Curve parametrizations and line integrals

Let r(t), for a≤t≤b, be a continuously differentiable regular curve, meaning r′(t) never vanishes. Its speed is the norm of r′, and its length is the integral of that speed. Piecewise regular curves are handled by adding the contributions of their regular pieces. A scalar line integral of density ρ along the curve inserts ρ(r(t)) before the speed factor. This computes mass when ρ is mass per unit length. It counts the parametrized traversal, so tracing a segment twice counts its contribution twice.

An increasing differentiable reparametrization with nonzero derivative changes the rate of travel but preserves the scalar integral through the ordinary substitution rule. Reversing orientation also preserves it because speed is a norm. That invariance does not mean that any formula with the same endpoints describes the same curve: different paths can pass through different densities and have different lengths. Nor can the speed factor be omitted unless the parameter happens to be arc length. A unit-speed parametrization is a useful special case, not a default assumption.

For a continuous vector field F, the work integral pairs its value with the infinitesimal oriented displacement. Reversing a curve changes the sign. Only the tangential field component contributes; a field normal to the instantaneous velocity does no work along that motion. On a closed curve, this integral is often called circulation. A nonzero circulation records a failure of global path independence, even when a plot of the field looks locally smooth. The actual parametrized traversal, including repeated turns, remains part of the data.

If F is the gradient of a C¹ potential φ on a neighborhood of the curve, the chain rule turns the integrand into the derivative of φ(r(t)). The fundamental theorem then gives only the endpoint potential difference, so every closed curve has zero work. This argument proves path independence for gradient fields without requiring the path to be straight. It also explains the sign convention: work by the gradient is final potential minus initial potential; physical force is sometimes defined as the negative gradient of potential energy, changing that interpretation.

In the plane, a C¹ field (P,Q) with Qx−Py=0 on a simply connected open domain has a potential. A hole can obstruct this conclusion. The field (−y/(x²+y²),x/(x²+y²)) is curl-free on the punctured plane, but its counterclockwise unit-circle work is 2π. The field is not defined at the origin, so the usual no-hole hypothesis is absent. This is a topological obstruction to a single-valued global potential, not an arithmetic contradiction in the local derivative test.

A practical line-integral solution begins by declaring whether the integral is scalar or vector and whether orientation matters. Then check the curve's parameter range, regular pieces, and any excluded field points. Search for a potential only after checking its domain; otherwise use the defining parameter integral. Endpoints alone are enough precisely when path independence has been established. A closed-curve result of zero for one chosen loop is not proof that every loop has zero circulation, just as one successful numerical test does not establish a global theorem.