In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus, differential geometry, and differential forms.[1]

Differentiation

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With a geometric algebra given, let   and   be vectors and let   be a multivector-valued function of a vector. The directional derivative 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 vectors   and consider the operators, denoted  , that perform directional derivatives in the directions of  :

 

Then, using the Einstein summation notation, 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, 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 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

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Although the partial derivative exhibits a product rule, the vector derivative only partially inherits this property. Consider two functions   and  :

 

Since the geometric product is not commutative with   in general, we need a new notation to proceed. A solution is to adopt the overdot 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

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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 and curl as

 
 

Unlike the vector derivative, neither the interior derivative operator nor the exterior derivative operator is invertible.

Multivector derivative

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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. With   a vector basis and   the corresponding dual basis, the multivector derivative is defined in terms of the directional derivative as[2]

 

This equation is just expressing   in terms of components in a reciprocal basis of blades, as discussed in the article section Geometric algebra#Dual basis.

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

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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, 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

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The reason for defining the vector derivative and integral as above is that they allow a strong generalization of Stokes' theorem. 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,

 

Covariant derivative

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A sufficiently smooth  -surface in an  -dimensional space is deemed a manifold. 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 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 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

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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, the Christoffel symbols, and the Riemann curvature tensor as follows:

 
 
 

These relations embed the theory of differential geometry within geometric calculus.

Relation to differential forms

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In a local coordinate system ( ), 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

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Following is a diagram summarizing the history of geometric calculus.

 
History of geometric calculus.

References and further reading

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  1. ^ David Hestenes, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984, ISBN 90-277-2561-6
  2. ^ Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge University press. p. 395. ISBN 978-0-521-71595-9.