Geometric calculus

The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus, differential geometry, and differential forms.

is defined as provided that the limit exists for all

This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued.

: 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 standard order of operations for the vector derivative is that it acts only on the function closest to its immediate right.

, then for example we have Although the partial derivative exhibits a product rule, the vector derivative only partially inherits this property.

In this case, if we define then the product rule for the vector derivative is Let

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.

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.

The multivector derivative finds applications in Lagrangian field theory.

If the basis vectors are orthonormal, then this is the unit pseudoscalar.

We are free to choose components as infinitesimally small as we wish as long as they remain nonzero.

The integral is taken with respect to this measure: More formally, consider some directed volume

as the average measure of the simplices sharing the vertex.

over this volume is obtained in the limit of finer partitioning of the volume into smaller simplices: The reason for defining the vector derivative and integral as above is that they allow a strong generalization of Stokes' theorem.

Then the fundamental theorem of geometric calculus relates the integral of a derivative over the volume

We find that Likewise, Thus we recover the divergence theorem, A sufficiently smooth

, locally defined on the manifold: (Note: The right hand side of the above may not lie in the tangent space to 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

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

Therefore, we can define the Riemann tensor to be the projection back onto the manifold: Lastly, if

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.

form a basic set of one-forms within the coordinate chart.

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.

Following is a diagram summarizing the history of geometric calculus.