Differentiation
With a geometric algebra given, let a and b be vectors and let F(a) be a multivector-valued function. The directional derivative of F(a) along b is defined as
provided that the limit exists, where the limit is taken for scalar ε. 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 let, where we use the Einstein summation notation. This allows the geometric derivative to be treated as an operator
which is independent of the choice of frame. 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 geometric derivative is that it acts only on the function closest to its immediate right. Given two functions F and G, then for example we have
Read more about this topic: Geometric Calculus