# Limits of overdot notation in geometric algebra

In geometric calculus the over dot notation is used to denote the proper way to do the vector differentiation of a multivector product - $$\nabla (AB) = (\nabla A)B + (\dot{\nabla}A)\dot{B}$$ The question is does make any sense to have more than one over dotted $$\dot{\nabla}$$ and one over dotted multivector in an expression. The reason for this question is that I am implementing the over dot symbolism in the galgebra symbolic geometric algebra python module.

Note that when it comes to multivectors I consider $$\dot{(ABC)}$$ to be a single multivector. The code would be (A*B*C).odot(). The program would first evaluate $$ABC$$ and then apply the over dot (actually just a flag on the result) to the product. Parenthesis are essential to the proper evaluation of expressiong with multivector differential operators and multivectors since in general $$\nabla(ABC) \ne (\nabla A)BC \ne (\nabla AB)C.$$

• I would rather not use overdot notation, and instead use brackets: $$\nabla[AB]=\nabla[A]B+\nabla A[B]$$ This can also work with several $\nabla$'s differentiating different things, by using numbers to pair each $\nabla$ with its brackets: $${_1\!\nabla}\,{_2\!\nabla} A\;{^2\![BC]}A\;{^1\![D]}$$ means the first $\nabla$ is differentiating $D$ and the second is differentiating $BC$. (I don't know if this is a good idea; I thought of it while studying tensors, hence the resemblance to "up-down" tensor contraction notation.) – mr_e_man Dec 10 '18 at 1:59