TITLE: Do line integrals along non-piecewise-smooth curves exist?
QUESTION [5 upvotes]: This article at Wolfram Mathworld has the following theorem on conservative vector fields:

Theorem. The following conditions are equivalent for a conservative vector field $ \mathbf{F} $ defined on an open subset $ U $ of $ \mathbb{R}^{n} $:

For any oriented, simple and closed curve $ C $ whose image lies in $ U $, the contour integral of $ \mathbf{F} $ along $ C $ equals $ 0 $, i.e., $ \displaystyle \oint_{C} \mathbf{F} \cdot \mathrm{d}{\mathbf{s}} = 0 $.

For any two oriented and simple curves $ C_{1} $ and $ C_{2} $ with the same endpoints whose images lie in $ U $, we have $ \displaystyle \int_{C_{1}} \mathbf{F} \cdot \mathrm{d}{\mathbf{s}} = \int_{C_{2}} \mathbf{F} \cdot \mathrm{d}{\mathbf{s}} $.

There exists a differentiable function $ f: U \to \mathbb{R} $, called the scalar potential function of $ \mathbf{F} $, such that $ \mathbf{F} = \nabla f $.



However, I have read in other places that the curves $ C $, $ C_{1} $ and $ C_{2} $ must be piecewise smooth. It seems like a necessary restriction on otherwise arbitrary curves.
Hence, my question is:

Question. Is the theorem valid for non-piecewise-smooth curves? What about non-piecewise-differentiable curves? I think that the line integral might not be defined then.

REPLY [2 votes]: I'm afraid Berrick Fillmore did not address the original poster's questions. His discussion pertains to line integrals of scalar fields, whereas the original question concerns line integrals of vector fields. These can not be defined in metric spaces -- one needs a vector space with an inner product.
Let me give a quick answer to the original poster's question. You can define a line integral of a continuous vector field over a rectifiable (continuous) curves. It is a sum of Riemann-Stieltjes integrals of the vector field components with respect to the corresponding components of the curve.  The crucial point is that a curve in n-dimensional Euclidean spaces is rectifiable iff its components are functions of bounded variation. A standard theorem about Riemann-Stieltjes integrals states that the integral exists when one integrates a continuous function with respect to a function of bounded variation. If the vector field has a potential, then this line integral only depends on the endpoints. To see this observe that a line integral of a conservative vector field over a rectifiable curve can be approximated arbitrarily closely by the same line integral over the piecewise linear curve specified by an appropriate partition. Since that line integral only depends on the endpoints, the same must be true for the line integral over the rectifiable curve.
There is a standard counter-example showing that the Riemann-Stieltjes integral of a continuous function over a continuous function of non-bounded variation need not be defined. It involves wild sine/cosine functions. I believe this example can be tweaked to show that a line integral of continuous vector field need not be defined when the curve in non-rectifiable, even when the vector field is conservative. However I have not checked that this works.
If there is interest, I can provide additional details.
If there is intere