What is Geometric Algebra?¶
Basics of Geometric Algebra¶
Geometric algebra is the Clifford algebra of a real finite dimensional vector space or the algebra that results when the 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 , . The elements of the geometric algebra are called multivectors and consist of the linear combination of scalars, vectors, and the geometric product of two or more vectors. The additional axioms for the geometric algebra are that for any vectors \(a\), \(b\), and \(c\) in the base vector space (,p85):
If the dot (inner) product of two vectors is defined by (,p86)
then we have
Thus \(a\cdot b\) is real. The objects generated from linear combinations of the geometric products of vectors are called multivectors. If a basis for the underlying vector space are the vectors \({\left \{{{{\eb}}_{1},\dots,{{\eb}}_{n}} \rbrc}\) (we use boldface \(\eb\)’s to denote basis vectors) a complete basis for the geometric algebra is given by the scalar \(1\), the vectors \({{\eb}}_{1},\dots,{{\eb}}_{n}\) and all geometric products of vectors
Each base of the complete basis is represented by a non-commutative symbol (except for the scalar 1) with name \({{\eb}}_{i_{1}}\dots {{\eb}}_{i_{r}}\) so that the general multivector \({\boldsymbol{A}}\) is represented by (\(A\) is the scalar part of the multivector and the \(A^{i_{1},\dots,i_{r}}\) are scalars)
The critical operation in setting up the geometric algebra is reducing the geometric product of any two bases to a linear combination of bases so that we can calculate a multiplication table for the bases. Since the geometric product is associative we can use the operation (by definition for two vectors \(a\cdot b \equiv (ab+ba)/2\) which is a scalar)
These processes are repeated until every basis list in \({\boldsymbol{A}}\) is in normal (ascending) order with no repeated elements. As an example consider the following
which results from repeated application of eq. (\(\ref{reduce}\)). If the product of basis vectors contains repeated factors eq. (\(\ref{reduce}\)) can be used to bring the repeated factors next to one another so that if \({{\eb}}_{i_{j}} = {{\eb}}_{i_{j+1}}\) then \({{\eb}}_{i_{j}}{{\eb}}_{i_{j+1}} = {{\eb}}_{i_{j}}\cdot {{\eb}}_{i_{j+1}}\) which is a scalar that commutes with all the terms in the product and can be brought to the front of the product. Since every repeated pair of vectors in a geometric product of \(r\) factors reduces the number of non-commutative factors in the product by \(r-2\). The number of bases in the multivector algebra is \(2^{n}\) and the number containing \(r\) factors is \({n\choose r}\) which is the number of combinations or \(n\) things taken \(r\) at a time (binomial coefficient).
The other construction required for formulating the geometric algebra is the outer or wedge product (symbol \({\wedge}\)) of \(r\) vectors denoted by \(a_{1}{\wedge}\dots{\wedge}a_{r}\). The wedge product of \(r\) vectors is called an \(r\)-blade and is defined by (,p86)
where \(\epsilon^{i_{j_{1}}\dots i_{j_{r}}}\) is the contravariant permutation symbol which is \(+1\) for an even permutation of the superscripts, \(0\) if any superscripts are repeated, and \(-1\) for an odd permutation of the superscripts. From the definition \(a_{1}{\wedge}\dots{\wedge}a_{r}\) is antisymmetric in all its arguments and the following relation for the wedge product of a vector \(a\) and an \(r\)-blade \(B_{r}\) can be derived
Using eq. (\(\ref{wedge}\)) one can represent the wedge product of all the basis vectors in terms of the geometric product of all the basis vectors so that one can solve (the system of equations is lower diagonal) for the geometric product of all the basis vectors in terms of the wedge product of all the basis vectors. Thus a general multivector \({\boldsymbol{B}}\) can be represented as a linear combination of a scalar and the basis blades.
Using the blades \({{\eb}}_{i_{1}}{\wedge}{{\eb}}_{i_{2}}{\wedge}\dots{\wedge}{{\eb}}_{r}\) creates a graded algebra where \(r\) is the grade of the basis blades. The grade-\(r\) part of \({\boldsymbol{B}}\) is the linear combination of all terms with grade \(r\) basis blades.
Grade Projection¶
The scalar part of \({\boldsymbol{B}}\) is defined to be grade-\(0\). Now that the blade expansion of \({\boldsymbol{B}}\) is defined we can also define the grade projection operator \({\left <{{\boldsymbol{B}}} \right >_{r}}\) by
and
Multivector Products¶
Then if \({\boldsymbol{A}}_{r}\) is an \(r\)-grade multivector and \({\boldsymbol{B}}_{s}\) is an \(s\)-grade multivector we have
and define (,p6)
where \({\boldsymbol{A}}_{r}\cdot{\boldsymbol{B}}_{s}\) is called the dot or inner product of two pure grade multivectors. For the case of two non-pure grade multivectors
Two other products, the left (\(\rfloor\)) and right (\(\lfloor\)) contractions, are defined by
Reverse of Multivector¶
A final operation for multivectors is the reverse. If a multivector \({\boldsymbol{A}}\) is the geometric product of \(r\) vectors (versor) so that \({\boldsymbol{A}} = a_{1}\dots a_{r}\) the reverse is defined by
where for a general multivector we have (the the sum of the reverse of versors)
note that if \({\boldsymbol{A}}\) is a versor then \({\boldsymbol{A}}{\boldsymbol{A}}^{{\dagger}}\in\Re\) and (\(AA^{{\dagger}} \ne 0\))
The reverse is important in the theory of rotations in \(n\)-dimensions. If \(R\) is the product of an even number of vectors and \(RR^{{\dagger}} = 1\) then \(RaR^{{\dagger}}\) is a composition of rotations of the vector \(a\). If \(R\) is the product of two vectors then the plane that \(R\) defines is the plane of the rotation. That is to say that \(RaR^{{\dagger}}\) rotates the component of \(a\) that is projected into the plane defined by \(a\) and \(b\) where \(R=ab\). \(R\) may be written \(R = e^{\frac{\theta}{2}U}\), where \(\theta\) is the angle of rotation and \(U\) is a unit blade \(\lp U^{2} = \pm 1\rp\) that defines the plane of rotation.
Reciprocal Frames¶
If we have \(M\) linearly independent vectors (a frame), \(a_{1},\dots,a_{M}\), then the reciprocal frame is \(a^{1},\dots,a^{M}\) where \(a_{i}\cdot a^{j} = \delta_{i}^{j}\), \(\delta_{i}^{j}\) is the Kronecker delta (zero if \(i \ne j\) and one if \(i = j\)). The reciprocal frame is constructed as follows:
Then
where \(\breve{a}_{i}\) indicates that \(a_{i}\) 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 set of reciprocal vectors will be calculated if a coordinate set is given when a geometric algebra is instantiated since they are required for geometric differentiation when the Ga member function Ga.mvr() is called to return the reciprocal basis in terms of the basis vectors.
Manifolds and Submanifolds¶
A \(m\)-dimensional vector manifold[4], \(\mathcal{M}\), is defined by a coordinate tuple (tuples are indicated by the vector accent “\(\vec{\;\;\;}\)”)
and the differentiable mapping (\(U^{m}\) is an \(m\)-dimensional subset of \(\Re^{m}\))
where \(\mathcal{V}\) is a vector space with an inner product[5] (\(\cdot\)) and is of \({{\dim}\lp {\mathcal{V}} \rp } \ge m\).
Then a set of basis vectors for the tangent space of \(\mathcal{M}\) at \(\vec{x}\), \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\), are
and
A \(n\)-dimensional (\(n\le m\)) submanifold \(\mathcal{N}\) of \(\mathcal{M}\) is defined by a coordinate tuple
and a differentiable mapping
Then the basis vectors for the tangent space \({{{\mathcal{T}_{\vec{u}}}\lp {\mathcal{N}} \rp }}\) are (using \({{{{\eb}}^{\mathcal{N}}}\lp {\vec{u}} \rp } = {{{{\eb}}^{\mathcal{M}}}\lp {{{\vec{x}}\lp {\vec{u}} \rp }} \rp }\) and the chain rule)[6]
and
Going back to the base manifold, \(\mathcal{M}\), note that the mapping \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{n}\subseteq\Re^{n}\rightarrow \mathcal{V}\) allows us to calculate an unnormalized pseudo-scalar for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\),
With the pseudo-scalar we can define a projection operator from \(\mathcal{V}\) to the tangent space of \(\mathcal{M}\) by
In fact for each tangent space \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) we can define a geometric algebra \({{\mathcal{G}}\lp {{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}} \rp }\) with pseudo-scalar \(I^{\mathcal{M}}\) so that if \(A \in {{\mathcal{G}}\lp {\mathcal{V}} \rp }\) then
and similarly for the submanifold \(\mathcal{N}\).
If the embedding \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{n}\subseteq\Re^{n}\rightarrow \mathcal{V}\) is not given, but the metric tensor \({{g_{ij}^{\mathcal{M}}}\lp {\vec{x}} \rp }\) is given the geometric algebra of the tangent space can be constructed. Also the derivatives of the basis vectors of the tangent space can be calculated from the metric tensor using the Christoffel symbols, \({{\Gamma_{ij}^{k}}\lp {\vec{u}} \rp }\), where the derivatives of the basis vectors are given by
If we have a submanifold, \(\mathcal{N}\), defined by eq. (\(\ref{eq_79}\)) we can calculate the metric of \(\mathcal{N}\) from eq. (\(\ref{eq_81}\)) and hence construct the geometric algebra and calculus of the tangent space, \({{{\mathcal{T}_{\vec{u}}}\lp {\mathcal{N}} \rp }}\subseteq {{{\mathcal{T}_{{{\vec{x}}\lp {\vec{u}} \rp }}}\lp {\mathcal{M}} \rp }}\).
Note:
If the base manifold is normalized (use the hat symbol to denote normalized tangent vectors, \(\hat{{\eb}}_{i}^{\mathcal{M}}\), and the resulting metric tensor, \(\hat{g}_{ij}^{\mathcal{M}}\)) we have \(\hat{{\eb}}_{i}^{\mathcal{M}}\cdot\hat{{\eb}}_{i}^{\mathcal{M}} = \pm 1\) and \(\hat{g}_{ij}^{\mathcal{M}}\) does not posses enough information to calculate \(g_{ij}^{\mathcal{N}}\). In that case we need to know \(g_{ij}^{\mathcal{M}}\), the metric tensor of the base manifold before normalization. Likewise, for the case of a vector manifold unless the mapping, \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{m}\subseteq\Re^{m}\rightarrow \mathcal{V}\), is constant the tangent vectors and metric tensor can only be normalized after the fact (one cannot have a mapping that automatically normalizes all the tangent vectors).
Geometric Derivative¶
The directional derivative of a multivector field \({{F}\lp {x} \rp }\) is defined by (\(a\) is a vector and \(h\) is a scalar)
Note that \(a\cdot\nabla_{x}\) is a scalar operator. It will give a result containing only those grades that are already in \(F\). \({\lp {a\cdot\nabla_{x}} \rp }F\) is the best linear approximation of \({{F}\lp {x} \rp }\) in the direction \(a\). Equation (\(\ref{eq_50}\)) also defines the operator \(\nabla_{x}\) which for the basis vectors, \({\left \{{{\eb}_{i}} \rbrc}\), has the representation (note that the \({\left \{{{\eb}^{j}} \rbrc}\) are reciprocal basis vectors)
If \(F_{r}\) is a \(r\)-grade multivector (if the independent vector, \(x\), is obvious we suppress it in the notation and just write \(\nabla\)) and \(F_{r} = F_{r}^{i_{1}\dots i_{r}}{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}\) then
Note that \({\eb}^{j}\lp {\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}} \rp\) can only contain grades \(r-1\) and \(r+1\) so that \(\nabla F_{r}\) also can only contain those grades. For a grade-\(r\) multivector \(F_{r}\) the inner (div) and outer (curl) derivatives are
and
For a general multivector function \(F\) the inner and outer derivatives are just the sum of the inner and outer derivatives of each grade of the multivector function.
Geometric Derivative on a Manifold¶
In the case of a manifold the derivatives of the \({\eb}_{i}\)’s are functions of the coordinates, \({\left \{{x^{i}} \rbrc}\), so that the geometric derivative of a \(r\)-grade multivector field is
where the multivector functions \({\eb}^{i}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp }\) are the connection for the manifold.[7]
The directional (material/convective) derivative, \({\lp {v\cdot\nabla} \rp }F_{r}\) is given by
so that the multivector connection functions for the directional derivative are \({{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp }\). Be careful and note that \({\lp {v\cdot\nabla} \rp } F_{r} \ne v\cdot {\lp {\nabla F_{r}} \rp }\) since the dot and geometric products are not associative with respect to one another (\(v\cdot\nabla\) is a scalar operator).
Normalizing Basis for Derivatives¶
The basis vector set, \({\left \{ {{\eb}_{i}} \rbrc}\), is not in general normalized. We define a normalized set of basis vectors, \({\left \{{{\boldsymbol{\hat{e}}}_{i}} \rbrc}\), by
This works for all \({\eb}_{i}^{2} \neq 0\). Note that \({\boldsymbol{\hat{e}}}_{i}^{2} = \pm 1\).
Thus the geometric derivative for a set of normalized basis vectors is (where \(F_{r} = F_{r}^{i_{1}\dots i_{r}} \bm{\hat{e}}_{i_{1}}\W\dots\W\bm{\hat{e}}_{i_{r}}\) and [no summation] \(\hat{F}_{r}^{i_{1}\dots i_{r}} = F_{r}^{i_{1}\dots i_{r}} \abs{\bm{\hat{e}}_{i_{1}}}\dots\abs{\bm{\hat{e}}_{i_{r}}}\)).
To calculate \({\eb}^{i}\) in terms of the \({\boldsymbol{\hat{e}}}_{i}\)’s we have
This is the general (non-orthogonal) formula. If the basis vectors are orthogonal then (no summation over repeated indexes)
Additionally, one can calculate the connection of the normalized basis as follows
where \({{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}}\) is expanded in terms of the \({\boldsymbol{\hat{e}}}_{i}\)’s.
Linear Differential Operators¶
First a note on partial derivative notation. We shall use the following notation for a partial derivative where the manifold coordinates are \(x_{1},\dots,x_{n}\):
If \(j_{k}=0\) the partial derivative with respect to the \(k^{th}\) coordinate is not taken. If \(j_{k} = 0\) for all \(1 \le k \le n\) then the partial derivative operator is the scalar one. If we consider a partial derivative where the \(x\)’s are not in normal order such as
and the \(i_{k}\)’s are not in ascending order. The derivative can always be put in the form in eq (\(\ref{eq_66a}\)) since the order of differentiation does not change the value of the partial derivative (for the smooth functions we are considering). Additionally, using our notation the product of two partial derivative operations is given by
A general general multivector linear differential operator is a linear combination of multivectors and partial derivative operators denoted by
Equation (\(\ref{eq_66b}\)) is the normal form of the differential operator in that the partial derivative operators are written to the right of the multivector coefficients and do not operate upon the multivector coefficients. The operator of eq (\(\ref{eq_66b}\)) can operate on mulitvector functions, returning a multivector function via the following definitions.
\(F\) as
, or
where the \(D^{j_{1}\dots j_{n}}\) are multivector functions and \(\circ\) is any of the multivector multiplicative operations.
Equations (\(\ref{eq_67a}\)) and (\(\ref{eq_68a}\)) are not the most general multivector linear differential operators, the most general would be
where \({{D^{j_{1}\dots j_{n}}}\lp {} \rp }\) are linear multivector functionals.
The definition of the sum of two differential operators is obvious since any multivector operator, \(\circ\), is a bilinear operator \({\lp {{\lp {D_{A}+D_{B}} \rp }\circ F = D_{A}\circ F+D_{B}\circ F} \rp }\), the product of two differential operators \(D_{A}\) and \(D_{B}\) operating on a multivector function \(F\) is defined to be (\(\circ_{1}\) and \(\circ_{2}\) are any two multivector multiplicative operations)
where we have used the fact that the \(\partial\) operator is a scalar operator and commutes with \(\circ_{1}\) and \(\circ_{2}\).
Thus for a pure operator product \(D_{A}\circ D_{B}\) we have
and the form of eq (\(\ref{eq_71a}\)) is the same as eq (\(\ref{eq_67a}\)). The basis of eq (\(\ref{eq_71a}\)) is that the \(\partial\) operator operates on all object to the right of it as products so that the product rule must be used in all differentiations. Since eq (\(\ref{eq_71a}\)) puts the product of two differential operators in standard form we also evaluate \(F\circ_{2}{\lp {D_{A}\circ_{1}D_{B}} \rp }\).
We now must distinguish between the following cases. If \(D\) is a differential operator and \(F\) a multivector function should \(D\circ F\) and \(F\circ D\) return a differential operator or a multivector. In order to be consistent with the standard vector analysis we have \(D\circ F\) return a multivector and \(F\circ D\) return a differential operator. Then we define the complementary differential operator \(\bar{D}\) which is identical to \(D\) except that \(\bar{D}\circ F\) returns a differential operator according to eq (\(\ref{eq_71a}\))[8] and \(F\circ\bar{D}\) returns a multivector according to eq (\(\ref{eq_68a}\)).
A general differential operator is built from repeated applications of the basic operator building blocks \({\lp {\bar{\nabla}\circ A} \rp }\), \({\lp {A\circ\bar{\nabla}} \rp }\), \({\lp {\bar{\nabla}\circ\bar{\nabla}} \rp }\), and \({\lp {A\pm \bar{\nabla}} \rp }\). Both \(\nabla\) and \(\bar{\nabla}\) are represented by the operator
but are flagged to produce the appropriate result.
In the our notation the directional derivative operator is \(a\cdot\nabla\), the Laplacian \(\nabla\cdot\nabla\) and the expression for the Riemann tensor, \(R^{i}_{jkl}\), is
We would use the complement if we wish a quantum mechanical type commutator defining
, or if we wish to simulate the dot notation (Doran and Lasenby)
Split Differential Operator¶
To implement the general “dot” notation for differential operators in python is not possible. Another type of symbolic notation is required. I propose what one could call the “split differential operator.” For \(\nabla\) denote the corresponding split operator by two operators \({{\nabla}_{\mathcal{G}}}\) and \({{\nabla}_{\mathcal{D}}}\) where in practice \({{\nabla}_{\mathcal{G}}}\) is a tuple of vectors and \({{\nabla}_{\mathcal{D}}}\) is a tuple of corresponding partial derivatives. Then the equivalent of the “dot” notation would be
We are using the \(\mathcal{G}\) subscript to indicate the geometric algebra parts of the multivector differential operator and the \(\mathcal{D}\) subscript to indicate the scalar differential operator parts of the multivector differential operator. An example of this notation in 3D Euclidean space is
To implement \({{\nabla}_{\mathcal{G}}}\) and \({{\nabla}_{\mathcal{D}}}\) we have in the example
Then the final evaluation is
which could be called the “dot” product of two tuples. Note that \(\nabla = {{\nabla}_{\mathcal{G}}}{{\nabla}_{\mathcal{D}}}\) and \(\dot{F}\dot{\nabla} = F\bar{\nabla} = {\lp {{{\nabla}_{\mathcal{D}}}F} \rp }{{\nabla}_{\mathcal{G}}}\).
For the general multivector differential operator, \(D\), the split operator parts are \({{D}_{\mathcal{G}}}\), a tuple of basis blade multivectors and \({{D}_{\mathcal{D}}}\), a tuple of scalar differential operators that correspond to the coefficients of the basis-blades in the total operator \(D\) so that
If the index set for the basis blades of a geometric algebra is denoted by \({\left \{{n} \rbrc}\) where \({\left \{{n} \rbrc}\) contains \(2^{n}\) indices for an \(n\) dimensional geometric algebra then the most general multivector differential operator can be written[9]
or
The implementation of equations [splitopV] and [splitopM] is described in sections [makeMV] and [makeMVD].
Linear Transformations/Outermorphisms¶
In the tangent space of a manifold, \(\mathcal{M}\), (which is a vector space) a linear transformation is the mapping \(\underline{T}\colon{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\rightarrow{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) (we use an underline to indicate a linear transformation) where for all \(x,y\in {{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) and \(\alpha\in\Re\) we have
The outermorphism induced by \(\underline{T}\) is defined for \(x_{1},\dots,x_{r}\in{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) where \(\newcommand{\f}[2]{{#1}\lp {#2} \rp } \newcommand{\Tn}[2]{\f{\mathcal{T}_{#2}}{#1}} r\le\f{\dim}{\Tn{\mathcal{M}}{\vec{x}}}\)
If \(I\) is the pseudo scalar for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) we also have the following definitions for determinate, trace, and adjoint (\(\overline{T}\)) of \(\underline{T}\)
If \({\left \{{{{\eb}}_{i}} \rbrc}\) is a basis for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) then we can represent \(\underline{T}\) with the matrix \(\underline{T}_{i}^{j}\) used as follows (Einstein summation convention as usual) -
The let \({\lp {\underline{T}^{-1}} \rp }_{m}^{n}\) be the inverse matrix of \(\underline{T}_{i}^{j}\) so that \({\lp {\underline{T}^{-1}} \rp }_{m}^{k}\underline{T}_{k}^{j} = \delta^{j}_{m}\) and
and calculate
Thus if eq \(\ref{eq_85a}\) is used to define the \(\underline{T}_{i}^{j}\) then the linear transformation defined by the matrix \({\lp {\underline{T}^{-1}} \rp }_{m}^{n}\) is the inverse of \(\underline{T}\).
In eq. (\(\ref{eq_85}\)) the matrix, \(\underline{T}_{i}^{j}\), only has it’s usual meaning if the \({\left \{{{{\eb}}_{i}} \rbrc}\) form an orthonormal Euclidean basis (Minkowski spaces not allowed). Equations (\(\ref{eq_82}\)) through (\(\ref{eq_84}\)) become
A important form of linear transformation with a simple representation is the spinor transformation. If \(S\) is an even multivector we have \(SS^{{\dagger}} = \rho^{2}\), where \(\rho^{2}\) is a scalar. Then \(S\) is a spinor transformation is given by (\(v\) is a vector)
if \({{S}\lp {v} \rp }\) is a vector and
Thus
One more topic to consider is whether or not \(T^{i}_{j}\) should be called the matrix representation of \(T\) ? The reason that this is a question is that for a general metric \(g_{ij}\) is that because of the dependence of the dot product on the metric \(T^{i}_{j}\) does not necessarily show the symmetries of the underlying transformation \(T\). Consider the expression
It is
that has the proper symmetry for self adjoint transformations \((a\cdot{{T}\lp {b} \rp } = b\cdot{{T}\lp {a} \rp })\) in the sense that if \(T = \overline{T}\) then \(T_{ij} = T_{ji}\). Of course if we are dealing with a manifold where the \(g_{ij}\)’s are functions of the coordinates then the matrix representation of a linear transformation will also be a function of the coordinates. Assuming we use \(T_{ij}\) for the matrix representation of the linear transformation, \(T\), then if we given the matrix representation, \(T_{ij}\), we can construct the linear transformation given by \(T^{i}_{j}\) as follows
Any program/code that represents \(T\) should allow one to define \(T\) in terms of \(T_{ij}\) or \(T_{j}^{l}\) and likewise given a linear transformation \(T\) obtain both \(T_{ij}\) and \(T_{j}^{l}\) from it. Please note that these considerations come into play for any non-Euclidean metric with respect to the trace and adjoint of a linear transformation since calculating either requires a dot product.
Multilinear Functions¶
A multivector multilinear function[10] is a multivector function \({{T}\lp {A_{1},\dots,A_{r}} \rp }\) that is linear in each of it arguments[11] (it could be implicitly non-linearly dependent on a set of additional arguments such as the position coordinates, but we only consider the linear arguments). \(T\) is a tensor of degree \(r\) if each variable \(A_{j}\) is restricted to the vector space \(\mathcal{V}_{n}\). More generally if each \(A_{j}\in{{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\) (the geometric algebra of \(\mathcal{V}_{n}\)), we call \(T\) an extensor of degree-\(r\) on \({{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\).
If the values of \({{T} \lp {a_{1},\dots,a_{r}} \rp }\) \(\lp a_{j}\in\mathcal{V}_{n}\;\forall\; 1\le j \le r \rp\) are \(s\)-vectors (pure grade \(s\) multivectors in \({{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\)) we say that \(T\) has grade \(s\) and rank \(r+s\). A tensor of grade zero is called a multilinear form.
In the normal definition of tensors as multilinear functions the tensor is defined as a mapping
so that the standard tensor definition is an example of a grade zero degree/rank$ r $ tensor in our definition.
Algebraic Operations¶
The properties of tensors are (\(\alpha\in\Re\), \(a_{j},b\in\mathcal{V}_{n}\), \(T\) and \(S\) are tensors of rank \(r\), and \(\circ\) is any multivector multiplicative operation)
Now let \(T\) be of rank \(r\) and \(S\) of rank \(s\) then the product of the two tensors is
where “\(\circ\)” is any multivector multiplicative operation.
Covariant, Contravariant, and Mixed Representations¶
The arguments (vectors) of the multilinear function can be represented in terms of the basis vectors or the reciprocal basis vectors
Equation ([vrep]) gives \(a_{j}\) in terms of the basis vectors and eq ([rvrep]) in terms of the reciprocal basis vectors. The index \(j\) refers to the argument slot and the indices \(i_{j}\) the components of the vector in terms of the basis. The covariant representation of the tensor is defined by
\(\newcommand{\indices}[1]{#1}\begin{aligned} T\indices{_{i_{1}\dots i_{r}}} \equiv& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{r}}} \rp } \\ {{T}\lp {a_{1},\dots,a_{r}} \rp } =& {{T}\lp {a^{i_{1}}{{\eb}}_{i_{1}},\dots,a^{i_{r}}{{\eb}}_{i_{r}}} \rp } \nonumber \\ =& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{r}}} \rp }a^{i_{1}}\dots a^{i_{r}} \nonumber \\ =& T\indices{_{i_{1}\dots i_{r}}}a^{i_{1}}\dots a^{i_{r}}.\end{aligned}\)$
Likewise for the contravariant representation
One could also have a mixed representation
In the representation of \(T\) one could have any combination of covariant (lower) and contravariant (upper) indexes.
To convert a covariant index to a contravariant index simply consider
Similarly one could lower an upper index with \(g_{i_{j}k_{j}}\).
Contraction and Differentiation¶
The contraction of a tensor between the \(j^{th}\) and \(k^{th}\) variables (slots) is
This operation reduces the rank of the tensor by two. This definition gives the standard results for metric contraction which is proved as follows for a rank \(r\) grade zero tensor (the circumflex “\(\breve{\:\:}\)” indicates that a term is to be deleted from the product).
Equation (\(\ref{eq108}\)) is the correct formula for the metric contraction of a tensor.
If we have a mixed representation of a tensor, \(T\indices{_{i_{1}\dots}{}^{i_{j}}{}_{\dots i_{k}\dots i_{r}}}\), and wish to contract between an upper and lower index (\(i_{j}\) and \(i_{k}\)) first lower the upper index and then use eq (\(\ref{eq108}\)) to contract the result. Remember lowering the index does not change the tensor, only the representation of the tensor, while contraction results in a new tensor. First lower index
Now contract between \(i_{j}\) and \(i_{k}\) and use the properties of the metric tensor.
Equation ([114a]) is the standard formula for contraction between upper and lower indexes of a mixed tensor.
Finally if \({{T}\lp {a_{1},\dots,a_{r}} \rp }\) is a tensor field (implicitly a function of position) the tensor derivative is defined as
assuming the \(a^{i_{j}}\) coefficients are not a function of the coordinates.
This gives for a grade zero rank \(r\) tensor
From Vector to Tensor¶
A rank one tensor is a vector since it satisfies all the axioms for a vector space, but a vector in not necessarily a tensor since not all vectors are multilinear (actually in the case of vectors a linear function) functions. However, there is a simple isomorphism between vectors and rank one tensors defined by the mapping \({{v}\lp {a} \rp }:\mathcal{V}\rightarrow\Re\) such that if \(v,a \in\mathcal{V}\)
So that if \(v = v^{i}{{\eb}}_{i} = v_{i}{{\eb}}^{i}\) the covariant and contravariant representations of \(v\) are (using \({{\eb}}^{i}\cdot{{\eb}}_{j} = \delta^{i}_{j}\))
Parallel Transport and Covariant Derivatives¶
The covariant derivative of a tensor field \({{T}\lp {a_{1},\dots,a_{r};x} \rp }\) (\(x\) is the coordinate vector of which \(T\) can be a non-linear function) in the direction \(a_{r+1}\) is (remember \(a_{j} = a_{j}^{k}{{\eb}}_{k}\) and the \({{\eb}}_{k}\) can be functions of \(x\)) the directional derivative of \({{T}\lp {a_{1},\dots,a_{r};x} \rp }\) where all the arguments of \(T\) are parallel transported. The definition of parallel transport is if \(a\) and \(b\) are tangent vectors in the tangent spaced of the manifold then
if \(b\) is parallel transported. Since \(b = b^{i}{{\eb}}_{i}\) and the derivatives of \({{\eb}}_{i}\) are functions of the \(x^{i}\)’s then the \(b^{i}\)’s are also functions of the \(x^{i}\)’s so that in order for eq (\(\ref{eq108a}\)) to be satisfied we have
Thus for \(b\) to be parallel transported we must have
The geometric meaning of parallel transport is that for an infinitesimal rotation and dilation of the basis vectors (cause by infinitesimal changes in the \(x^{i}\)’s) the direction and magnitude of the vector \(b\) does not change.
If we apply eq (\(\ref{eq121a}\)) along a parametric curve defined by \({{x^{j}}\lp {s} \rp }\) we have
and if we define the initial conditions \({{b^{j}}\lp {0} \rp }{{\eb}}_{j}\). Then eq (\(\ref{eq122a}\)) is a system of first order linear differential equations with initial conditions and the solution, \({{b^{j}}\lp {s} \rp }{{\eb}}_{j}\), is the parallel transport of the vector \({{b^{j}}\lp {0} \rp }{{\eb}}_{j}\).
An equivalent formulation for the parallel transport equation is to let \({{\gamma}\lp {s} \rp }\) be a parametric curve in the manifold defined by the tuple \({{\gamma}\lp {s} \rp } = {\lp {{{x^{1}}\lp {s} \rp },\dots,{{x^{n}}\lp {s} \rp }} \rp }\). Then the tangent to \({{\gamma}\lp {s} \rp }\) is given by
and if \({{v}\lp {x} \rp }\) is a vector field on the manifold then
Thus eq (\(\ref{eq124a}\)) is equivalent to eq (\(\ref{eq122a}\)) and parallel transport of a vector field along a curve is equivalent to the directional derivative of the vector field in the direction of the tangent to the curve being zero.
If the tensor component representation is contra-variant (superscripts instead of subscripts) we must use the covariant component representation of the vector arguments of the tensor, \(a = a_{i}{{\eb}}^{i}\). Then the definition of parallel transport gives
and we need
To satisfy equation (\(\ref{eq111a}\)) consider the following
Now dot eq (\(\ref{eq111a}\)) into \({{\eb}}_{k}\) giving
Thus if we have a mixed representation of a tensor
the covariant derivative of the tensor is
From eq (\(\ref{eq126a}\)) we obtain the components of the covariant derivative to be
The component free form of the covariant derivative (the one used to calculate it in the code) is
Representation of Multivectors in sympy¶
The sympy python module offers a simple way of representing multivectors using linear combinations of commutative expressions (expressions consisting only of commuting sympy objects) and non-commutative symbols. We start by defining \(n\) non-commutative sympy symbols as a basis for the vector space
(e_1,...,e_n) = symbols(’e_1,...,e_n’,commutative=False,real=True)
Several software packages for numerical geometric algebra calculations are available from Doran-Lasenby group and the Dorst group. Symbolic packages for Clifford algebra using orthogonal bases such as \({{\eb}}_{i}{{\eb}}_{j}+{{\eb}}_{j}{{\eb}}_{i} = 2\eta_{ij}\), where \(\eta_{ij}\) is a numeric array are available in Maple and Mathematica. The symbolic algebra module, ga, developed for python does not depend on an orthogonal basis representation, but rather is generated from a set of \(n\) arbitrary symbolic vectors \({{\eb}}_{1},{{\eb}}_{2},\dots,{{\eb}}_{n}\) and a symbolic metric tensor \(g_{ij} = {{\eb}}_{i}\cdot {{\eb}}_{j}\) (the symbolic metric can be symbolic constants or symbolic function in the case of a manifold).
In order not to reinvent the wheel all scalar symbolic algebra is handled by the python module sympy and the abstract basis vectors are encoded as non-commuting sympy symbols.
The basic geometric algebra operations will be implemented in python by defining a geometric algebra class, Ga, that performs all required geometric algebra an calculus operations on sympy expressions of the form (Einstein summation convention)
where the \(F\)’s are sympy symbolic constants or functions of the coordinates and a multivector class, Mv, that wraps Ga and overloads the python operators to provide all the needed multivector operations as shown in Table [ops] where \(A\) and \(B\) are any two multivectors (In the case of \(+\), \(-\), \(*\), \({\wedge}\), \(|\), \(<\), and \(>\) the operation is also defined if \(A\) or \(B\) is a sympy symbol or a sympy real number).
| \(A+B\) | sum of multivectors |
| \(A-B\) | difference of multivectors |
| \(A*B\) | geometric product of multivectors |
| \(A{\wedge}B\) | outer product of multivectors |
| $A | B$ |
| \(A<B\)|left contraction of multivectors| |\(A>B\) | right contraction of multivectors |
| \(A/B\) | division of multivectors |
Multivector operations for GA¶
Since < and > have no r-forms (in python for the < and > operators there are no __rlt__() and __rgt__() member functions to overload) we can only have mixed modes (sympy scalars and multivectors) if the first operand is a multivector.
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 number).
Operator Precedence¶
Note that the operator order precedence is determined by python and is not necessarily 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 -.
Additional care has to be used when using the operators != and == with the operators < and >. All these operators have the same precedence and are evaluated chained from left to right. To be completely safe for expressions such as A == B or A != B always user (A) == (B) and (A) != (B) if A or B contains a left, <, or right, >, contraction.
For those users who wish to define a default operator precedence the functions def_prec() and GAeval() are available in the module printer.
def_prec(gd,op_ord='<>|,^,*')
Define the precedence of the multivector operations. The functiondef_prec()must be called from the main program and the first argumentgdmust be set toglobals(). The second argumentop_orddetermines the operator precedence for expressions input to the functionGAeval(). The default value ofop_ordis'<>|,^,*'. For the default value the<,>, and|operations have equal precedence followed by^, and^is followed by*.
GAeval(s,pstr=False)
The function
GAeval()returns a multivector expression defined by the stringswhere the operations in the string are parsed according to the precedences defined bydef_prec().pstris a flag to print the input and output ofGAeval()for debugging purposes.GAeval()works by adding parenthesis to the input stringswith the precedence defined byop_ord=’<>|,,*’. Then the parsed string is converted to a sympy expression using the pythoneval()function. For example consider whereX,Y,Z, andWare multivectorsdef_prec(globals()) V = GAeval('X|Y^Z*W')The sympy variable
Vwould evaluate to((X|Y)^Z)*W.
Vector Basis and Metric¶
The two structures that define the metric class (inherited by the geometric algebra 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 geometric algebra class instantiation function, Ga(). 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
where each of the \(g_{ij}\) is a symbol representing all of the dot products of the basis vectors. Note that the symbols are named so that \(g_{ij} = g_{ji}\) since for the symbol function \((a0.a1) \ne (a1.a0)\).
Note that the strings shown in the above equation are only used when the values of \(g_{ij}\) are output (printed). In the ga module (library) the \(g_{ij}\) symbols are stored in a member of the geometric algebra instance so that if o3d is a geometric algebra then o3d.g is the metric tensor ( \(g_{ij} =\) o3d.g[i,j]) for that algebra.
The default definition of \(g\) can be overwritten by specifying a string that will define \(g\). As an example consider a symbolic representation for conformal geometry. Define for a basis
[ ]:
basis = 'a0 a1 a2 n nbar'
and for a metric
[ ]:
g = '# # # 0 0, # # # 0 0, # # # 0 0, 0 0 0 0 2, 0 0 0 2 0'
then calling cf3d = Ga(basis,g=g) would initialize the metric tensor
for the cf3d (conformal 3-d) geometric algebra.
Here we have specified that n and nbar are orthogonal to all the a’s, (n.n) = (nbar.nbar) = 0, and (n.nbar) = 2. Using # in the metric definition string just tells the program to use the default symbol for that value.
When Ga is called multivector representations of the basis local to the program are instantiated. For the case of an orthogonal 3-d vector space that means the symbolic vectors named a0, a1, and a2 are created. We can instantiate the geometric algebra and obtain the basis vectors with -
[ ]:
o3d = Ga('a_1 a_2 a_3',g=[1,1,1])
(a_1,a_2,a_3) = o3d.mv()
or use the Ga.build() function -
[ ]:
(o3d,a_1,a_2,a_3) = Ga.build('a_1 a_2 a_3',g=[1,1,1])
Note that the python variable name for a basis vector does not have to correspond to the name give in Ga() or Ga.build(), one may wish to use a shortened python variable name to reduce programming (typing) errors, for example one could use -
[ ]:
(o3d,a1,a2,a3) = Ga.build('a_1 a_2 a_3',g=[1,1,1])
or
[ ]:
(st4d,g0,g1,g2,g3) = Ga.build('gamma_0 gamma_1 gamma_2 gamma_3',\
g=[1,-1,-1,-1])
for Minkowski space time.
If the latex printer is used e1 would print as \({\boldsymbol{e_{1}}}\) and g1 as \({\boldsymbol{\gamma_{1}}}\).
Representation and Reduction of Multivector Bases¶
In our symbolic geometric algebra all multivectors 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 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 \(a_{i_{1}}a_{i_{2}}\dots a_{i_{r}}\) where \(i_{1}<i_{2}<\dots <i_{r}\) and \(r \le n\). We call these multivectors bases and represent them internally with non-commutative symbols so for example \(a_{1}a_{2}a_{3}\) is represented by
[ ]:
Symbol('a_1*a_2*a_3',commutative=False)
In the simplest case of two basis vectors a_1 and a_2 we have a list of bases
[ ]:
self.bases = [[Symbol('a_1',commutative=False,real=True),\
Symbol('a_2',commutative=False,real=True)],\
[Symbol('a_1*a_2',commutative=False,real=True)]]
For the case of the basis blades we have
[ ]:
self.blades = [[Symbol('a_1',commutative=False,real=True),\
Symbol('a_2',commutative=False,real=True)],\
[Symbol('a_1^a_2',commutative=False,real=True)]]
The index tuples for the bases of each pseudo grade and each grade for the case of dimension 3 is
[ ]:
self.indexes = (((0,),(1,),(2,)),((0,1),(0,2),(1,2)),((0,1,2)))
Then the non-commutative symbol representing each base is constructed from each index tuple. For example for self.indexes[1][1] the symbol is Symbol('a_1*a_3',commutative=False).
Base Representation of Multivectors¶
In terms of the bases defined as non-commutative sympy symbols the general multivector is a linear combination (scalar sympy coefficients) of bases so that for the case of two bases the most general multivector is given by -
[ ]:
A = A_0+A__1*self.bases[1][0]+A__2*self.bases[1][1]+\
A__12*self.bases[2][0]
If we have another multivector B to multiply with A we can calculate the product in terms of a linear combination of bases if we have a multiplication table for the bases.
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
where \(A_{r}\) is a multivector of grade \(r\) and \(b\) is a vector. For our example basis the result is shown in Table [bladexpand].
[h]
[ ]:
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
[bladexpand]
The important thing to notice about Table [bladexpand] is that it is a triagonal (lower triangular) system of equations so that using a simple back substitution algorithm we can solve for the pseudo bases in terms of the blades giving Table [baseexpand].
[ ]:
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
[baseexpand]
Using Table [baseexpand] and simple substitution we can convert from a base multivector representation to a blade representation. Likewise, using Table [bladexpand] 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.is_blade_rep, that is set to False for a base representation and True 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.
Module Components¶
The geometric algebra module consists of the following files and classes
| File | Classes | Usage |
|---|---|---|
metric.py |
Metric |
Instantiates metric tensor and derivatives of basis vectors. Normalized basis if required. |
ga.py |
Ga |
Instantiates geometric algebra (inherits \(\T{Metric}\)), generates bases, blades, multiplication tables, reciprocal basis, and left and right geometric derivative operators. |
Sm |
Instantiates geometric algebra for submainfold (inherits \(\T{Ga}\)). | |
mv.py |
Mv |
Instantiates multivector. |
Dop |
Instantiates linear multivector differential operator. | |
lt.py |
Lt |
Instantiates multivector linear transformation. |
printer.py |
Eprint |
Starts enhanced text printing on ANSI terminal (requires \(\T{ConEmu}\) on Windows). |
GaPrinter |
Text printer for all geometric algebra classes (inherits from \(\T{sympy}\) \(\T{StringPrinter}\)). | |
GaLatexPrinter |
\(\LaTeX\)printer for all geometric algebra classes (inherits from\(\T{sympy}\) \(\T{LatexPrinter}\)). |
Instantiating a Geometric Algebra¶
The geometric algebra class is instantiated with
Ga(basis,g=None,coords=None,X=None,norm=False,sig=’e’,Isq=’-’,wedge=True,debug=False)
The
basisandgparameters were described in section [BasisMetric]. If the metric is a function of position, if we have multivector fields, or we wish to calculate geometric derivatives a coordinate set,coords, is required.coordsis a list of sympy symbols. For the case of instantiating a 3-d geometric algebra in spherical coordinates we have(r, th, phi) = coords = symbols('r,theta,phi', real=True) basis = 'e_r e_theta e_phi' g = [1, r**2, r**2*sin(th)**2] sp3d = Ga(basis,g=g,coords=coords,norm=True)The input
Xallows the metric to be input as a vector manifold.Xis a list of functions ofcoordsof dimension, \(m\), equal to or greater than the number of coordinates. Ifg=Noneit is assumed thatXis a vector in an \(m\)-dimensional orthonormal Euclidean vector space. If it is wished the embedding vector space to be non-Euclidean that condition is specified withg. For example if we wish the embedding space to be a 5-dimensional Minkowski space theng=[-1,1,1,1,1]. Then the Ga class usesXto calculate the manifold basis vectors as a function of the coordinates and from them the metric tensor.[12]If
norm=Truethe basis vectors of the manifold are normalized so that the absolute values of the squares of the basis vectors are one. *Currently you should only use this option for diagonal metric tensors, and even there due so with caution, due to the possible problems with taking the square root of a general *sympy* expression (one that has an unknown sign).*When a geometric algebra is created the unnormalized metric tensor is always saved so that submanifolds created from the normalized manifold can be calculated correctly.
sigindicates the signature of the vector space in the following ways.[13]
- If the metric tensor is purely numerical (the components are not symbolic or functions of the coordinates) and is diagonal (orthogonal basis vectors) the signature is computed from the metric tensor.
- If the metric tensor is not purely numerical and orthogonal the following hints are used (dimension of vector space is \(n\))
sig=’e’the default hint assumes the signature is for a Euclidean space with signature \((n,0)\).sig=’m+’assumes the signature if for the Minkowski space \((n-1,1)\).sig=’m-’assumes the signature if for the Minkowski space \((1,n-1)\).sig=pwherepis an integer \(p\le n\) and the signature it \((p,n-p)\).If the metric tensor contains no symbolic constants, but is a function of the coordinates, it is possible to determine the signature of the metric numerically by specifying a allowed numerical coordinate tuple due to the invariance of the signature. This will be implemented in the future.
Currently one need not be concerned about inputting
sigunless one in using the Ga member functionGa.I()or the functionsMv.dual()orcross()which also useGa.I().If \(I^{2}\) is numeric it is calculated if it is not numeric then
Isq=’-’is the sign of the square of the pseudo-scalar. This is needed for some operations. The default is chosen for the case of a general 3D Euclidean metric.If
wedge=Truethe basis blades of a multivector are printed using the^symbol between basis vectors. Ifwedge=Falsethe subscripts of each individual basis vector (assuming that the basis vector symbols are of the form root symbol with a subscript[14]). For example in three dimensions if the basis vectors are \({{\eb}}_{x}\), \({{\eb}}_{y}\), and \({{\eb}}_{z}\) the grade 3 basis blade would be printed as \({{\eb}}_{xyz}\).If
debug=Truethe data structures required to initialize the Ga class are printed out.To get the basis vectors for
sp3dwe would have to use the member functionGa.mv()in the form(er,eth,ephi) = sp3d.mv()
To access the reciprocal basis vectors of the geometric algebra use the member function mvr()
Ga.mvr(norm=’True’)
Ga.mvr(norm)returns the reciprocal basis vectors as a tuple. This allows the programmer to attach any python variable names to the reciprocal basis vectors that is convenient. For example (demonstrating the use of bothmv()andmvr())(e_x,e_y,e_z) = o3d.mv() (e__x,e__y,e__z) = o3d.mvr()If
norm=’True’or the basis vectors are orthogonal the dot product of the basis vector and the corresponding reciprocal basis vector is one \({\lp {e_{i}\cdot e^{j}=\delta_{i}^{j}} \rp }\). Ifnorm=’False’and the basis is non-orthogonal The dot product of the basis vector and the corresponding reciprocal basis vector is the square of the pseudo scalar, \(I^{2}\), of the geometric algebra \({\lp {e_{i}\cdot e^{j}=E^{2}\delta_{i}^{j}} \rp }\).
In addition to the basis vectors, if coordinates are defined for the geometric algebra, the left and right geometric derivative operators are calculated and accessed with the Ga member function grads().
Ga.grads()
Ga.grads()returns a tuple with the left and right geometric derivative operators. A typical usage would be(grad,rgrad) = sp3d.grads()for the spherical 3-d geometric algebra. The left derivative \({\lp {{\texttt{grad}} ={\boldsymbol{\nabla}}} \rp }\) and the right derivative \({\lp {{\texttt{rgrad}} = {\boldsymbol{\bar{\nabla}}}} \rp }\) have been explained in section [ldops]. Again the names
gradandrgradused in a program are whatever the user chooses them to be. In the previous examplegradandrgradare used.
an alternative instantiation method is
Ga.build(basis, g=None, coords=None, X=None, norm=False, debug=False)
The input parameters for
Ga.build()are the same as forGa(). The difference is that in addition to returning the geometric algebraGa.build()returns the basis vectors at the same time. UsingGa.build()in the previous example gives(r, th, phi) = coords = symbols('r,theta,phi', real=True) basis = 'e_r e_theta e_phi' g = [1, r**2, r**2*sin(th)**2] (sp3d,er,eth,ephi) = Ga.build(basis,g=g,coord=coords,norm=True)
To access the pseudo scalar of the geometric algebra use the member function I().
Ga.I()
Ga.I()returns the normalized pseudo scalar \({\lp {{\left |{I^{2}}\right |}=1} \rp }\) for the geometric algebra. For example \(I = \mbox{{\texttt{o3d.I()}}}\) for theo3dgeometric algebra. This function requires the signature of the vector space (see instantiating a geometric algebra).
Ga.E()
Ga.E()returns the unnormalized pseudo scalar \(E_{n} = {\eb}_{1}{\wedge}\dots{\wedge}{\eb}_{n}\) for the geometric algebra.
In general we have defined member functions of the Ga class that will instantiate objects of other classes since the objects of the other classes are all associated with a particular geometric algebra object. Thus we have
| Object | Class | Ga method |
|---|---|---|
| multivector | Mv |
mv |
| submanifold | Sm |
sm |
| linear transformation | Lt |
lt |
| differential operator | Dop |
dop |
for the instantiation of various objects from the Ga class. This means that in order to instantiate any of these objects we need only to import Ga into our program.
Instantiating a Multivector¶
Since we need to associate each multivector with the geometric algebra that contains it we use a member function of Ga to instantiate every multivector[15] The multivector is instantiated with:
Ga.mv(name, mode, f=False)
As an example of both instantiating a geometric algebra and multivectors consider the following code fragment for a 3-d Euclidean geometric algebra.
from sympy import symbols from ga import Ga (x, y, z) = coords = symbols('x,y,z',real=True) o3d = Ga('e_x e_y e_z', g=[1,1,1], coords=coords) (ex, ey, ez) = o3d.mv() V = o3d.mv('V','vector',f=True) f = o3d.mv(x*y*z) B = o3d.mv('B',2)First consider the multivector instantiation in line 6,
V = o3d.mv(’V’,’vector’,f=True).Here a 3-dimensional multivector field that is a function of
x,y, andz(f=True) is being instantiated. If latex output were used (to be discussed later) the multivectorVwould be displayed as\[\be V^{x}\eb_{x} + V^{y}\eb_{y} + V^{z}\eb_{z} \ee\]Where the coefficients of the basis vectors are generalized sympy functions of the coordinates. If
f=(x,y)then the coefficients would be functions ofxandy. In general isfis a tuple of symbols then the coefficients of the basis would be functions of those symbols. The superscripts[16] are formed from the coordinate symbols or if there are no coordinates from the subscripts of the basis vectors. The types of name and modes available for multivector instantiation are
namemoderesult string s scalarsymbolic scalar of value Symbol(s) string s vectorsymbolic vector string s grade2orbivectorsymbolic bivector string s r(integer)symbolic r-grade multivector string s pseudosymbolic pseudoscalar string s spinorsymbolic even multivector string s mvsymbolic general multivector scalar c None zero grade multivector with coefficient value c Line 5 of the previous listing illustrates the case of using the
mvmember function with no arguments. The code does not return a multivector, but rather a tuple or the basis vectors of the geometric algebrao3d. The elements of the tuple then can be used to construct multivectors, or multivector fields through the operations of addition, subtraction, multiplication (geometric, inner, and outer products and left and right contraction). As an example we could construct the vector functionF = x**2*ex + z*ey + x*y*ezor the bivector function
B = z*(ex^ey) + y*(ey^ez) + y*(ex^ez).Line 7 is an example of instantiating a multivector scalar function (a multivector with only a scalar part). If we print
fthe result isx*y*z. Line 8 is an example of instantiating a grade \(r\) (in the example a grade 2) multivector where\[\be B = B^{xy}{\eb}_{x}{\wedge}{\eb}_{y}+B^{yz}{\eb}_{y}{\wedge}{\eb}_{z}+B^{xz}{\eb}_{x}{\wedge}{\eb}_{z}. \ee\]
If one wished to calculate the left and right geometric derivatives of F and B the required code would be
[ ]:
(grad,rgrad) = o3d.grads()
dF = grad*F
dB = grad*B
dFr = F*rgrad
dBr = B*rgrad
dF, dB, dFr, and dBr are all multivector functions. For the code where the order of the operations are reversed
[ ]:
(grad,rgrad) = o3d.grads()
dFop = F*grad
dBop = B*grad
dFrop = rgrad*F
dBrop = rgrad*B
dFop, dBop, dFrop, and dBrop are all multivector differential operators (again see section [ldops]).
Backward Compatibility Class MV¶
In order to be backward compatible with older versions of galgebra we introduce the class MV which is inherits it’s functions from then class Mv. To instantiate a geometric algebra using MV use the static function
MV.setup(basis, metric=None, coords=None, rframe=False, debug=False, curv=(None,None))
This function allows a single geometric algebra to be created. If the function is called more than once the old geometric algebra is overwritten by the new geometric algebra. The named inputmetricis the same as the named inputgin the current version of galgebra. Likewise,basis,coords, anddebugare the same in the old and current versions of galgebra[17]. Due to improvements in sympy the inputsrframeandcurv[1]are no longer required.curv[0]is the vector function (list or tuple of scalar functions) of the coordinates required to define a vector manifold. For compatibility with the old version of galgebra ifcurvis usedmetricshould be a orthonormal Euclidean metric of the same dimension ascurv[0]. It is strongly suggested that one use the new methods of defining a geometric algebra on a manifold.
MV(base, mvtype, fct=False, blade_rep=True)
For the instantiation of multivector usingMVthebaseandmvtypearguments are the same as for new methods of multivector instantiation. Thefctinput is the same and theginput in the new methods.blade_repis not used in the new methods so settingblade_rep=Falsewill do nothing. Effectivelyblade_rep=Falsewas not used in the old examples.
Fmt(self, fmt=1, title=None)
FmtinMVhas inputs identical toFmtinMvexcept that ifAis a multivector thenA.Fmt(2,’A’)executes a print statement fromMVand returnsNone, while fromMv,A.Fmt(2,’A’)returns a string so that the function is compatible with use in ipython notebook.
Basic Multivector Class Functions¶
If we can instantiate multivectors we can use all the multivector class functions as described as follows.
blade_coefs(self,basis_lst)
Find coefficients (sympy expressions) of multivector basis blade expansion corresponding to basis blades inbasis_lst. For example if \(V = V^{x}{{\eb}}_{x}+V^{y}{{\eb}}_{x}+V^{z}{{\eb}}_{x}\) Then \(V\text{.blade_coefs}([{{\eb}}_{z},{{\eb}}_{x}]) = [V^{z},V^{x}]\) or if \(B = B^{xy}{{\eb}}_{x}{\wedge}{{\eb}}_{y}+V^{yz}{{\eb}}_{y}{\wedge}{{\eb}}_{z}\) then \(B\text{.blade_coefs}([{{\eb}}_{x}{\wedge}{{\eb}}_{y}]) = [B^{xy}]\).
convert_to_blades(self)
Convert multivector from the base representation to the blade representation. If multivector is already in blade representation nothing is done.
convert_from_blades(self)
Convert multivector from the blade representation to the base representation. If multivector is already in base representation nothing is done.
diff(self,var)
Calculate derivative of each multivector coefficient with respect to variablevarand form new multivector from coefficients.
dual(self)
The mode of the
dual()function is set by theGaclass static member function,GA.dual_mode(mode=’I+’)of theGAgeometric galgebra which sets the following return values (\(I\) is the pseudo-scalar for the geometric algebraGA)
modeReturn Value ’+I’\(IA\) ’I+’\(AI\) ’-I’\(-IA\) ’I-’\(-AI\) ’+Iinv’\(I^{-1}A\) ’Iinv+’\(AI^{-1}\) ’-Iinv’\(-I^{-1}A\) ’Iinv-’\(-AI^{-1}\) For example if the geometric algebra is
o3d,Ais a multivector ino3d, and we wish to usemode=’I-’. We set the mode with the functiono3d.dual(’I-’)and get the dual ofAwith the functionA.dual()which returns \(-AI\).If
o3d.dual(mode)is not called the default for the dual mode ismode=’I+’andA*Iis returned.Note that
Ga.dual(mode)used the functionGa.I()to calculate the normalized pseudoscalar. Thus if the metric tensor is not numerical and orthogonal the correct hint for thensiginput of the Ga constructor is required.
even(self)
Return the even grade components of the multivector.
exp(self,hint=’-’)
If \(A\) is a multivector then \(e^{A}\) is defined for any \(A\) via the series expansion for \(e\). However as a practical matter we only have a simple closed form formula for \(e^{A}\) if \(A^{2}\) is a scalar.[18] If \(A^{2}\) is a scalar and we know the sign of \(A^{2}\) we have the following formulas for \(e^{A}\).
\[\begin{split}$\begin{aligned} A^{2} > 0 : & & &\\ A &= \sqrt{A^{2}} {\displaystyle\frac{A}{\sqrt{A^{2}}}} ,& e^{A} &= {{\cosh}\lp {\sqrt{A^{2}}} \rp }+{{\sinh}\lp {\sqrt{A^{2}}} \rp }{\displaystyle\frac{A}{\sqrt{A^{2}}}} \\ A^{2} < 0 : & & &\\ A &= \sqrt{-A^{2}} {\displaystyle\frac{A}{\sqrt{-A^{2}}}} ,& e^{A} &= {{\cos}\lp {\sqrt{-A^{2}}} \rp }+{{\sin}\lp {\sqrt{-A^{2}}} \rp }{\displaystyle\frac{A}{\sqrt{-A^{2}}}} \\ A^{2} = 0 : & & &\\ A &=0 ,& e^{A} &= 1 + A \end{aligned}\end{split}\]The hint is required for symbolic multivectors \(A\) since in general sympy cannot determine if \(A^{2}\) is positive or negative. If \(A\) is purely numeric the hint is ignored since the sign can be calculated.
expand(self)
Return multivector in which each coefficient has been expanded using sympyexpand()function.
factor(self)
Apply thesympyfactorfunction to each coefficient of the multivector.
Fmt(self, fmt=1,title=None)
Fuction to print multivectors in different formats where
fmt1 Print entire multivector on one line. 2 Print each grade of multivector on one line. 3 Print each base of multivector on one line.
titleappends a title string to the beginning of the output. An equal sign in the title string is not required, but is added as a default. Note thatFmtonly overrides the the global multivector printing format for the particular instance being printed. To reset the global multivector printing format use the functionFmt()in the printer module.
func(self,fct)
Apply thesympyscalar functionfctto each coefficient of the multivector.
grade(self,igrade=0)
Return a multivector that consists of the part of the multivector of grade equal toigrade. If the multivector has noigradepart return a zero multivector.
inv(self)
Return the inverse of the multivector \(M\) (
M.inv()). If \(M\) is a non-zero scalar return \(1/M\). If \(M^{2}\) is a non-zero scalar return \(M/{\lp {M^{2}} \rp }\), If \(MM^{{\dagger}}\) is a non-zero scalar return \(M^{{\dagger}}/{\lp {MM^{{\dagger}}} \rp }\). Otherwise exit the program with an error message.All division operators (
/,/=) use right multiplication by the inverse.
norm(self,hint=’+’)
Return the norm of the multivector \(M\) (M.norm()) defined by \(\sqrt{{\left |{MM^{{\dagger}}}\right |}}\). If \(MM^{{\dagger}}\) is a scalar (a sympy scalar is returned). If \(MM^{{\dagger}}\) is not a scalar the program exits with an error message. If \(MM^{{\dagger}}\) is a number sympy can determine if it is positive or negative and calculate the absolute value. If \(MM^{{\dagger}}\) is a sympy expression (function) sympy cannot determine the sign of the expression so thathint=’+’orhint=’-’is needed to determine if the program should calculate \(\sqrt{MM^{{\dagger}}}\) or \(\sqrt{-MM^{{\dagger}}}\). For example if we are in a Euclidean space andMis a vector thenhint=’+’, ifMis a bivector then lethint=’-’. Ifhint=’0’and \(MM^{{\dagger}}\) is a symbolic scalarsqrt(Abs(M*M.rev()))is returned whereAbs()is the sympy symbolic absolute value function.
norm2(self)
Return the the scalar defined by \(MM^{{\dagger}}\) if \(MM^{{\dagger}}\) is a scalar. If \(MM^{{\dagger}}\) is not a scalar the program exits with an error message.
proj(self,bases_lst)
Return the projection of the multivector \(M\) (M.proj(bases_lst)) onto the subspace defined by the list of bases (bases_lst).
proj(self,lst)
Return the projection of the mutivector \(A\) onto the list, \(lst\), of basis blades. For example if \(A = A^{x}{{\eb}}_{x}+A^{y}{{\eb}}_{y}+A^{z}{{\eb}}_{z}\) then \(A.proj{\lp {[{{\eb}}_{x},{{\eb}}_{y}]} \rp } = A^{x}{{\eb}}_{x}+A^{y}{{\eb}}_{y}\). Similarly if \(A = A^{xy}{{\eb}}_{x}{\wedge}{{\eb}}_{y}+A^{yz}{{\eb}}_{y}{\wedge}{{\eb}}_{z}\) then \(A.proj{\lp {[{{\eb}}_{x}{\wedge}{{\eb}}_{y}]} \rp } = A^{xy}{{\eb}}_{x}{\wedge}{{\eb}}_{y}\).
project_in_blade(self,blade)
Return the projection of the mutivector \(A\) in subspace defined by the blade, \(B\), using the formula \({\lp {A\rfloor B} \rp }B^{-1}\) in , page 121.
pure_grade(self)
If the multivector \(A\) is pure (only contains one grade) return, \(A.pure\_grade()\), the index (’0’ for a scalar, ’1’ for vector, ’2’ for a bi-vector, etc.) of the non-zero grade. If \(A\) is not pure return the negative of the highest non-zero grade index.
odd(self)
Return odd part of multivector.
reflect_in_blade(self,blade)
Return the reflection of the mutivector \(A\) in the subspace defined by the \(r\)-grade blade, \(B_{r}\), using the formula (extended to multivectors) \(\sum_{i} {\lp {-1} \rp }^{r{\lp {i+1} \rp }}{B}_{r}{\left < {A} \right >}_{i}B_{r}^{-1}\) in , page 129.
rev(self)
Return the reverse of the multivector.
rotate_multivector(self,itheta,hint=’-’)
Rotate the multivector \(A\) via the operation \(e^{-\theta i/2}Ae^{\theta i/2}\) where itheta = \(\theta i\), \(\theta\) is a scalar, and \(i\) is a unit, \(i^{2} = \pm 1\), 2-blade. If \({\lp {\theta i} \rp }^{2}\) is not a numberhintis required to determine the sign of the square ofitheta. The default chosen,hint=’-’, is correct for any Euclidean space.
scalar(self)
Return the coefficient (sympy scalar) of the scalar part of a multivector.
simplify(self,mode=simplify)
modeis a sympy simplification function of a list/tuple of sympy simplification functions that are applied in sequence (if more than one function) each coefficient of the multivector. For example if we wished to appliedtrigsimpandratsimpsympy functions to the multivectorFthe code would beFsimp = F.simplify(mode=[trigsimp,ratsimp]).Actually
simplifycould be used to apply any scalar sympy function to the coefficients of the multivector.
set_coef(self,grade,base,value)
Set the multivector coefficient of index(grade,base)tovalue.
subs(self,x)
Return multivector where sympy subs function has been applied to each coefficient of multivector for argument dictionary/listx.
trigsimp(self,**kwargs)
Apply thesympytrigonometric simplification functiontrigsimpto each coefficient of the multivector.**kwargsare the arguments of trigsimp. Seesympydocumentation ontrigsimpfor more information.
Basic Multivector Functions¶
com(A,B)
Calculate commutator of multivectors \(A\) and \(B\). Returns \((AB-BA)/2\).
Additionally, commutator and anti-commutator operators are defined by
\[\begin{split}$\begin{aligned} \texttt{A >> B} \equiv & {\displaystyle\frac{AB - BA}{2}} \\ \texttt{A << B} \equiv & {\displaystyle\frac{AB + BA}{2}}. \end{aligned}\end{split}\]
cross(v1,v2)
Ifv1andv2are 3-dimensional Euclidean vectors the vector cross product is returned, \(v_{1}\times v_{2} = -I{\lp {v_{1}{\wedge}v_{2}} \rp }\).
def_prec(gd,op_ord='<>|,^,*')[19]
This is used with theGAeval()function to evaluate a string representing a multivector expression with a revised operator precedence.def_prec()redefines the operator precedence for multivectors.def_prec()must be called in the main program an the argumentgdmust beglobals(). The argumentop_orddefines the order of operator precedence from high to low with groups of equal precedence separated by commas. the default precedenceop_ord='<>|,^,\*'is that used by Hestenes (,p7,,p38).
dual(A,mode=’I+’)
Return the dual of the multivectorA. The default operation is \(AI\). For other modes see member functionMv.dual(mode)
even(A)
Return even part of \(A\).
exp(A,hint=’-’)
If \(A\) is a multivector thenA.exp(hint)is returned. If \(A\) is a sympy expression the sympy expression \(e^{A}\) is returned (seesympy.exp(A)member function).
GAeval(s,pstr=False)[20]
Returns multivector expression for stringswith operator precedence for stringsdefined by inputs to functiondef_prec(). ifpstr=Truesandswith parenthesis added to enforce operator precedence are printed.
grade(A,r=0)
If \(A\) is a multivector \({\left < {A} \right >}_{r}\) is returned.
inv(A)
If \(A\) is a multivector and \(AA^{{\dagger}}\) is a non-zero scalar then \(A^{-1} = A^{{\dagger}}/(AA^{{\dagger}})\) is returned otherwise an exception is returned.
Nga(x,prec=5)
Ifxis a multivector with coefficients that contain floating point numbers,Nga()rounds all these numbers to a precision ofprecand returns the rounded multivector.
norm(A,hint=’-’)
If \(A\) is a multivector and \(AA^{{\dagger}}\) is a number (not a scalar function) then \(\sqrt{{\left |{AA^{{\dagger}}}\right |}}\) is returned. If \(AA^{{\dagger}}\) is a scalar sympy expression, but not a number, andhint=’-’then return \(\sqrt{-AA^{{\dagger}}}\) otherwise return \(\sqrt{AA^{{\dagger}}}\).
norm2(A)
If \(A\) is a multivector and \(AA^{{\dagger}}\) is a scalar return \({\left |{AA^{{\dagger}}}\right |}\).
odd(A)
Return odd part of \(A\).
proj(B,A)
Project bladeAon bladeBreturning \({\lp {A\rfloor B} \rp }B^{-1}\).
ReciprocalFrame(basis,mode=’norm’)
Ifbasisis a list/tuple of vectors,ReciprocalFrame()returns a tuple of reciprocal vectors. Ifmode=normthe vectors are normalized. Ifmodeis anything other thannormthe vectors are unnormalized and the normalization coefficient is added to the end of the tuple. One must divide by this coefficient to normalize the vectors.
refl(B,A)
Reflect multivector \(A\) in blade \(B\). If \(s\) is grade of \(B\) returns \(\sum_{r}(-1)^{s(r+1)}B{\left < {A} \right >}_{r}B^{-1}\).
rev(A)
If \(A\) is a multivector return \(A^{{\dagger}}\).
rot(itheta,A,hint=’-’)
IfAis a multivector returnA.rotate_multivector(itheta,hint)whereithetais the bi-vector blade defining the rotation. For the use ofhintsee the member functionMv.rotate_multivector(self,itheta,hint).
Multivector Derivatives¶
The various derivatives of a multivector function is accomplished by multiplying the gradient operator vector with the function. The gradient operation vector is returned by the Ga.grads() function if coordinates are defined. For example if we have for a 3-D vector space
[ ]:
X = (x,y,z) = symbols('x y z')
o3d = Ga('e*x|y|z',metric='[1,1,1]',coords=X)
(ex,ey,ez) = o3d.mv()
(grad,rgrad) = o3d.grads()
Then the gradient operator vector is grad (actually the user can give it any name he wants to). The derivatives of the multivector function F = o3d.mv(’F’,’mv’,f=True) are given by multiplying by the left geometric derivative operator and the right geometric derivative operator (\(\T{grad} = \nabla\) and \(\T{rgrad} = \bar{\nabla}\)). Another option is to use the radiant operator members of the geometric algebra directly where we have \(\nabla = {\texttt{o3d.grad}}\) and
\(\bar{\nabla} = {\texttt{o3d.rgrad}}\).
The preceding list gives examples of all possible multivector derivatives of the multivector function F where the operation returns a multivector function. The complementary operations
all return multivector linear differential operators.
Submanifolds¶
In general the geometric algebra that the user defines exists on the tangent space of a manifold (see section [sect_manifold]). The submanifold class, Sm, is derived from the Ga class and allows one to define a submanifold of a manifold by defining a coordinate mapping between the submanifold coordinates and the manifold coordinates. What is returned as the submanifold is the geometric algebra of the tangent space of the submanifold. The submanifold for a geometric algebra is
instantiated with
Ga.sm(map,coords,root=’e’,norm=False)
To define the submanifold we must def a coordinate map from the coordinates of the submanifold to each of the coordinates of the base manifold. Thus the arguments
mapandcoordsare respectively lists of functions and symbols. The list of symbols,coords, are the coordinates of the submanifold and are of length equal to the dimension of the submanifold. The list of functions,map, define the mapping from the coordinate space of the submanifold to the coordinate space of the base manifold. The length ofmapis equal to the dimension of the base manifold and each function inmapis a function of the coordinates of the submanifold.rootis the root of the string that is used to name the basis vectors of the submanifold. The default value ofrootise. The result of this is that if the sympy symbols for the coordinates areuandv(two dimensional manifold) the text symbols for the basis vectors aree_uande_vor in LaTeX \(e_{u}\) and \(e_{v}\). As a concrete example consider the following code. The output of this program (using LaTeX) is
The base manifold,
sp3d, is a 3-d Euclidean space using standard spherical coordinates. The submanifoldsph2dofsp3dis a spherical surface of radius \(1\). To take the sumanifold operation one step further the submanifoldcir1dofsph2dis a circle insph2dwhere the latitude of the circle is \(\pi/8\).In each case, for demonstration purposes, a scalar and vector function on each manifold is defined (
fandFfor the 2-d manifold andhandHfor the 1-d manifold) and the geometric derivative of each function is taken. The manifold mapping and the metric tensor forcir1dofsph2dare also shown. Note that if the submanifold basis vectors are not normalized[21] the program output is
Linear Transformations¶
The mathematical background for linear transformations is in section [Ltrans]. Linear transformations on the tangent space of the manifold are instantiated with the Ga member function lt (the actual class being instantiated is Lt) as shown in lines 12, 20, 26, and 44 of the code listing Ltrans.py. In all of the examples in Ltrans.py the default instantiation is used which produces a general (all the coefficients of the linear transformation are symbolic constants) linear
transformation. Note that to instantiate linear transformations coordinates, :math:`{left { {{eb}_{i}} rbrc}`, must be defined when the geometric algebra associated with the linear transformation is instantiated. This is due to the naming conventions of the general linear transformation (coordinate names are used) and for the calculation of the trace of the linear transformation which requires taking a divergence. To instantiate a specific linear transformation the usage of lt() is
Ga.lt(M,f=False,mode=’g’)
Mis an expression that can define the coefficients of the linear transformation in various ways defined as follows.
MResult string MCoefficients are symbolic constants with names \(\T{M}^{x_{i}x_{j}}\) where \(x_{i}\) and \(x_{j}\) are the names of the \(i^{th}\) and \(j^{th}\) coordinates (see output of \(\T{Ltrans.py}\)). char modeIf \(\T{M}\) is a string then \(\T{mode}\)determines whether the linear transformation is general,\(\T{mode='g'}\), symmetric, \(\T{mode='s'}\), or antisymmetric, \(\T{mode='a'}\). The default is \(\T{mode='g'}\). list MIf \(\T{M}\) is a list of vectors equal in length to the dimension of the vector space then the linear transformation is \(\f{L}{\ebf_{i}} = \T{M}\mat{i}\). If \(\T{M}\)is a list of lists of scalars where all lists are equal in length to the dimension of the vector space then the linear transformation is\(\f{L}{\ebf_{i}} = \T{M}\mat{i}\mat{j}\ebf_{j}\). dict MIf \(\T{M}\) is a dictionary the linear transformation is defined by \(\f{L}{\ebf_{i}} = \T{M}\mat{\ebf_{i}}\). If \(\ebf_{i}\) is not in the dictionary then \(\f{L}{\ebf_{i}} =0\). rotor MIf \(\T{M}\) is a rotor, \(\T{M}\T{M}^{\R}=1\), the linear transformation is defined by \(\f{L}{{\ebf}_{i}} = \T{M}{\ebf}_{i}\T{M}^{\R}\) . multivector function MIf \(\T{M}\) is a general multivector function, the function is tested for linearity, and if linear the coefficients of the linear transformation are calculated from \(\f{L}{\ebf_{i}} = \f{\T{M}}{\ebf_{i}}\).
fisTrueorFalse. IfTruethe symbolic coefficients of the general linear transformation are instantiated as functions of the coordinates.
The different methods of instantiation are demonstrated in the code LtransInst.py with output
The member function of the Lt class are
Lt(A)
Returns the image of the multivector \(A\) under the linear transformation \(L\) where \({{L}\lp {A} \rp }\) is defined by the linearity of \(L\), the vector values \({{L}\lp {{{\eb}}_{i}} \rp }\), and the definition \({{L}\lp {{{\eb}}_{i_{1}}{\wedge}\dots{\wedge}{{\eb}}_{i_{r}}} \rp } = {{L}\lp {{{\eb}}_{i_{1}}} \rp }{\wedge}\dots{\wedge}{{L}\lp {{{\eb}}_{i_{r}}} \rp }\).
Lt.det()
Returns the determinant (a scalar) of the linear transformation, \(L\), defined by \({{\det}\lp {L} \rp }I = {{L}\lp {I} \rp }\).
Lt.adj()
Returns the adjoint (a linear transformation) of the linear transformation, \(L\), defined by \(a\cdot{{L}\lp {b} \rp } = b\cdot{{\bar{L}}\lp {a} \rp }\) where \(a\) and \(b\) are any two vectors in the tangent space and \(\bar{L}\) is the adjoint of \(L\).
Lt.tr()
Returns the trace (a scalar) of the linear transformation, \(L\), defined by \({{\operatorname{tr}}\lp {L} \rp }=\nabla_{a}\cdot{{L}\lp {a} \rp }\) where \(a\) is a vector in the tangent space.
Lt.matrix()
Returns the matrix representation (sympyMatrix) of the linear transformation, \(L\), defined by \({{L}\lp {{{\eb}}_{i}} \rp } = L_{ij}{{\eb}}_{j}\) where \(L_{ij}\) is the matrix representation.
The Ltrans.py demonstrate the use of the various Lt member functions and operators. The operators that can be used with linear transformations are +, -, and *. If \(A\) and \(B\) are linear transformations, \(V\) a multivector, and \(\alpha\) a scalar then \({{{\lp {A\pm B} \rp }}\lp {V} \rp } = {{A}\lp {V} \rp }\pm{{B}\lp {V} \rp }\), \({{{\lp {AB} \rp }}\lp {V} \rp } = {{A}\lp {{{B}\lp {V} \rp }} \rp }\), and
\({{{\lp {\alpha A} \rp }}\lp {V} \rp } = \alpha{{A}\lp {V} \rp }\).
The matrix() member function returns a sympy Matrix object which can be printed in IPython notebook. To directly print an linear transformation in ipython notebook one must implement (yet to be done) a printing method similar to mv.Fmt().
Note that in Ltrans.py lines 30 and 49 are commented out since the latex output of those statements would run off the page. The use can uncomment those statements and run the code in the “LaTeX docs” directory to see the output. The output of this code is.
Differential Operators¶
For the mathematical treatment of linear multivector differential operators see section [ldops]. The is a differential operator class Dop. However, one never needs to use it directly. The operators are constructed from linear combinations of multivector products of the operators Ga.grad and Ga.rgrad as shown in the following code for both orthogonal rectangular and spherical 3-d coordinate systems. The output of this code is.
Note that for print an operator in the IPython notebook one must implement (yet to be done) a printing method similar to mv.Fmt().
Instantiating a Multi-linear Functions (Tensors)¶
The mathematical background for multi-linear functions is in section [MLtrans]. To instantiate a multi-linear function use
Mlt(self, f, Ga, nargs=None, fct=False)
Where the arguments are
fEither a string for a general tensor (this option is included mainly for debugging of the \(\T{Mlt}\) class) or a multi-linear function of manifold tangent vectors (multi-vectors of grade one) to scalar. For example one could generate a custom python function such as shown in \(\T{TensorDef.py}\) . GaGeometric algebra that tensor is associated with. nargsIf \(\T{f}\) is a string then \(\T{nargs}\) is the number of vector arguments of the tensor. If \(\T{f}\) is anything other than a string \(\T{nargs}\) is not required since \(\T{Mlt}\) determines the number of vector arguments from \(\T{f}\). fctIf \(\T{f}\) is a string then \(\T{fct=True}\) forces the tensor to be a tensor field (function of the coordinates. If \(\T{f}\) anything other than a string \(\T{fct}\) is not required since \(\T{Mlt}\) determines whether the tensor is a tensor field from \(\T{f}\) .
Basic Multilinear Function Class Functions¶
If we can instantiate multilinear functions we can use all the multilinear function class functions as described as follows. See section [MLtrans] for the mathematical description of each operation.
self(kargs)
Calling function to evaluates multilinear function forkargslist of vector arguments and returns a value. Note that a sympy scalar is returned, not a multilinear function.
self.contract(slot1,slot2)
Returns contraction of tensor betweenslot1andslot2whereslot1is the index of the first vector argument andslot2is the index of the second vector argument of the tensor. For example if we have a rank two tensor,T(a1,a2), thenT.contract(1,2)is the contraction ofT. For this case since there are only two slots there can only be one contraction.
self.pdiff(slot)
Returns gradient of tensor,T, with respect to slot vector. For example if the tensor is \({{T}\lp {a_{1},a_{2}} \rp }\) thenT.pdiff(2)is \(\nabla_{a_{2}}T\). SinceTis a scalar function,T.pdiff(2)is a vector function.
self.cderiv()
Returns covariant derivative of tensor field. IfTis a tensor of rank \(k\) thenT.cderiv()is a tensor of rank \(k+1\). The operation performed is defined in section [MLtrans].
Standard Printing¶
Printing of multivectors is handled by the module printer which contains a string printer class derived from the sympy string printer class and a latex printer class derived from the sympy latex printer class. Additionally, there is an Eprint class that enhances the console output of sympy to make the printed output multivectors, functions, and derivatives more readable. Eprint requires an ansi console such as is supplied in linux or the program ConEmu replaces cmd.exe.
For a windows user the simplest way to implement ConEmu is to use the geany editor and in the Edit\(\rightarrow\)Preferences\(\rightarrow\)Tools menu replace cmd.exe with[22]
"C:\Program Files\ConEmu\ConEmu64.exe" /WndW 180 /cmd %c
and then run an example galgeba program that used Eprint. The default background and foreground colors make the output unreadable. To change these parameters to reasonable values:[23]
- Right click on title bar of console.
- Open setting window.
- Open colors window.
- Set the following parameters to the indicated values:
- Text: #0
- Back: #7
- Popup: #0
- Back: #7
- \(\rlap{ \checkmark }\square\) Extend foreground colors with background #13
If Eprint is called in a program (linux) when multivectors are printed the basis blades or bases are printed in bold text, functions are printed in red, and derivative operators in green.
For formatting the multivector output there is the member function Fmt(self,fmt=1,title=None) which is documented in the multivector member functions. This member function works in the same way for LaTeX printing.
There are two functions for returning string representations of multivectors. If A is a multivector then str(A) returns a string in which the scalar coefficients of the multivector bases have been simplified (grouped, factored, etc.). The member function A.raw_str() returns a string in which the scalar coefficients of the multivector bases have not been simplified.
Latex Printing¶
For latex printing one uses one functions from the ga module and one function from the printer module. The functions are
Format(Fmode=True,Dmode=True)
This function from the
gamodule turns on latex printing with the following options
Argument Value Result FmodeTruePrint functions without argument list, \(f\) FalsePrint functions with standard sympy latex formatting, \({{f}\lp {x,y,z} \rp }\) DmodeTruePrint partial derivatives with condensed notation, \(\partial_{x}f\) FalsePrint partial derivatives with standard sympy latex formatting, \(\pdiff{f}{x}\)
Format()is also required for printing from ipython notebook (note thatxpdf()is not needed to print from ipython notebook).
Fmt(obj,fmt=1)
Fmt()can be used to set the global multivector printing format or to print a tuple, list, of dictionary.[24] The modes and operation ofFmt()are as follows:
objEffect obj=1,2,3Global multivector format is set to 1, 2, or 3 depending on \(\T{obj}\). See multivector member function \(\T{Fmt()}\)for effect of\(\T{obj}\) value. obj=tuple/list/dict The printing format of an object that is a tuple, list, or dict is controlled by the \(\T{fmt}\) argument in \(\T{Fmt}\) : \(\T{fmt=1}\): Print complete \(\T{obj}\) on one line. \(\T{fmt=2}\): Print one element of \(\T{obj}\) on each line.
xpdf(filename=None,debug=False,paper=(14,11),crop=False)
This function from the
printermodule post-processes the output captured from print statements, writes the resulting latex strings to the filefilename, processes the file with pdflatex, and displays the resulting pdf file. All latex files except the pdf file are deleted. Ifdebug = Truethe filefilenameis printed to standard output for debugging purposes andfilename(the tex file) is saved. Iffilenameis not entered the default filename is the root name of the python program being executed with.texappended. Thepaperoption defines the size of the paper sheet for latex. The format for thepaperis
paper=(w,h)wis paper width in inches andhis paper height in inchespaper=’letter’paper is standard letter size 8.5 in \(\times\) 11 in paper=’landscape’paper is standard letter size but 11 in \(\times\) 8.5 in The default of
paper=(14,11)was chosen so that long multivector expressions would not be truncated on the display.If the
cropinput isTruethe linuxpdfcropprogram is used to crop the pdf output (if output is one page). This only works for linux installations (wherepdfcropis installed).The
xpdffunction requires that latex and a pdf viewer be installed on the computer.
xpdfis not required when printing latex in IPython notebook.
As an example of using the latex printing options when the following code is executed
[ ]:
from printer import Format, xpdf
from ga import Ga
Format()
g3d = Ga('e*x|y|z')
A = g3d.mv('A','mv')
print r'\bm{A} =',A
print A.Fmt(2,r'\bm{A}')
print A.Fmt(3,r'\bm{A}')
xpdf()
The following is displayed
For the cases of derivatives the code is
[ ]:
from printer import Format, xpdf
from ga import Ga
Format()
X = (x,y,z) = symbols('x y z')
o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
f = o3d.mv('f','scalar',f=True)
A = o3d.mv('A','vector',f=True)
B = o3d.mv('B','grade2',f=True)
print r'\bm{A} =',A
print r'\bm{B} =',B
print 'grad*f =',o3d.grad*f
print r'grad|\bm{A} =',o3d.grad|A
(o3d.grad*A).Fmt(2,r'grad*\bm{A}')
print r'-I*(grad^\bm{A}) =',-o3g.mv_I*(o3d.grad^A)
print (o3d.grad*B).Fmt(2,r'grad*\bm{B}')
print r'grad^\bm{B} =',o3d.grad^B
print r'grad|\bm{B} =',o3d.grad|B
xpdf()
and the latex displayed output is (\(f\) is a scalar function)
This example also demonstrates several other features of the latex printer. In the case that strings are input into the latex printer such as r'grad*\bm{A}', r'grad^\bm{A}', or r'grad*\bm{A}'. The text symbols grad, ^, |, and * are mapped by the xpdf() post-processor as follows if the string contains an =.
| original | replacement | displayed latex |
grad*A |
\bm{\nabla}A |
\({\boldsymbol{\nabla}}A\) |
A^B |
A\wedge B |
\(A\wedge B\) |
A|B |
A\cdot B |
\(A\cdot B\) |
A*B |
AB |
\(AB\) |
A<B |
A\rfloor B |
\(A\rfloor B\) |
A>B |
A\lfloor B |
\(A\lfloor B\) |
A>>B |
A\times B |
\(A\times B\) |
A<<B |
A\bar{\times} B |
\(A\bar{\times} B\) |
If the first character in the string to be printed is a % none of the above substitutions are made before the latex processor is applied. In general for the latex printer strings are assumed to be in a math environment (equation or align) unless the first character in the string is a #.[25]
There are two member functions for returning LaTeX string representations of multivectors. If A is a multivector then A.Mv_latex_str() returns a LaTeX string in which the scalar coefficients of the multivector bases have been simplified (grouped, factored, etc.). This function is used when using print in the LaTeX mode. The member function A.raw_latex_str() returns a LaTeX string in which the scalar coefficients of the multivector bases have not been simplified.
Printing Lists/Tuples of Multivectors/Differential Operators¶
Since the expressions for multivectors or differential operators can be very long printing lists or tuples of such items can easily exceed the page with when printing in LaTeX or in “ipython notebook.” I order to alleviate this problem the function Fmt can be used.
Fmt(obj,fmt=0)
This function from the
printermodule allows the formatted printing of lists/tuples or multivectors/differential operators.
objobjis a list or tuple of multivectors and/or differential operators.fmt=0fmt=0prints each element of the list/tuple on an individual lines[26].fmt=1prints all elements of the list/tuple on a single line.If l is a list or tuple to print in the LaTeX environment use the command
print Fmt(l) # One element of l per lineor
print Fmt(l,1) # All elements of l on one lineIf you are printing in “ipython notebook” then enter
Fmt(l) # One element of l per lineor
Fmt(l,1) # All elements of l on one line
Bibliography¶
- Chris Doran and Anthony Lasenby, “Geometric Algebra for Physicists,” Cambridge University Press, 2003. http://www.mrao.cam.ac.uk/~clifford
- David Hestenes and Garret Sobczyk, “Clifford Algebra to Geometric Calculus,” Kluwer Academic Publishers, 1984. http://geocalc.clas.asu.edu/html/CA_to_GC.html
- Alan Macdonald, “Linear and Geometric Algebra,” 2010. http://faculty.luther.edu/~macdonal/laga
- Alan Macdonald, “Vector and Geometric Calculus,” 2012. http://faculty.luther.edu/~macdonal/vagc
- Hestenes, “New Foundations for Classical Mechanics,” Kluwer Academic Publishers, 1999. http://geocalc.clas.asu.edu/html/NFCM.html
- Dorst, D. Fontijne, S. Mann, “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry*,” Morgan Kaufmann, \(2^{\text{nd}}\) printing, 2009. http://www.geometricalgebra.net/
- Christian Perwass, “Geometric Algebra with Applications in Engineering,” Springer, 2008
- John W. Arthur, “Understanding Geometric Algebra for Electromagnetic Theory,” Wiley-IEEE Press, 2011.
[4] By the manifold embedding theorem any \(m\)-dimensional manifold is isomorphic to a \(m\)-dimensional vector manifold
[5] This product in not necessarily positive definite.
[6] In this section and all following sections we are using the Einstein summation convention unless otherwise stated.
[7] We use the Christoffel symbols of the first kind to calculate the derivatives of the basis vectors and the product rule to calculate the derivatives of the basis blades where (http://en.wikipedia.org/wiki/Christoffel_symbols)
and
The Christoffel symbols of the second kind,
could also be used to calculate the derivatives in term of the original basis vectors, but since we need to calculate the reciprocal basis vectors for the geometric derivative it is more efficient to use the symbols of the first kind.
[8] In this case \(D_{B}^{j_{1}\dots j_{n}} = F\) and \(\partial_{j_{1}\dots j_{n}} = 1\).
[9] For example in three dimensions \({\left \{{3} \rbrc} = (0,1,2,3,(1,2),(2,3),(1,3),(1,2,3))\) and as an example of how the superscript would work with each grade \({{\eb}}^{0}=1\), \({{\eb}}^{1}={{\eb}}^{1}\), \({{\eb}}^{{\lp {1,2} \rp }}={{\eb}}^{1}{\wedge}{{\eb}}^{2}\), and \({{\eb}}^{{\lp {1,2,3} \rp }}={{\eb}}^{1}{\wedge}{{\eb}}^{2}{\wedge}{{\eb}}^{3}\).
[10] We are following the treatment of Tensors in section 3–10 of .
[11] We assume that the arguments are elements of a vector space or more generally a geometric algebra so that the concept of linearity is meaningful.
[12] Since X or the metric tensor can be functions of coordinates the vector space that the geometric algebra is constructed from is not necessarily flat so that the geometric algebra is actually constructed on the tangent space of the manifold which is a vector space.
[13] The signature of the vector space, \((p,q)\), is required to determine whether the square of the normalized pseudoscalar, \(I\), is \(+1\) or \(-1\). In the future the metric tensor would be required to create a generalized spinor .
[14] Using LaTeX output if a basis vector is denoted by \({{\eb}}_{x}\) then \({{\eb}}\) is the root symbol and \(x\) is the subscript
[15] There is a multivector class, Mv, but in order the insure that every multivector is associated with the correct geometric algebra we always use the member function Ga.mv to instantiate the multivector.
[16] Denoted in text output by A__x, etc. so that for text output A would be printed as A__x*e_x+A__y*e_y+A__z*e_z.
[17] If the metric is input as a list or list or lists the object is no longer quoted (input as a string). For example the old metric=’[1,1,1]’ becomes metric=[1,1,1].
[18] In the future it should be possible to generate closed form expressions for \(e^{A}\) if \(A^{r}\) is a scalar for some interger \(r\).
[19] See footnote [fn_6].
[20] GAeval is in the printer module.[fn_6]
[21] Remember that normalization is currently supported only for orthogonal systems (diagonal metric tensors).
[22] The 180 in the ConEmu command line is the width of the console you wish to display in characters. Change the number to suit you.
[23] I am not exactly sure what the different parameter setting do. I achieved the result I wished for by trial and error. I encourage the users to experiment and share their results.
[24] In Ipython notebook tuples, or lists, or dictionarys of multivectors do print correctly. One mode of Fmt() corrects this deficiency.
[25] Preprocessing do not occur for the Ipython notebook and the string post processing commands % and # are not used in this case.
[26] The formatting of each element is respected as applied by A.Fmt(fmt=1,2, or 3) where A is an element of objso that if multivector/differential operation have been formatted to print on multiple lines it will printed on multiple lines.[Fmt_format]