Additivity of line integrals under concatenation

Statement

Let $\gamma_1 \colon [a, b] \to \mathbb{R}^n$ and $\gamma_2 \colon [b, c] \to \mathbb{R}^n$ be piecewise $C^1$ curves with $\gamma_1(b) = \gamma_2(b)$. Let $\gamma \colon [a, c] \to \mathbb{R}^n$ be their concatenation: $\gamma = \gamma_1$ on $[a, b]$ and $\gamma = \gamma_2$ on $[b, c]$. For each continuous vector field $F$ and each continuous function $f$ on the image of $\gamma$, $$\int_\gamma F \cdot d\mathbf{r} = \int_{\gamma_1} F \cdot d\mathbf{r} + \int_{\gamma_2} F \cdot d\mathbf{r}, \qquad \int_\gamma f \, ds = \int_{\gamma_1} f \, ds + \int_{\gamma_2} f \, ds.$$

Remarks

Proof: apply the parametric formula on $[a, c]$ and split the integral at $t = b$. If the parameter intervals do not match, first shift the parameter of $\gamma_2$. A shift does not change line integrals. Together with path reversal, this lets one compute line integrals over the boundary of a polygon, side by side. The concatenation is often written $\gamma_1 + \gamma_2$ or $\gamma_1 * \gamma_2$.

Explanations

L1 human

Walking from $A$ to $B$ and then from $B$ to $C$ is one walk from $A$ to $C$. The line integral over the whole walk is the sum of the line integrals over the two parts.