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.

Differentiation

be vectors be a multivector-valued function of a vector. The directional derivative 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

This operator is independent of the choice of frame, and can thus be used to define what in geometric calculus is called the

This is similar to the usual definition of the gradient, but it, too, extends to functions that are not necessarily scalar-valued.

can be written is often noted .

The standard order of operations and

Product rule

Although the partial derivative exhibits a product rule and

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

Interior and exterior derivative

be an

and identify the divergence and curl

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 .

in the direction is the scalar product the corresponding dual basis, the multivector derivative is defined in terms of the directional derivative as in terms of components in a reciprocal basis of blades, as discussed in the article section Geometric algebra#Dual basis.

onto the grades contained in .

The multivector derivative finds applications in Lagrangian field theory.

Integration

be a set of basis vectors that span an -dimensional vector space. From geometric algebra, we interpret the pseudoscalar to be the signed volume-parallelotope subtended by these basis vectors. If the basis vectors are orthonormal, then this is the unit pseudoscalar.

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 multivector .

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

The measure is therefore always proportional to the unit pseudoscalar of a -dimensional subspace of the vector space. Compare the Riemannian volume form

of the subspace. We may divide this volume into a sum of 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 be a multivector-valued function of and general position

and a ()-grade multivector

Thus we recover the divergence theorem

Covariant derivative

A sufficiently smooth -surface in an -dimensional space is deemed a manifold. To each point on the manifold, we may attach a acts as a pseudoscalar of the -dimensional space. This blade defines a projection

is defined over the entire -dimensional space, we may wish to define an

(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.)

itself is not necessarily on the manifold. Therefore, we define the

we introduce a new function, the shape tensor is the commutator product

Importantly, on a general manifold, the covariant derivative does not commute. In particular, the commutator

is of interest. However it, like the intrinsic derivative, is not necessarily on the manifold. Therefore, we can define the Riemann tensor

is of grade 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, 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

In a local coordinate system (), the coordinate differentials form a basic set of one-forms within the coordinate chart. Given a multi-index for , we can define a

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 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