Editing Geometric algebra (section)

== Geometric calculus ==
{{main|Geometric calculus}}

Geometric calculus extends the formalism to include differentiation and integration including differential geometry and [[differential form]]s.{{sfn|ps=|Hestenes|Sobczyk|1984}}

Essentially, the vector derivative is defined so that the GA version of [[Green's theorem]] is true,
: <math>\int_A dA \,\nabla f = \oint_{\partial A} dx \, f</math>
and then one can write
: <math>\nabla f = \nabla \cdot f + \nabla \wedge f</math>
as a geometric product, effectively generalizing [[Stokes' theorem]] (including the differential form version of it).

In 1D when {{tmath|1= A }} is a curve with endpoints {{tmath|1= a }} and {{tmath|1= b }}, then
: <math>\int_A dA \,\nabla f = \oint_{\partial A} dx \, f</math>
reduces to
: <math>\int_a^b dx \, \nabla f = \int_a^b dx \cdot \nabla f = \int_a^b df = f(b) -f(a)</math>
or the fundamental theorem of integral calculus.

Also developed are the concept of [[Vector Manifold|vector manifold]] and geometric integration theory (which generalizes differential forms).