For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals

Given: A piecewise-$C^1$ contour $\gamma =x+iy$, a continuous $f=u+iv$, and an admissible partition $(t_j)$. Apply all Euclidean path suppliers below to $\mathrm{\Gamma }=\mathrm{\Phi }\circ \gamma =(x,y)$. By Complex path length equals coordinate length and piecewise speed integral, complex length agrees with coordinate length on every subinterval, the complex velocity notation is ${\gamma }_j^{\mathrm{\prime }}=x_j^{\mathrm{\prime }}+i{y}_j^{\mathrm{\prime }}$, and $\mathrm{\mid }{\gamma }_j^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\parallel }{\mathrm{\Gamma }}_j^{\mathrm{\prime }}{\mathrm{\parallel }}_2$. The absolute-integral definition The absolute line integral over a rectifiable path using its arc-length function gives $s_{\gamma }=s_{\mathrm{\Gamma }}$; on each smooth piece this differs from its local arc-length function by a constant. In [L2] use the real vector fields $(u\circ {\mathrm{\Phi }}^{-1},-v\circ {\mathrm{\Phi }}^{-1})$ and $(v\circ {\mathrm{\Phi }}^{-1},u\circ {\mathrm{\Phi }}^{-1})$ along $\mathrm{\Gamma }$, and the analogous scalar field.

Let $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Suppose $\alpha$ is continuous on $[a,b]$, differentiable on $(a,b)$, and ${\alpha }^{\mathrm{\prime }}$ extends continuously to $[a,b]$. Then $f$ is Riemann–Stieltjes integrable with respect to $\alpha$ and ${\int }_a^{b}fd\alpha ={\int }_a^{b}f(x){\alpha }^{\mathrm{\prime }}(x)dx$ (A continuously differentiable integrator reduces Stieltjes integration to ordinary integration).

The published scalar and vector line integrals are the sums of $\int f(\gamma )\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }$ and $\int ⟨F(\gamma ),{\gamma }^{\mathrm{\prime }}⟩$ over the smooth pieces (Scalar line integrals with respect to arc length and vector-field line integrals).

A piecewise-$C^1$ path has length equal to the sum of the speed integrals, with corners and singleton intervals allowed (A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

Vector-valued derivatives and integrals are defined componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

Let $a<b$ be reals and let $f:[a,b]\to \mathbb{R}$ be continuous. Then $f$ is bounded and Riemann integrable on $[a,b]$ (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

Let $a<b$ and let $\gamma :[a,b]\to {\mathbb{R}}^n$ be $C^1$, and set $s_{\gamma }(t):=L_{[a,t]}(\gamma {\mathrm{\mid }}_{[a,t]})$. Then $s_{\gamma }$ is differentiable on $[a,b]$ in the relative sense and $s_{\gamma }^{\mathrm{\prime }}(t)=\parallel {\gamma }^{\mathrm{\prime }}(t){\parallel }_2$, the values at $a$ and $b$ being the relative one-sided derivatives (For a C1 path the arc-length accumulation function has derivative equal to speed).