For simplifying things, suppose we are working on $\mathbb{R}^3$. We have a surface $S$, a path $C$, and a path $D$ in the $xy$ plane where a line integral is going to be taken.
I would start defining the variables I will use: "Let $x,y,z,t \in \mathbb{R}$.
I would then define the surface $S$ as $S=\left\{ (x,y,z)\mid z=f(x,y) \right\}$, and then add to 1. $f:\mathbb{R}^2\mapsto \mathbb{R}$ "The function $f$ is a mapping from one vector field to a scalar field"
The path $C$, I want to specify that the path is parametrized by $t$, I would define it as $C=\left\{(x,y)\mid x \rightarrow x(t), y\rightarrow y(t) \right\}$
On 3, I am stating that $x$ and $y$ are both function of $t$, but in 2, $x$ and $y$ shall not depend on the parameter $t$ to build the surface, so to avoid this conflict, I would rewrite $C$ as $C = \left\{(x_1,y_1)\mid x_1 \rightarrow x_1(t), y_1 \rightarrow y_1 (t)\right\}$... Then, do I need to add to the statement 1. $x_1(t), y_1(t) \in \mathbb{R}$ OR $x_1 (t) ,x_2 (t): \mathbb{R} \mapsto \mathbb{R}$
- I will define $D$ the same way as $C$, but using $x_2(t)$ and $y_2 (t)$. Also, I want $D$ to be the same collection of points as $C$, but the direction of the path is inverse. If the path $C$ is going from point A to B, then the path $D$ is going from B to A. So, May I say $C=D$, or instead of $C=D$ something like $C \subset D, D \subset C$? And the functions $x_1(t)\ne x_2(t), y_1 (t) \ne y_2$, But at some point on the path, we may have a value of $t=t_a$ such that $x_1(t_a) = x_2(t_a)$, and the same for $y_1(t_a) = y_2 (t_a)$ breaking with $x_1(t) \ne x_2(t), y_1(t) \ne y_2 (t)$.
Since now, the final statement looks like this:
Let $x,y,z,t \in \mathbb{R}$. Consider the function $f:\mathbb{R}^2\mapsto \mathbb{R}$ and the functions $x_1(t), x_2 (t), y_1 (t) y_2 (t) : \mathbb{R} \mapsto \mathbb{R}$. Consider the surface $S=\left\{ (x,y,z)\mid z=f(x,y) \right\}$ and the path $C = \left\{(x_1,y_1)\mid x_1 \rightarrow x_1(t), y_1 \rightarrow y_1 (t)\right\}$ and $D = \left\{(x_2,y_2)\mid x_2 \rightarrow x_2(t), y_2 \rightarrow y_2 (t)\right\}$ where $D=C$ and $x_1(t) \ne x_2(t)$ and $y_1(t) \ne y_2(t)$, \begin{eqnarray} \int_C f(x,y)ds=\int_Df(x,y)ds \end{eqnarray} Which leads to \begin{eqnarray} \int_C f(x_1 (t) , y_1 (t) ) \sqrt{\left(\frac{dx_1(t)}{dt}\right)^2 + \left(\frac{dy_1(t)}{dt} \right)^2} dt = \int_D f(x_2 (t) , y_2 (t) ) \sqrt{\left(\frac{dx_2(t)}{dt}\right)^2 + \left(\frac{dy_2(t)}{dt} \right)^2} dt \end{eqnarray}
I want someone to point out all errors by defining all the elements, and how would it be better to define them! Also, the thing I am most worried about is the last part of the 4th point.
Thanks in advance :)