Question 9.9.3: Evaluate ∫C ydx + xdy, where C is a path with initial point ......
Evaluate \int_{C} y d x+x d y, where C is a path with initial point (0,0) and terminal point (1,1).
The path C shown in FIGURE 9.9.2 represents any piecewise-smooth curve with initial and terminal points (0,0) and (1,1). We have just seen that \mathrm{F}=y \mathrm{i}+x \mathrm{j} is a conservative vector field defined at each point of the x y-plane and that \phi(x, y)=x y is a potential function for \mathrm{F}. Thus, in view of (2) of Theorem 9.9.1 and (3), we can write
\int_C F \cdot d r =\int_C \nabla \phi \cdot d r =\phi(B)-\phi(A), (2)
\int_A^B F \cdot d r =\int_A^B \nabla \phi \cdot d r . (3)
\begin{aligned} \int_{C} y d x+x d y & =\int_{(0,0)}^{(1,1)} \mathrm{F} \cdot d \mathrm{r}=\int_{(0,0)}^{(1,1)} \nabla \phi \cdot d \mathrm{r} \\ & =x y\bigg]_{(0,0)}^{(1,1)} \\ & =1 \cdot 1-0 \cdot 0=1 . \end{aligned}
