Let $\alpha: [a,b] \rightarrow \mathbb{R}^2$ be a smooth path (i.e. $\alpha'$ is continuous on $[a,b]$), and let $f$ be a continuous vector field. The line integral of $f$ along $\alpha$ is defined as $\int_a^b f[\alpha(t)]\cdot \alpha'(t) dt$. In a typical calculus textbook, it is written as $\int_C f_1 dx + f_2 dy$, suggesting that the line integral depends only on $C$ – the image of the path, not the actual parameterization $\alpha$.
In general however, this is not true. For example, $\oint_C -ydx + x dy =2\pi$ when integrated along the path $\alpha: [0,2\pi]\rightarrow \mathbb{R}^2, \alpha(t) = (\cos(t), \sin(t))$. The value becomes $4\pi$ when integrated along $\beta: [0,4\pi]\rightarrow \mathbb{R}^2, \beta(t)= (\cos(t),\sin(t))$, even though $\alpha$ and $\beta$ have the same image.
My question: what additional assumption is required to make the notation $\int_C f_1dx +f_2 dy$ unambiguous?
Best Answer
More generally, you can say this: Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a continuous function. Let $\alpha:[0,1]\rightarrow\mathbb{R}^n$ and $\beta:[0,1]\rightarrow\mathbb{R}^n$ be two smooth curves. Suppose there is a smooth curve $\theta:[0,1]\rightarrow[0,1]$ such that $\theta(0)=0, \theta(1)=1$, and $\beta(t) = \alpha(\theta(t))$ for all $t \in [0,1]$. Then: $$ \int_0^1 f(\alpha(t))\cdot \alpha'(t)dt = \int_0^1 f(\beta(t))\cdot \beta'(t)dt $$
Proof: By the chain rule we have $\beta'(t) = \alpha'(\theta(t))\theta'(t)$. Thus: \begin{align} \int_0^1 f(\beta(t))\cdot \beta'(t) dt &= \int_0^1 f(\alpha(\theta(t)))\cdot \alpha'(\theta(t))\theta'(t)dt \\ &= \int_0^1 f(\alpha(u))\cdot \alpha'(u)du \end{align} where the second equality is by the change of variables rule for integration of real-valued functions. $\Box$
In your example of a loop traversed once by a curve $\alpha$ and then twice by a curve $\beta$, you cannot write $\beta(t) = \alpha(\theta(t))$ for all $t \in [0,1]$. Defining $\theta(t) = 2t$ does not work because that would only be defined for $t \in [0,1/2]$. In another example of a clockwise loop versus a counter-clockwise loop, you again cannot find such a $\theta(t)$ function.
Edit: Some additional observations:
Allowing for non-injectivity:
The condition on $\theta$ above does not require the functions $\alpha, \beta$, or $\theta$ to be injective. For example, it works for the following: Let $\alpha(t)$ be any smooth curve defined over $t \in [0,1]$ (possibly having repeated values over that interval) and define $\beta(t) = \alpha(\theta(t))$, where $$ \theta(t) = \sin^2\left(\frac{5\pi t}{2}\right) $$ The $\beta$ curve retraces the $\alpha$ curve 5 times, and the first four times cancel each other out.
Path independence when $f$ is a gradient
Suppose there is a smooth function $H:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $f(x) = \nabla H(x)$ for all $x \in \mathbb{R}^n$. Then path integrals of $f$ depend only on the endpoints of the path, not on the intermediate values of the path. Indeed: $$ \int_0^1 f(\alpha(t))\cdot \alpha'(t) dt = \int_0^1\frac{d}{dt}\left[H(\alpha(t))\right] dt = H(\alpha(1))-H(\alpha(0)) $$
Cauchy integral formula
There are also connections to complex numbers and the Cauchy integral formula. For example, in some cases the path integral of a loopy curve depends only on the number of times the curve winds clockwise around the origin.