Answer
See proof
Work Step by Step
Let $C$ be a piecewise smooth curve composed of smooth curves $C_1, C_2, \ldots, C_n$ in which $C_1: \vec{r}(\tau): \tau_0 \leq \tau \leq \tau_1$ $C_2: \vec{r}(\tau): \tau_1 \leq \tau \leq \tau_2$ $C_3: \vec{r}(\tau): \tau_2 \leq \tau \leq \tau_3$ $\ldots$ $C_n: \vec{r}(\tau): \tau_{n-1} \leq \tau \leq \tau_n$ where $\vec{r}(t): a_0 \leq t \leq a_1$, $\vec{r}(t): a_1 \leq t \leq a_2$, $\vec{r}(t): a_2 \leq t \leq a_3$, $\vec{r}(t): a_{n-1} \leq t \leq a_n$ in order to be consistent with theorem (17.3.3). Thus \begin{align*} \int_C \mathbf{F} \cdot d\mathbf{r} &= \sum_{k=1}^n \int_{C_k} \mathbf{F} \cdot d\mathbf{r} \\ &= \sum_{k=1}^n \int_{C_k} \frac{\partial x}{\partial \phi} dx + \frac{\partial y}{\partial \phi} dy \\ &= \sum_{k=1}^n \int_{a_{k-1}}^{a_k} \left(\frac{\partial x}{\partial \phi} \frac{dx}{dt} + \frac{\partial y}{\partial \phi} \frac{dy}{dt}\right) dt \\ &\approx \sum_{k=1}^n \int_{a_{k-1}}^{a_k} \left[\frac{d}{dt} \phi(x(t),y(t))\right] dt \\ &= \sum_{k=1}^n \left[\phi(x(a_k),y(a_k)) - \phi(x(a_{k-1}),y(a_{k-1}))\right] \\ &= \phi(x_1,y_1) - \phi(x_0,y_0) \end{align*} Q.E.D
