Geometric Calculus
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, geometric calculus extends the
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
to include differentiation and
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
. The formalism is powerful and can be shown to encompass other mathematical theories including
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s.


Differentiation

With a geometric algebra given, let a and b be vectors and let F be a
multivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
-valued function of a vector. The
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
of F along b at a is defined as :(\nabla_b F)(a) = \lim_, provided that the limit exists for all b, where the limit is taken for scalar \epsilon. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued. Next, choose a set of
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s \ and consider the operators, denoted \partial_i, that perform directional derivatives in the directions of e_i: :\partial_i : F \mapsto (x\mapsto (\nabla_ F)(x)). Then, using the
Einstein summation notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
, consider the operator: :e^i\partial_i, which means :F \mapsto e^i\partial_i F, where the geometric product is applied after the directional derivative. More verbosely: :F \mapsto (x\mapsto e^i(\nabla_ F)(x)). This operator is independent of the choice of frame, and can thus be used to define what in geometric calculus is called the ''vector derivative'': :\nabla = e^i\partial_i. This is similar to the usual definition of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
, but it, too, extends to functions that are not necessarily scalar-valued. The directional derivative is linear regarding its direction, that is: :\nabla_ = \alpha\nabla_a + \beta\nabla_b. From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction a can be written a = (a\cdot e^i) e_i, so that: :\nabla_a = \nabla_ = (a\cdot e^i)\nabla_ = a\cdot(e^i\nabla_) = a\cdot \nabla. For this reason, \nabla_a F(x) is often noted a\cdot \nabla F(x). The standard
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...
for the vector derivative is that it acts only on the function closest to its immediate right. Given two functions F and G, then for example we have :\nabla FG = (\nabla F)G.


Product rule

Although the partial derivative exhibits a
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, the vector derivative only partially inherits this property. Consider two functions F and G: :\begin\nabla(FG) &= e^i\partial_i(FG) \\ &= e^i((\partial_iF)G+F(\partial_iG)) \\ &= e^i(\partial_iF)G+e^iF(\partial_iG). \end Since the geometric product is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
with e^iF \ne Fe^i in general, we need a new notation to proceed. A solution is to adopt the ''
overdot When used as a diacritic mark, the term dot is usually reserved for the '' interpunct'' ( · ), or to the glyphs "combining dot above" ( ◌̇ ) and "combining dot below" ( ◌̣ ) which may be combined with some letters of t ...
notation'', in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define :\dotF\dot=e^iF(\partial_iG), then the product rule for the vector derivative is :\nabla(FG) = \nabla FG+\dotF\dot.


Interior and exterior derivative

Let F be an r-grade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives, :\nabla \cdot F = \langle \nabla F \rangle_ = e^i \cdot \partial_i F, :\nabla \wedge F = \langle \nabla F \rangle_ = e^i \wedge \partial_i F. In particular, if F is grade 1 (vector-valued function), then we can write :\nabla F = \nabla \cdot F + \nabla \wedge F and identify the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
and
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
as :\nabla \cdot F = \operatorname F, :\nabla \wedge F = I \, \operatorname F. Unlike the vector derivative, neither the interior derivative operator nor the exterior derivative operator is invertible.


Multivector derivative

The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the multivector derivative. Let F be a multivector-valued function of a multivector. The directional derivative of F with respect to X in the direction A, where X and A are multivectors, is defined as :A*\partial_X F(X)=\lim_\frac\ , where A* B=\langle A B\rangle is the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. With \ a vector basis and \ the corresponding
dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...
, the multivector derivative is defined in terms of the directional derivative as :\frac=\partial_X=\sum_J e^J (e_J*\partial_X)\ , where J is denoting an ordered set of basis vector indices, as in the article section Geometric algebra#Dual basis. This equation is just expressing \partial_X in terms of components in a reciprocal basis of blades, as discussed in that article section. A key property of the multivector derivative is that :\partial_X\langle X A\rangle=P_X(A)\ , where P_X(A) is the projection of A onto the grades contained in X. The multivector derivative finds applications in
Lagrangian field theory Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
.


Integration

Let \ be a set of basis vectors that span an n-dimensional vector space. From geometric algebra, we interpret the
pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
e_1 \wedge e_2 \wedge\cdots\wedge e_n to be the
signed volume In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector ...
of the n- parallelotope subtended by these basis vectors. If the basis vectors are
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
, then this is the unit pseudoscalar. More generally, we may restrict ourselves to a subset of k of the basis vectors, where 1 \le k \le n, to treat the length, area, or other general k-volume of a subspace in the overall n-dimensional vector space. We denote these selected basis vectors by \. A general k-volume of the k-parallelotope subtended by these basis vectors is the grade k multivector e_ \wedge e_ \wedge\cdots\wedge e_. Even more generally, we may consider a new set of vectors \ proportional to the k basis vectors, where each of the \ is a component that scales one of the basis vectors. We are free to choose components as infinitesimally small as we wish as long as they remain nonzero. Since the outer product of these terms can be interpreted as a k-volume, a natural way to define a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
is :\begind^kX &= \left(dx^ e_\right) \wedge \left(dx^e_\right) \wedge\cdots\wedge \left(dx^e_\right) \\ &= \left( e_\wedge e_\wedge\cdots\wedge e_ \right) dx^ dx^ \cdots dx^.\end The measure is therefore always proportional to the unit pseudoscalar of a k-dimensional subspace of the vector space. Compare the
Riemannian volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
in the theory of differential forms. The integral is taken with respect to this measure: :\int_V F(x)\,d^kX = \int_V F(x) \left( e_\wedge e_\wedge\cdots\wedge e_ \right) dx^ dx^ \cdots dx^. More formally, consider some directed volume V of the subspace. We may divide this volume into a sum of
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
. Let \ be the coordinates of the vertices. At each vertex we assign a measure \Delta U_i(x) as the average measure of the simplices sharing the vertex. Then the integral of F(x) with respect to U(x) over this volume is obtained in the limit of finer partitioning of the volume into smaller simplices: :\int_V F\,dU = \lim_ \sum_^n F(x_i)\,\Delta U_i(x).


Fundamental theorem of geometric calculus

The reason for defining the vector derivative and integral as above is that they allow a strong generalization of
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
. Let \mathsf(A;x) be a multivector-valued function of r-grade input A and general position x, linear in its first argument. Then the fundamental theorem of geometric calculus relates the integral of a derivative over the volume V to the integral over its boundary: \int_V \dot \left(\dot dX;x \right) = \oint_ \mathsf (dS;x). As an example, let \mathsf(A;x)=\langle F(x) A I^ \rangle for a vector-valued function F(x) and a (n-1)-grade multivector A. We find that :\begin\int_V \dot \left(\dot dX;x \right) &= \int_V \langle\dot(x)\dot\,dX\,I^ \rangle \\ &= \int_V \langle\dot(x)\dot\,, dX, \rangle \\ &= \int_V \nabla \cdot F(x)\,, dX, . \end Likewise, :\begin\oint_ \mathsf (dS;x) &= \oint_ \langle F(x)\,dS\,I^ \rangle \\ &= \oint_ \langle F(x) \hat\,, dS, \rangle \\ &= \oint_ F(x) \cdot \hat\,, dS, . \end Thus we recover the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
, :\int_V \nabla \cdot F(x)\,, dX, = \oint_ F(x) \cdot \hat\,, dS, .


Covariant derivative

A sufficiently smooth k-surface in an n-dimensional space is deemed a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. To each point on the manifold, we may attach a k-blade B that is tangent to the manifold. Locally, B acts as a pseudoscalar of the k-dimensional space. This blade defines a
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
of vectors onto the manifold: :\mathcal_B (A) = (A \cdot B^) B. Just as the vector derivative \nabla is defined over the entire n-dimensional space, we may wish to define an ''intrinsic derivative'' \partial, locally defined on the manifold: :\partial F = \mathcal_B (\nabla )F. (Note: The right hand side of the above may not lie in the tangent space to the manifold. Therefore, it is not the same as \mathcal_B (\nabla F), which necessarily does lie in the tangent space.) If a is a vector tangent to the manifold, then indeed both the vector derivative and intrinsic derivative give the same directional derivative: :a \cdot \partial F = a \cdot \nabla F. Although this operation is perfectly valid, it is not always useful because \partial F itself is not necessarily on the manifold. Therefore, we define the ''covariant derivative'' to be the forced projection of the intrinsic derivative back onto the manifold: :a \cdot DF = \mathcal_B (a \cdot \partial F) = \mathcal_B (a \cdot \mathcal_B (\nabla) F). Since any general multivector can be expressed as a sum of a projection and a rejection, in this case :a \cdot \partial F = \mathcal_B (a \cdot \partial F) + \mathcal_B^ (a \cdot \partial F), we introduce a new function, the shape tensor \mathsf(a), which satisfies :F \times \mathsf(a) = \mathcal_B^ (a \cdot \partial F), where \times is the commutator product. In a local coordinate basis \ spanning the tangent surface, the shape tensor is given by :\mathsf(a) = e^i \wedge \mathcal_B^ (a \cdot \partial e_i). Importantly, on a general manifold, the covariant derivative does not commute. In particular, the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
is related to the shape tensor by : \cdot D, \, b \cdot D=-(\mathsf(a) \times \mathsf(b)) \times F. Clearly the term \mathsf(a) \times \mathsf(b) is of interest. However it, like the intrinsic derivative, is not necessarily on the manifold. Therefore, we can define the
Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
to be the projection back onto the manifold: :\mathsf(a \wedge b)=-\mathcal_B (\mathsf(a) \times \mathsf(b)). Lastly, if F is of grade r, then we can define interior and exterior covariant derivatives as :D \cdot F = \langle DF \rangle_, :D \wedge F = \langle D F \rangle_, and likewise for the intrinsic derivative.


Relation to differential geometry

On a manifold, locally we may assign a tangent surface spanned by a set of basis vectors \. We can associate the components of a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
, and the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
as follows: :g_=e_i \cdot e_j, :\Gamma^k_=(e_i \cdot De_j) \cdot e^k, :R_=(\mathsf(e_i \wedge e_j) \cdot e_k) \cdot e_l. These relations embed the theory of differential geometry within geometric calculus.


Relation to differential forms

In a
local coordinate system In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
(x^1, \ldots, x^n), the coordinate differentials dx^1, ..., dx^n form a basic set of one-forms within the
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
. Given a
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
I = (i_1, \ldots, i_k) with 1 \le i_p \le n for 1 \le p \le k, we can define a k-form :\omega = f_I\,dx^I=f_\,dx^\wedge dx^\wedge\cdots\wedge dx^. We can alternatively introduce a k-grade multivector A as :A = f_e^\wedge e^\wedge\cdots\wedge e^ and a measure :\begind^kX &= \left(dx^ e_\right) \wedge \left(dx^e_\right) \wedge\cdots\wedge \left(dx^e_\right) \\ &= \left( e_\wedge e_\wedge\cdots\wedge e_ \right) dx^ dx^ \cdots dx^.\end Apart from a subtle difference in meaning for the exterior product with respect to differential forms versus the exterior product with respect to vectors (in the former the ''increments'' are covectors, whereas in the latter they represent scalars), we see the correspondences of the differential form :\omega \cong A^ \cdot d^kX = A \cdot \left(d^kX \right)^, its derivative :d\omega \cong (D \wedge A)^ \cdot d^X = (D \wedge A) \cdot \left(d^X \right)^, and its
Hodge dual In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
:\star\omega \cong (I^ A)^ \cdot d^kX, embed the theory of differential forms within geometric calculus.


History

Following is a diagram summarizing the history of geometric calculus.


References and further reading

* {{Industrial and applied mathematics Applied mathematics Calculus Geometric algebra