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 ...
to include
differentiation and
integration. 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 mult ...
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 application ...
s.
Differentiation
With a geometric algebra given, let
and
be
vectors and let
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 ...
-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 ...
of
along
at
is defined as
:
provided that the limit exists for all
, where the limit is taken for scalar
. 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
, that perform directional derivatives in the directions of
:
:
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:
:
which means
:
where the geometric product is applied after the directional derivative. More verbosely:
:
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'':
:
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 gr ...
, but it, too, extends to functions that are not necessarily scalar-valued.
The directional derivative is linear regarding its direction, that is:
:
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
can be written
, so that:
:
For this reason,
is often noted
.
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 examp ...
for the vector derivative is that it acts only on the function closest to its immediate right. Given two functions
and
, then for example we have
:
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
and
:
:
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 of ...
with
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
:
then the product rule for the vector derivative is
:
Interior and exterior derivative
Let
be an
-grade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives,
:
:
In particular, if
is grade 1 (vector-valued function), then we can write
:
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 ...
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 ...
as
:
:
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
be a multivector-valued function of a multivector. The directional derivative of
with respect to
in the direction
, where
and
are multivectors, is defined as
:
where
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 alge ...
. 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 the ...
, the multivector derivative is defined in terms of the directional derivative as
:
where
is denoting an ordered set of basis vector indices, as in the article section
Geometric algebra#Dual basis. This equation is just expressing
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
:
where
is the projection of
onto the grades contained in
.
The multivector derivative finds applications in
Lagrangian field theory.
Integration
Let
be a set of basis vectors that span an
-dimensional vector space. From geometric algebra, we interpret the
pseudoscalar to be the
signed volume of the
-
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 ...
, then this is the unit pseudoscalar.
More generally, we may restrict ourselves to a subset of
of the basis vectors, where
, to treat the length, area, or other general
-volume of a subspace in the overall
-dimensional vector space. We denote these selected basis vectors by
. A general
-volume of the
-parallelotope subtended by these basis vectors is the grade
multivector
.
Even more generally, we may consider a new set of vectors
proportional to the
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
-volume, a natural way to define a
measure is
:
The measure is therefore always proportional to the unit pseudoscalar of a
-dimensional subspace of the vector space. Compare the
Riemannian volume form in the theory of differential forms. The integral is taken with respect to this measure:
:
More formally, consider some directed volume
of the subspace. We may divide this volume into a sum of
simplices. Let
be the coordinates of the vertices. At each vertex we assign a measure
as the average measure of the simplices sharing the vertex. Then the integral of
with respect to
over this volume is obtained in the limit of finer partitioning of the volume into smaller simplices:
:
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
be a multivector-valued function of
-grade input
and general position
, linear in its first argument. Then the fundamental theorem of geometric calculus relates the integral of a derivative over the volume
to the integral over its boundary:
As an example, let
for a vector-valued function
and a (
)-grade multivector
. We find that
:
Likewise,
:
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 ...
,
:
Covariant derivative
A sufficiently smooth
-surface in an
-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
-blade
that is tangent to the manifold. Locally,
acts as a pseudoscalar of the
-dimensional space. This blade defines a
projection of vectors onto the manifold:
:
Just as the vector derivative
is defined over the entire
-dimensional space, we may wish to define an ''intrinsic derivative''
, locally defined on the manifold:
:
(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
, which necessarily does lie in the tangent space.)
If
is a vector tangent to the manifold, then indeed both the vector derivative and intrinsic derivative give the same directional derivative:
:
Although this operation is perfectly valid, it is not always useful because
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:
:
Since any general multivector can be expressed as a sum of a projection and a rejection, in this case
:
we introduce a new function, the
shape tensor , which satisfies
:
where
is the
commutator product. In a local coordinate basis
spanning the tangent surface, the shape tensor is given by
:
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, ...
is related to the shape tensor by
:
Clearly the term
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:
:
Lastly, if
is of grade
, then we can define interior and exterior covariant derivatives as
:
:
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 allow ...
, 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 distanc ...
, 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:
:
:
:
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 ...
(
), the coordinate differentials
, ...,
form a basic set of one-forms within the
coordinate chart. Given a
multi-index with
for
, we can define a
-form
:
We can alternatively introduce a
-grade multivector
as
:
and a measure
:
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
:
its derivative
:
and its
Hodge dual
:
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