Questions about Green's Theorem?

Question

Background Information:

Now let us talk about integrals that relate to closed curves. They enjoy a large number of remarkable properties, among which we must point out those which are stated in the following theorems:

First Theorem: The position of a mobile point $P$ being determined in space using rectilinear, or polar coordinates, or of any other nature, let us name $x, y, z,\ldots$ quantities which vary in a way that varies with the position of that point. Let $S$ also be an area which is measured in a given plane, or on a given surface, and which limits a single closed curve on all sides. Let us conceive then that the mobile point $P$ is subjected to traverse this curve while turning around the area $S$ in a determined direction.

Let us name $s$ the arc of the same curve, measured positively in the direction in question, from a fixed origin, or at least a variable which constantly increases with this arc. Finally, let $k$ be a function of the variables $x, y, z, \ldots$ and of their derivatives relating to $s$; and denote by $(S)$ the value acquired by the integral $$\int k\, ds$$ when the mobile point $P$, having traversed the entire contour of the area $S$, returns to its original position.

Question:

This might look like a normal single integral, but there's a bit more going on here. What does it mean for $k$ to be a function of the variables $x, y, z,\ldots$ and of their derivatives relating to $s$?"

Answer 1

$k$ is known as a differential $1$-form. In general it looks something like

$$ k = a(x,y,z,\dots)dx + b(x,y,z,\dots)dy + c(x,y,z,\dots)dz + \dots$$

so the integral $$\int k $$

is equivalent to the integral of the "pull-back" of the $1$-form. The pull-back (basically given by chain rule) looks like

$$ \int \bigg(a(x,y,z,\dots)\frac{dx}{ds} + b(x,y,z,\dots)\frac{dy}{ds} + c(x,y,z,\dots)\frac{dz}{ds} + \dots\bigg) ds\text{ .}$$

In short, $k = k(x,y,z,\frac{dx}{ds},\frac{dy}{ds},\frac{dz}{ds}).$

Answer 2

Let $\gamma: \mathbb{R} \to \mathbb{R}^n$ be the curve you discuss in your paragraphs, mapping $s$ to a point $(x,y,z,\ldots)$ in space on the boundary of your region.

Then all the text is saying is that $k$ is a function of $\gamma$ and/or $\frac{d\gamma}{ds}$, i.e., at every point along the boundary, $k$ depends on the position of that boundary point, and the tangent vector to the boundary curve at that point.