Winding number in geometric algebra

And so we can adapt it to geometric algebra in 2 dimensions.

Let \(\partial M\) be a curve (we are calling it \(\delta M\) because we will later perform the analysis using the fundamental theorem of calculus for geometric algebra), and let \(j\), be the pseudo scalar in 2 dimensions.

The winding number is: \[\frac{1}{2\pi j} \oint_{\partial M} d\mathbf{x} \frac{\mathbf{x}}{|x|^{-2}} = \frac{1}{2\pi j}\oint_{\partial M} d\mathbf{x} \mathbf{x}^{-1} \]

The scalar component will cancel itself, and just the bivector component is left, therefore we divide by the pseudoscalar. And 2π is the length of the boundary of a circle.

Using the fundamental theorem of calculus for geometric calculus (Alan Macdonald - Vector and Geometric Calculus, section 10.1), supposing that \(M\) is a surface.

\[\frac{1}{2\pi j}\oint_{\partial M} d\mathbf{x} \mathbf{x}^{-1} = \frac{1}{2\pi j} \int_{ M} d\mathbf{x^2} \partial\left(\mathbf{x}^{-1}\right)\]

\(\partial\left(\mathbf{x}^{-1}\right)\) is equal to 0 at every place except the origin, but we can "include the origin" by using a kronecker delta

\[\frac{1}{2\pi j}\oint_{\partial M} d\mathbf{x} \mathbf{x}^{-1} = \int_{ M} d\mathbf{x^2} \delta_\mathbf{x}(\mathbf{x}) = \begin{cases} 1,& \text{if the volume includes the origin}\\ 0, & \text{otherwise} \end{cases} \]

We can extend this to more dimensions. Let \(n\) be the dimensions, and let \(A_n\) be the \(n-1\) dimensional area of the \(n\) dimensional unit sphere. Then: \[\frac{1}{A_n}\oint_{\partial M} d\mathbf{x^{n-1}} \frac{\mathbf{x}}{|x|^n} = \int_{ M} d\mathbf{x^n} \delta_\mathbf{x}(\mathbf{x}) = \begin{cases} 1,& \text{if the volume includes the origin}\\ 0, & \text{otherwise} \end{cases} \]

for 3 dimensions you should recognize this equation from the electrical field of a point charge in the origin.

Let say you wanna find the \(0\)s of a function, and you are using the winding number algorithm https://www.youtube.com/watch?v=b7FxPsqfkOY.

TODO