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I think that the answer to (2), generally speaking is 'no'. For example, consider the case of $\mathcal{A}$ being the Cauchy-Riemann operator: $$ \mathcal{A}(u,v) = (u_x - v_y,\ u_y+v_x). $$ The kern …

I don't know what you might mean by 'uniformly accelerated surface', but I think that, by 'uniformly accelerated curve', you mean a curve in the plane parametrized in such a way that its velocity at t …

There exist $F$ for which there is no global solution $f$ to the above equation. Here is how you can construct an example: First, regard $F$ as a vector field on $\mathbb{R}^3$ and consider its dual …

Here is what you should try: Consider the function $$ p(x,\lambda) = K_0(x,0,\ldots,0)+\lambda\ K_1(x,0,\ldots,0) + \cdots + \lambda^n\ K_n(x,0,\ldots,0) $$ For any analytic solution $\lambda=L(x)$ t …

You are asking about the subject of orthogonal (coordinate) systems. There is an extensive literature on this subject, in particular by Darboux when $n=3$, and if you search on "triply orthogonal sys …

Edited on May 2, 2020: The OP pointed out that I had not addressed a special case (namely $C=1$ below), so I am amending my answer to address this and reorganizing so that the $C=1$ case gets addresse …

Ah, so you just mean that, when you regard $\mathbb{R}^2$ as $\mathbb{C}$ and you have a complex-valued function $f$ on $Q$, the closed first quadrant of $\mathbb{C}$, that satisfies the Cauchy-Rieman …

The answer is 'no', because polynomial mappings with polynomial inverses preserve volumes up to a constant multiple. To see why this property holds, suppose that $p:\mathbb{R}^d\to\mathbb{R}^d$ is a p …

See the following Wikipedia page: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)

An approach that should work is to derive the differential equation that any minimizer would have to satisfy and check that its solutions are the known ones for which equality holds. To fill in the d …

The answer is 'yes, there always is such a vector field $X$' and, in particular, the answer to your 'new question' is also 'yes'. (In fact, the first 'yes' implies the second 'yes', but the second 'y …

Here are a few comments that you might find useful, though they don't completely solve the problem. First, using symmetries of the problem, you can easily reduce to the case that $f$ is mapping the …

Here is a simple example: Take $F = w^3 +3 w u^2 -v^3$ on $\mathbb{R}^3$ with coordinates $(u,v,w)$. At the point $p=(u,v,w)=(1,0,0)$, we have that $F=0$ can be solved for $w$ as a function of $(u,v …

The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them: First, it helps to …

I think that there is a smooth (or analytic) result of the kind that you are seeking: Let $M^m$ be a connected smooth (or analytic) manifold, and let $P:M\times\mathbb{R}\to\mathbb{R}$ be a smooth …