];   X.grade(r) = A
        ;                  [int r, string s];             Symbolic grade r multivector
        'vector';          1-D symbol array A;            X.grade(1) = A
        ;                  string s;                      Symbolic vector
        'grade2';          1-D symbol array A;            X.grade(2) = A
        'pseudo';          symbol x;                      X.grade(n) = x
        'spinor';          string s;                      Symbolic even multivector


        default;           string s;                      Symbolic general multivector


   If the ``value`` argument has the option of being a string s then a general symbolic
   multivector will be constructed constrained by the value of ``mvtype``.  The string
   s will be the base name of the multivector symbolic coefficients.  If ``coords`` is
   not defined in :func:`MV.setup` the indices of the multivector bases are appended to
   the base name with a double underscore (superscript notation).  If ``coords`` is defined
   the coordinate names will replace the indices in the coefficient names.  For example if
   the base string is *A* and the coordinates *(x,y,z)* then the
   coefficients of a spinor in 3d space would be *A*, *A__xy*, *A__xz*, and *A__yz*.  If
   the :mod:`latex_ex` is used to print the multivector the coefficients would print as
   :math:`A`, :math:`A^{xy}`, :math:`A^{xz}`, and :math:`A^{yz}`.

   If the ``fct`` argrument of :func:`MV` is set to ``True`` and the ``coords`` argument in
   :func:`MV.setup` is defined the symbolic coefficients of the multivector are functions
   of the coordinates.


Basic Multivector Class Functions
---------------------------------


.. function:: __call__(self,igrade=0,ibase=0)

   ``__call__`` returns the the ``igrade``, ``ibase`` coefficient of the
   multivector.  The defaults return the scalar component of the multivector.


.. function:: convert_to_blades(self)

   Convert multivector from the base representation to the blade representation.
   If multivector is already in blade representation nothing is done.


.. function:: convert_from_blades(self)

   Convert multivector from the blade representation to the base representation.
   If multivector is already in base representation nothing is done.


.. function:: project(self,r)

   If r is a integer return the grade-:math:`r` components of the multivector. If
   r is a multivector return the grades of the multivector that correspond to the
   non-zero grades of r. For example if one is projecting a general multivector and
   r is a spinor, ``A.project(r)`` will return only the even grades of the multivector
   A since a spinor only has even grades that are non-zero.


.. function:: even(self)

   Return the even grade components of the multivector.


.. function:: odd(self)

   Return the odd grade components of the multivector.


.. function:: rev(self)

   Return the reverse of the multivector.  See section :ref:`reverse`.


.. function:: is_pure(self)

   Return true if multivector is pure grade (all grades but one are zero).


.. function:: diff(self,x)

   Return the partial derivative of the multivector function with respect to
   variable :math:`x`.


.. function:: grad(self)

   Return the geometric derivative of the multivector function.


.. function:: grad_ext(self)

   Return the outer (curl) derivative of the multivector function. Equivalent to
   :func:`curl`.


.. function:: grad_int(self)

   Return the inner (div) derivative of the multivector function. Equivalent to
   :func:`div`.

.. warning::

    If *A* is a vector field in three dimensions :math:`\nabla\cdot {\bf A}` = A.grad_int() = A.div(), but
    :math:`\nabla\times {\bf A}` = -MV.I*A.grad_ext() = -MV.I*A.curl(). Note that grad_int() lowers the grade
    of all blades by one grade and grad_ext() raises the grade of all blades by one.

.. function:: set_coef(self,grade,base,value)

   Set the multivector coefficient of index ``(grade,base)`` to ``value``.


SymPy Functions Applied Inplace to Multivector Coefficients
-----------------------------------------------------------

All the following fuctions belong to the :class:`MV` class and apply the
corresponding :mod:`sympy` function to each component of a multivector.  All
these functions perform the operations inplace (``None`` is returned)  on each
coefficient.  For example if you wished to simplify all the components of the
multivector ``A`` you would invoke ``A.simplify()``.  The argument list for each
function is the same as for the corresponding :mod:`sympy` function.  The only
function that differs in its action from the :mod:`sympy` version is
:func:`trigsimp` which is applied using the recursive option.


.. function:: collect(self,lst)


.. function:: sqrfree(self,lst)


.. function:: subs(self,*args)


.. function:: simplify(self)


.. function:: trigsimp(self)


.. function:: cancel(self)


.. function:: expand(self)


Helper Functions
----------------

These are functions in :mod:`GA`, but not in the multivector (:class:`MV`)
class.


.. function:: set_names(var_lst,var_str)

    :func:`set_names` allows one to name a list, ``var_lst``, of multivectors enmass.
    The names are in ``var_str``, a blank separated string of names.  An error is
    generated if the number of name is not equal to the length of ``var_lst``.


.. function:: reciprocal_frame(vlst,names='')

    :func:`reciprocal_frame` implements the proceedure described in section
    :ref:`recframe`.  ``vlst`` is a list of independent vectors that you wish the
    reciprocal frame calculated for. ``names`` is a blank separated string of names
    for the reciprocal vectors if names are required by you application.  The
    function returns a list containing the reciprocal vectors.


.. function:: trigsimp(f)

    In general :func:`sympy.trigsimp` will not catch all the trigonometric
    simplifications in an :mod:`sympy` expression, but by using the recursive
    option, more of them are caught. So :func:`trigsimp` uses this option by
    default.

.. function:: S(x)

    :func:`S` instantiates a scaler multivector of value ``x``, where ``x`` can be a
    :mod:`sympy` variable or integer.  This is just a shorthand method for
    constructing scalar multivectors and can be used when there is any ambiguity
    in a multivector expression as to whether a symbol or constant should be
    treated as a scalar multivector or not.

Examples
========


Algebra
-------

The following examples of geometric algebra (not calculus) are all in the file
:program:`testsymbolicGA.py` which is included in the sympy distribution
examples under the galbebra directory.  The section of code in the program for
each example is show with the respective output following the code section.


Example Header
^^^^^^^^^^^^^^

This is the header of :program:`testsymbolicGA.py` that allows access to the
required modules and also allow variables and certain multivectors to be
broadcast from the :mod:`GA` module to the main program.


.. literalinclude:: headerGAtest.py


Basic Geometric Algebra Operations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Example of basic geometric algebra operation of geometric, outer, and inner
products.


.. literalinclude:: BasicGAtest.py


Examples of Conformal Geometry
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Examples of comformal geometry [Lasenby,Chapter 10]. The examples show that
basic geometric entities (lines, circles, planes, and spheres) in three
dimensions can be represented by blades in a five dimensional (conformal) space.


.. literalinclude:: conformalgeometryGAtest.py


Calculation of Reciprocal Frame
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Example shows the calculation of the reciprocal frame for three arbitrary
vectors and verifies that the calculated reciprocal vectors have the correct
properties.


.. literalinclude:: reciprocalframeGAtest.py


Hyperbolic Geometry
^^^^^^^^^^^^^^^^^^^

Examples of calculation of distance in hyperbolic geometry [Lasenby,pp373-375].
This is a good example of the utility of not restricting the basis vector to be
orthogonal. Note that most of the calculation is simplifying a scalar
expression.


.. literalinclude:: hyperbolicGAtest.py


Calculus
--------

The calculus examples all use the extened LaTeX output module, ``latex_ex``, for
clarity.


Maxwell's Equations
^^^^^^^^^^^^^^^^^^^

In geometric calculus the equivalent of the electromagnetic tensor is
:math:`F = E\gamma_{0}+IB\gamma_{0}` where a spacetime vector is given by
:math:`x = x^{0}\gamma_{0}+x^{1}\gamma_{1}+x^{2}\gamma_{2}+x^{3}\gamma_{3}`
where :math:`x^{0} = ct`, the pseudoscalar
:math:`I = \gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}`, :math:`E` and :math:`B`
are four vectors where the time component is zero and the spatial components
equal to the electric and magnetic field components.  Then Maxwell's equations
can be all written as :math:`\nabla F = J` with :math:`J` the four current.
This example shows that this equations generates all of Maxwell's equations
correctly  (in our units:math:`c=1`) [Lasenby,pp229-231].

``Begin Program Maxwell.py``


.. literalinclude:: Maxwell.py

``End Program Maxwell.py``

``Begin Program Output``

:math:`I` Pseudo-Scalar

.. math::
  :nowrap:

  \begin{equation*}
  I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
  \end{equation*}


:math:`B` Magnetic Field Bi-Vector

.. math::
  :nowrap:

  \begin{equation*}
  B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z}
  \end{equation*}


:math:`F` Electric Field Bi-Vector

.. math::
  :nowrap:

  \begin{equation*}
  E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}
  \end{equation*}


:math:`E+IB` Electo-Magnetic Field Bi-Vector

.. math::
  :nowrap:

  \begin{equation*}
  F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z}
  \end{equation*}


:math:`J` Four Current

.. math::
  :nowrap:

  \begin{equation*}
  J =  {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z}
  \end{equation*}


Geometric Derivative of Electo-Magnetic Field Bi-Vector

.. math::
  :nowrap:

  \begin{align*}
  \nabla F & = \left( \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
           & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
           & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
           & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\
           & + \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
           & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
           & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
           & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
  \end{align*}


All Maxwell Equations are

.. math::
  :nowrap:

  \begin{equation*}
  \nabla F = J
  \end{equation*}


Div :math:`E` and Curl :math:`H` Equations

.. math::
  :nowrap:

  \begin{align*}
  <\nabla F>_1 -J & = \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
                  & + \left(-{J^{x}} - \partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
                  & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} -{J^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
                  & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z}
  \end{align*}




.. math::
  :nowrap:

  \begin{equation*}
    = 0
  \end{equation*}


Curl :math:`E` and Div :math:`B` equations

.. math::
  :nowrap:

  \begin{align*}
  <\nabla F>_3 & = \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\
               & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\
               & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\
               & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z}
  \end{align*}




.. math::
  :nowrap:

  \begin{equation*}
    = 0
  \end{equation*}


``End Program Output``


Dirac's Equation
^^^^^^^^^^^^^^^^

The geometric algebra/calculus allows one to formulate the Dirac equation in
real terms (no :math:`\sqrt{-1}`).  Spinors  :math:`\left(\Psi\right)` are even
multivectors in space time (Minkowski space with signature (1,-1,-1,-1)) and the
Dirac equation becomes
:math:`\nabla \Psi I \sigma_{z}-eA\Psi = m\Psi\gamma_{t}`.  All the terms in the
real equation are explined in Doran and Lasenby [Lasenby,pp281-283].

``Begin Program Dirac.py``


.. literalinclude:: Dirac.py

``End Program Dirac.py``

``Begin Program Output``

:math:`A` is 4-vector potential

.. math::
  :nowrap:

  \begin{equation*}
  A =  {A^{t}}{\gamma}_{t}+ {A^{x}}{\gamma}_{x}+ {A^{y}}{\gamma}_{y}+ {A^{z}}{\gamma}_{z}
  \end{equation*}


:math:`\boldsymbol{\psi}` is 8-component real spinor (even multi-vector)

.. math::
  :nowrap:

  \begin{equation*}
  {}\boldsymbol{{\psi}} = {{\psi}}+ {{\psi}^{tx}}{\gamma}_{t}{\gamma}_{x}+ {{\psi}^{ty}}{\gamma}_{t}{\gamma}_{y}+ {{\psi}^{xy}}{\gamma}_{x}{\gamma}_{y}+ {{\psi}^{tz}}{\gamma}_{t}{\gamma}_{z}+ {{\psi}^{xz}}{\gamma}_{x}{\gamma}_{z}+ {{\psi}^{yz}}{\gamma}_{y}{\gamma}_{z}+ {{\psi}^{txyz}}{\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
  \end{equation*}


Dirac equation in terms of real geometric algebra/calculus
:math:`\left(\nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi} = m\boldsymbol{\psi}\gamma_{t}\right)`
Spin measured with respect to :math:`z` axis

.. math::
  :nowrap:

  \begin{align*}
  \nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi}-m\boldsymbol{\psi}\gamma_{t}
    & = \left(-e {A^{y}} {{\psi}^{ty}} - m {{\psi}} - \partial_{z} {{\psi}^{txyz}} - \partial_{y} {{\psi}^{tx}} - e {A^{z}} {{\psi}^{tz}} - e {A^{x}} {{\psi}^{tx}} + \partial_{x} {{\psi}^{ty}} - e {A^{t}} {{\psi}} + \partial_{t} {{\psi}^{xy}}\right){\gamma}_{t} \\
    & + \left(-e {A^{y}} {{\psi}^{xy}} + \partial_{z} {{\psi}^{yz}} - e {A^{z}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tx}} + m {{\psi}^{tx}} - \partial_{x} {{\psi}^{xy}} + \partial_{y} {{\psi}} - e {A^{x}} {{\psi}} - \partial_{t} {{\psi}^{ty}}\right){\gamma}_{x} \\
    & + \left(-e {A^{y}} {{\psi}} + e {A^{x}} {{\psi}^{xy}} - e {A^{t}} {{\psi}^{ty}} - \partial_{x} {{\psi}} + \partial_{t} {{\psi}^{tx}} - \partial_{z} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{yz}} + m {{\psi}^{ty}} - \partial_{y} {{\psi}^{xy}}\right){\gamma}_{y} \\
    & + \left(-\partial_{z} {{\psi}^{xy}} + e {A^{x}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tz}} - \partial_{x} {{\psi}^{yz}} + \partial_{y} {{\psi}^{xz}} + e {A^{y}} {{\psi}^{yz}} + \partial_{t} {{\psi}^{txyz}} + m {{\psi}^{tz}} - e {A^{z}} {{\psi}}\right){\gamma}_{z} \\
    & + \left( \partial_{y} {{\psi}^{ty}} - e {A^{y}} {{\psi}^{tx}} + e {A^{x}} {{\psi}^{ty}} + \partial_{z} {{\psi}^{tz}} - e {A^{z}} {{\psi}^{txyz}} + \partial_{x} {{\psi}^{tx}} - e {A^{t}} {{\psi}^{xy}} - \partial_{t} {{\psi}} - m {{\psi}^{xy}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
    & + \left(-\partial_{y} {{\psi}^{tz}} + \partial_{x} {{\psi}^{txyz}} - e {A^{t}} {{\psi}^{xz}} + e {A^{x}} {{\psi}^{tz}} + e {A^{y}} {{\psi}^{txyz}} + \partial_{z} {{\psi}^{ty}} - m {{\psi}^{xz}} - e {A^{z}} {{\psi}^{tx}} - \partial_{t} {{\psi}^{yz}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
    & + \left(-e {A^{t}} {{\psi}^{yz}} - \partial_{z} {{\psi}^{tx}} + \partial_{x} {{\psi}^{tz}} - m {{\psi}^{yz}} + \partial_{y} {{\psi}^{txyz}} + \partial_{t} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{ty}} - e {A^{x}} {{\psi}^{txyz}} + e {A^{y}} {{\psi}^{tz}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
    & + \left(\partial_{z} {{\psi}} + e {A^{y}} {{\psi}^{xz}} + m {{\psi}^{txyz}} -\partial_{t} {{\psi}^{tz}} -\partial_{x} {{\psi}^{xz}} -e {A^{x}} {{\psi}^{yz}} -e {A^{t}} {{\psi}^{txyz}} -\partial_{y} {{\psi}^{yz}} -e {A^{z}} {{\psi}^{xy}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
  \end{align*}




.. math::
  :nowrap:

  \begin{equation*}
    = 0
  \end{equation*}


``End Program Output``


Spherical Coordinates
^^^^^^^^^^^^^^^^^^^^^

Curvilinear coodinates are implemented as shown in section :ref:`deriv`.  The
gradient of a scalar function and the divergence and curl of a vector function
(:math:`-I\left(\nabla\wedge A\right)` is the curl in three dimensions in the notation of
geometric algebra) to demonstrate the formulas derived in section :ref:`deriv`.

``Begin Program coords.py``


.. literalinclude:: coords.py

``End Program coords.py``

``Begin Program Output``

Gradient of Scalar Function :math:`\psi`

.. math::
  :nowrap:

  \begin{equation*}
  \nabla\psi =  \partial_{r} {{\psi}}{}\boldsymbol{e}_{r}+\frac{1 \partial_{{\theta}} {{\psi}}}{r}{}\boldsymbol{e}_{{\theta}}+\frac{1 \partial_{{\phi}} {{\psi}}}{r \operatorname{sin}\left({\theta}\right)}{}\boldsymbol{e}_{{\phi}}
  \end{equation*}


Div and Curl of Vector Function :math:`A`

.. math::
  :nowrap:

  \begin{equation*}
  A =  {A^{r}}{}\boldsymbol{e}_{r}+ {A^{{\theta}}}{}\boldsymbol{e}_{{\theta}}+ {A^{{\phi}}}{}\boldsymbol{e}_{{\phi}}
  \end{equation*}




.. math::
  :nowrap:

  \begin{equation*}
  \nabla \cdot A =  \left(\frac{{A^{{\theta}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} + 2 \frac{{A^{r}}}{r} + \partial_{r} {A^{r}} + \frac{\partial_{{\theta}} {A^{{\theta}}}}{r} + \frac{\partial_{{\phi}} {A^{{\phi}}}}{r \operatorname{sin}\left({\theta}\right)}\right)
  \end{equation*}




.. math::
  :nowrap:

  \begin{align*}
  -I\left(\nabla \wedge A\right) & = \left(\frac{\partial_{{\theta}} {A^{{\phi}}}}{r} + \frac{{A^{{\phi}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} -\frac{\partial_{{\phi}} {A^{{\theta}}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{r} \\
                                 & - \left(\frac{{A^{{\phi}}}}{r} + \partial_{r} {A^{{\phi}}} -\frac{\partial_{{\phi}} {A^{r}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{{\theta}} \\
                                 & + \left(\frac{{A^{{\theta}}}}{r} + \partial_{r} {A^{{\theta}}} -\frac{\partial_{{\theta}} {A^{r}}}{r}\right){}\boldsymbol{e}_{{\phi}}
  \end{align*}


``End Program Output``


.. seealso::

   `Hestenes <http://geocalc.clas.asu.edu/html/CA_to_GC.html>`_
      ``Clifford Algebra to Geometric Calculus`` by D.Hestenes and G. Sobczyk, Kluwer
      Academic Publishers, 1984.

   `Lasenby <http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm>`_
      ``Geometric Algebra for Physicists`` by C. Doran and A. Lasenby, Cambridge
      University Press, 2003.


