Chapter 16.3, Problem 30E

(a)

To determine

Find the work done by the vector field $F=e^{yz}i+\left(xz{e}^{yz}+z\cos y\right)j+\left(xy{e}^{yz}+\sin y\right)k$ over the line segment $x=1,y=\frac{\pi t}{2},z=1-t,0\le t\le 1$ from $\left(1,0,1\right)$ to $\left(1,\frac{\pi }{2},0\right)$.

(b)

To determine

Find the work done by the vector field $F=e^{yz}i+\left(xz{e}^{yz}+z\cos y\right)j+\left(xy{e}^{yz}+\sin y\right)k$ over the line segment $\left(1,0,1\right)$ to the origin followed by the line segment from the origin to $\left(1,\frac{\pi }{2},0\right)$.

(c)

To determine

Find the work done by the vector field $F=e^{yz}i+\left(xz{e}^{yz}+z\cos y\right)j+\left(xy{e}^{yz}+\sin y\right)k$ over the line segment from $\left(1,0,1\right)$ to $\left(1,0,0\right)$ followed by $x\text{-axis}$ from $\left(1,0,0\right)$ to the origin, followed by the parabola $y=\frac{\pi x^2}{2},z=0$ from there to $\left(1,\frac{\pi }{2},0\right)$.