Chapter 16.3, Problem 31E

(a)

To determine

Find the parameterizations for the line segment from $\left(-1,1\right)$ to $\left(0,0\right)$ followed by the line segment from $\left(0,0\right)$ to $\left(1,1\right)$ and evaluate the integral ${\displaystyle \underset{C}{\int }F\cdot dr}$ for the vector field $F=\nabla \left(x^3{y}^2\right)$.

(b)

To determine

Evaluate the integral ${\displaystyle \underset{C}{\int }F\cdot dr}$ for the vector field $F=\nabla \left(x^3{y}^2\right)$ with the potential function $f\left(x,y\right)=x^3{y}^2$.