Geometric calculus

In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms. ∫ V L ˙ ( ∇ ˙ d X ; x ) = ∮ ∂ V ⁡ L ( d S ; x ) . {displaystyle int _{V}{dot {mathsf {L}}}left({dot { abla }}dX;x ight)=oint _{partial V}{mathsf {L}}(dS;x).} In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms. With a geometric algebra given, let a {displaystyle a} and b {displaystyle b} be vectors and let F ( a ) {displaystyle F(a)} be a multivector-valued function. The directional derivative of F ( a ) {displaystyle F(a)} along b {displaystyle b} is defined as provided that the limit exists, where the limit is taken for scalar ϵ {displaystyle epsilon } . 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 { e i } {displaystyle {e_{i}}} and consider the operators, denoted ∂ i {displaystyle partial _{i}} , that perform directional derivatives in the directions of e i {displaystyle e_{i}} : Then, using the Einstein summation notation, consider the operator: