The typical parametrization of the line segment from \((0,1)\) to \((3,3)\) (the oriented curve \(C_3\) in Example 13.3.9) is \(\vr(t) = \langle 3t,1+2t\rangle\) where \(0 \leq t\leq 1\text{.}\) 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 Example 13.3.9.
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 \((0,0,2)\text{.}\)
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 Activity 13.3.4?
