The notion of a curve in the context of line integrals

Question

For brevity I'm making the following assumption: I'm only talking about regular curves on $\left[a,b\right]$ with values in $\mathbb{R}^{n}$, and line integrals of scalar fields.

[Since there are a lot of questions at the end and you have to dig through the text below to make sense of them, I'm willing to offer 150 bounty points for a complete and thorough answer by a knowledgeable person in either differential geometry or vector analysis (or related fields) of all the questions.

There are three common ways to define curves:

1. Here they are defined as mappings $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$.

2. Here a curve is used as a subset $C$ of $\mathbb{R}^{n}$ that's the image of a (regular) $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$- which clearly doesn't work with the concept from 1. (although Wikipedia has linked to that - has no one spotted this so far ?)

3. Here curves are equivalence classes of mappings $\left[a,b\right]\rightarrow\mathbb{R}^{n}$ (which are equivalent if they are obtained from reparametrisations of eachother).

Now these definitions relate in different ways to the concept of a line integral over scalar field $f$ along $\gamma$.

If I take 1. as my definition, I can define $$ \int_{\gamma}fd s:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$ without mathematical problems, but I have the "psychological" problem that I would like my line integral to not be dependent on all the information $\gamma$ contains (since, for example, different reparametrisations of $\gamma$ give me the same $\int_{\gamma}f$) - of course I can show that this holds in a separate theorem, but this just seems ugly).

If I take 2. as my definition, I can define $$ \int_{C}fds:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$ where $\gamma$ is any parametrisation $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$ that has image $C$. This has mathematical problems: There are (regular) curves that have the same image , but aren't equivalent, so have different lengths. For example $$ t\mapsto\left(\begin{array}{c} \cos t\\ \sin t \end{array}\right)\ \text{and}\ t\mapsto\left(\begin{array}{c} \cos2t\\ \sin2t \end{array}\right) $$ where $t\in\left[0,2\pi\right]$. So $\int_{C}f s$ isn't well defined since it depends on the choice of the parametrisation of $C$. But from a "psychological" perspective I like this the most, since it only has a geometric content (since $C\subseteq\mathbb{R}^{n})$ and not a dynamic one (I don't know anything about the "speed" with which $C$ is traced) and my personal view is, that line integral (or arc lengths, since I could have discussed this issue in the same matter with arc lengths instead of line integrals) should be purely geometrical.

If I take 3. as my definition, I can define $$ \int_{\hat{\gamma}}fd s:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$ where $\hat{\gamma}$ is the equivalance class of $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$. This seems to me to be a compromise between 1. and 2.: The line integral is (from the start) independent of reparametrisations - but it isn't purely geometric, as the definitions from 2.
But other weird issues arise in this case: Since a curve is a set of mappings (which makes up the equivalence class) I loose a comfortable way of speaking about curves by the following subtle point: I can't say anymore that a smooth curve is also a continuous curve, since for example the equivalence class of the identity on $[0,1]$ (taken as a curve in $\mathbb{R}$), viewed as a smooth curve (i.e. as the set $\{f\mid f:[0,1] \rightarrow \mathbb{R} \text{ is smooth and can be reparametrised to be the identity}\}$) does not contain $$t\mapsto\begin{cases} 2t, & 0\leq t\leq\frac{1}{2}\\ 1, & \frac{1}{2}<t\leq1 \end{cases}$$which is is in the equivalence class of the identity, viewed as a continuous curve (i.e. as the set $\{f\mid f:[0,1] \rightarrow \mathbb{R} \text{ is continuous and can be reparametrised to be the identity}\}$) .

Questions:

A. Is there a standart definition of what a curve (and thus a line integral) is ? If there isn't a definition that's valid for the whole of mathematics, is the definition at least separately standarised in subfields (like differential geometric, vector analysis etc.) ?

B. Is my view that line integrals and arc lengths should only depend on a purely geometric object of a curve "correct" ? (You may understand what you wish by "correct".)

C. Could I perhaps save the definition of $\int_{C}f ds$ from 2., by modifying its definition so that it says $$ \int_{C}fds:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$only for those $\gamma$ that are injective on $\left(a,b\right)$ ? (In this case I would need a proposition, that for every $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$ there exists an injective $\gamma':\left[a,b\right]\rightarrow\mathbb{R}^{n}$, such that $\gamma$ and $\gamma'$ have the same image. Does such a proposition exist ?)

Would I exclude important physical phenomena by this alternative definition of 2.?

D. I've know that there also a fourth definition if a curve, namely as a topological space locally homeomorphic to a line. How does this definition reduce to each of the three definitions above (as Wikipedia says at the beginning of the article http://en.wikipedia.org/wiki/Curve about curves) and how do I define a line integral (or arc length) by this definition ?

Note: I've already read this in case you wanted to direct me there.

Answer 1

This is a nice question IMHO, and my take is as follows (not a full answer maybe and not bounty worth btw):

• The image definition is certainly ill-posed, as your example illustrates. Possible fix: $\forall p\in C$, define $t_p = \min\{t: \gamma(t) = p\}$, and redefine $\gamma = \gamma|_{D}$ where $D = \cup_{p\in C}t_{p}$. BTW Wikipedia says the following in the entry of Curve:

  The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

• Mathematically, I prefer to define in the first way, i.e., as a mapping to avoid ambiguity so that different parametrizations lead to the same value for the line integral. A definition restricting an oriented curve to have a designated parameterisation is good for me too. "Psychologically", I prefer the physical way, i.e., view $f = F\cdot \gamma'/\|\gamma'\|$, provided there exists such field $F$. Then the scalar integral is the same as the work done by the field $F$ when a unit mass point particle traverses the whole curve $\gamma$ from $\gamma(a)$ to $\gamma(b)$. In your first example, if using the second parametrization, the particle rotates the circle twice, and we have twice the work done.

• I agree with you on that "the line integrals and arc lengths should only depend on a purely geometric object of a curve", so it is "subjectively correct" for myself :) The reason I prefer the physical way, is that, we can then associate the line integral with the bilinear dual pair between forms and chains. Bilinearity is in that $$ \langle \gamma, a_1 \omega_1 + a_2\omega_2\rangle = \int_{\gamma} (a_1 \omega_1 + a_2\omega_2) = a_1 \int_{\gamma}\omega_1 + a_2\int_{\gamma}\omega_2, $$ and $$ \langle c_1\gamma_1 + c_2 \gamma_2, \omega\rangle = \int_{c_1\gamma_1 + c_2 \gamma_2} \omega = c_1 \int_{\gamma_1}\omega + c_2\int_{\gamma_2}\omega. $$ The linearity in $\gamma$ essentially rules out the ambiguity in definition 2. For in this view, the "curve" is viewed as an element in the 1-chain vector space, which includes the piecewise smooth curves being continuous not smooth. By your example in the third definition, reparametrization seems working fine "psychologically" on for curves such that $\gamma'(t)$ not vanishing for all $t$, for now smooth curves are continuous curves when the equivalence restricts on $\gamma$ of which $\gamma'(t)\neq 0$.