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 vector calculus, differential geometry, and differential forms. With a geometric algebra given, let and be vectors and let be a multivector-valued function of a vector. The directional derivative of along at is defined as 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, 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 directional derivative is linear regarding its direction, that is: From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction can be written , so that: For this reason, is often noted . The standard order of operations for the vector derivative is that it acts only on the function closest to its immediate right. Given two functions and , then for example we have Although the partial derivative exhibits a product rule, the vector derivative only partially inherits this property. Consider two functions and : Since the geometric product is not commutative with in general, we need a new notation to proceed. A solution is to adopt the overdot notation, in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot.