Activity 13.4.5

The typical parametrization of the line segment from $(0,1)$ to $(3,3)$ (the oriented curve $C_3$ in this example) is $\vr(t) = \langle 3t,1+2t\rangle$ where $0 \leq t\leq 1$. Use this parametrization to calculate $\int_{C_3}\vF\cdot d\vr$ for the vector field $\vF = x\vi$ and compare your answer to the result of this example.

Calculate $\int_C \langle (3xy+e^z), x^2, (4z+xe^z)\rangle\cdot d\vr$ where $C$ is the oriented curve consisting of the line segment from the origin to $(1,1,1)$ followed by the line segment from $(1,1,1)$ to $(0,0,2)$.

Calculate $\int_{C'} \langle 3xy+e^z, x^2, 4z+xe^z\rangle\cdot d\vr$ where $C_3$ is the line segment from $(0,0,0)$ to $(0,0,2)$.

Is the vector field you considered in the previous two parts a gradient vector field? Why or why not? How does this compare to the vector field $\vF$ of this activity?