In Activity 13.3.4, we considered the vector field \(\vF(x,y) = \langle y^2,2xy+3\rangle\) and two different oriented curves from \((-2,5)\) to \((3,30)\text{.}\) We found that the value of the line integral of \(\vF\) was the same along those two oriented curves.
Verify that \(\vF(x,y) = \langle y^2,2xy+3\rangle\) is a gradient vector field by showing that \(\vF = \nabla f\) for the function \(f(x,y) = xy^2 + 3y\text{.}\)
Calculate the change in the output of the scalar function \(f\) over the curves \(C_1\) and \(C_2\text{.}\) In other words, what is the difference in the output of \(f\) at the start of the curve and the end of the curve? How does this value compare to the value of the line integral \(\int_{C_1}\vF\cdot d\vr\) you found in Activity 13.3.4?
Let \(C_3\) be the line segment from \((1,1)\) to \((3,4)\text{.}\) Calculate \(\int_{C_3}\vF\cdot d\vr\) as well as \(f(3,4)-f(1,1)\text{.}\) Write a sentence that compares your answer to this part to your result for part 13.4.1.b.
