Reading through the linked answer to Can a non-zero vector field have zero divergence and zero curl? left me with some follow up interrelated questions:
- To clarify the first part of the answer, which states that
we can write any vector field $F$ in terms of its value on a boundary curve $\partial M$ and its divergence and curl within a region $M$.
$$iF(p) = \oint_{\partial M} G(p-p') \, d\ell' \, F(p') + \int_M G(p-p') \, dA' \, \nabla F|_{p'}$$
where $G(p) = p/2\pi p^2$ is the 2d Green's function for $\nabla$. If $\nabla \cdot F = 0$ and $\nabla \wedge F= 0$, then $\nabla F = 0$ everywhere, and the area integral goes to zero.
Is $\nabla\wedge\vec{F} = i\nabla\times\vec{F}$, analogous to $\vec{F}\wedge\vec{G}=i\vec{F}\times\vec{G}$? I've only just learned parts of geometric algebra, so I want to make sure I'm not incorrectly assuming here.
My current interpretation I have of this section of the answer is as follows: you can write some $\vec{F}$ as the sum of a irrotational, compressible vector field and a rotational, incompressible vector field, but then you must also introduce some 'boundary condition'/'constant of integration'/'winding number' which is what comes from that closed line integral. Is this correct/reasonable?
To follow up on this interpretation - I'm aware the terms boundary condition, constant of integration, and winding number all mean different things. Is one the 'closest' to what is meant here?
The answer concludes by mentioning
one word for this in the geometric calculus literature is monogenic, which is used to distinguish from the weaker condition of being harmonic
I'm a little unsure of what 'this' is referring to - the idea that you can decompose $\vec{F}$ into three components, or the case when the third component is 0? And if the former, how is this distinct from harmonic?
Happy to split this into multiple questions if that's cleaner, but they seemed sufficiently interrelated to post together. Thank you!
First, to make sure its clear, $\nabla F = \nabla\cdot F + \nabla\wedge F$.
Yes, in fact it is the same formula since you can treat $\nabla$ as a vector, hence $\nabla F$ is essentially the multivector sum of divergence and curl. Keep in mind the cross product is only defined in 3D (and subspaces of 3D), but the outer product is defined in arbitrary dimension.
I'm not sure where you're getting that from and don't think you can say that about either integral.
In case it's not clear, $p'$ is the variable being integrated over.
The first integral is a boundary condition since you only need to specify the values of $F$ on $\partial M$ to compute it, and you could interpret it as a "constant" of integration analogous to the 1D case, but it's not a constant in general. The case $\nabla F = 0$ identically is when $F$ is irrotational and incompressible, and the second integral disappears; so now $F$ inside of a region is completely determined by its values on any boundary around that region. Cauchy's integral formula from complex analysis is exactly a special case of this.
It's worth noting that what the general formula, with both integrals, is doing is solving the equation $\nabla F(x) = H(x)$ given any $H$ (probably with some reasonable restrictions I can't recite off the top of my head). This is what is meant when it's said that $\nabla$ is invertible.
For winding number, if $F$ had singularities but still $\nabla F = 0$ everywhere else (which makes it analogous to a meromorphic function), then this would compute something analogous to the winding number times residue, just like in complex analysis.
A monogenic function is a function such that $\nabla F = 0$, and harmonic is $\nabla^2 F = \nabla(\nabla F) = 0$. Hence every monogenic function is harmonic, but not all harmonic functions are monogenic; monogenic is stronger than harmonic.