Geometric Calculus - Differentiation - Product Rule

Product Rule

Although the partial derivative exhibits a product rule, the geometric derivative only partially inherits this property. Consider two functions F and G:

\begin{align}\nabla(FG) &= e^i\partial_i(FG) \\
&= e^i((\partial_iF)G+F(\partial_iG)) \\
&= e^i(\partial_iF)G+e^iF(\partial_iG) \end{align}

Since the geometric product is not commutative with in general, we cannot proceed further without new notation. A solution is to adopt the overdot notation, in which the scope of a geometric derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define

then the product rule for the geometric derivative is

Read more about this topic:  Geometric Calculus, Differentiation

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