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When ${\bf A}=\nabla f$ in some region $\Omega\subset{\mathbb R}^3$ then for any curve $\gamma\subset\Omega$ with initial point ${\bf p}$ and endpoint ${\bf q}$ one has $$f({\bf q})-f({\bf p})=\int_\gamma {\bf A}\cdot d{\bf x}\ .\tag{1}$$ This is the $3$-dimensional version of the familiar formula $$f(q)-f(p)=\int_p^q f'(t)\>dt$$ from elementary calculus. ...


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Don't know about MAPLE, but MATHEMATICA's algorithms are given in the following (hard to find) link "Some Notes on Internal Implementation" (Scroll down to "Algebra and Calculus") Some of the algorithms (e.g. Gauss Elimination) are textbook stuff, but others are more esoteric, and might be unknown to people outside a specific research community - e.g. ...


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Here's an approach which, while naive, can be improved considerably. I'll simply define the functions $f(w)$ and $g(w)$ and then the functions $x(w)$, $y(w)$, and $z(w)$ in terms of the integrals, making sure the values at zero are zero. I changed the complex variable $z$ to $w$ to avoid conflict with the Cartesian coordinate $z$. The input parameters ...


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$\newcommand{\Cpx}{\mathbf{C}}$Not exactly an answer, but a collection of miscellanea too extensive for a comment. Caveat: I've never done any of this myself; YMMV. :) You can fix $z_{0}$ arbitrarily: a different choice amounts to an additive constant (i.e., a translation of the surface). The complex quantity $z_{1} = u + iv$ comprises the surface ...



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