Solenoidal and conservative fields

[Trifis]

@Ben Niehoff Yes you're right, I didn't say there aren't any such fields. I did say that we do not attribute any physical meaning to their vector potentials, as we do with the megnetic vector potential for example...
(Btw I hate the θ convention for the azimuth :P)

@Muphrid hmmm aren't monogenic functions just the generalization of analytic ones in higher dimensions? I might have missed sth, but where exactly do you prove that both of their derivatives must be zero in this decomposition?

 

[Muphrid]

@Muphrid hmmm aren't monogenic functions just the generalization of analytic ones in higher dimensions? I might have missed sth, but where exactly do you prove that both of their derivatives must be zero in this decomposition?

Geometric algebra allows us to represent complex numbers as being part of an exterior algebra. Basically, $w(x,y) = u(x,y) + e^{xy} v(x,y)$, where $e^{xy}$ is a bivector.

Now, take the vector derivative of this object.

$$\nabla w= (e^x \partial_x + e^y \partial_y)w = e^x \left[\partial_x u - \partial_y v\right] + e^y \left[\partial_y u + \partial_x v\right]$$

Setting $\nabla w = 0$ enforces the Cauchy-Riemann conditions for complex differentiability. However, instead of working in the realm of complex analysis, one can factor out $e^x$ on the right to get

$$f = we^x = u e^x - v e^y \implies \nabla f = \nabla w e^x = (\nabla w) e^x = 0$$

(This explains the sign change to the y-component that is often necessary when converting between complex analysis and vector fields.) Regardless, $f$ is a vector field, and condition for analyticity--for integrability--still holds. As I showed in the decomposition posts above, as long as $\nabla f = 0$, the function is entirely determined by its values on a closed surface, and there is no need for volume integrals to account for source terms. This is exactly in analogy to the properties of complex analytic functions. Hence, $\nabla f = 0$ is the generalization of the Cauchy-Riemann condition not only to a real 2d vector space but to arbitrary dimensions.