Consider the vector field $\mathbf{F}=\langle x^2 y+y^3-y,3x+2y^2 x+e^y\rangle$. For which closed non-self-intersecting curve in the plane does the line integral over this vector field have the maximal value? What is this value?
Let $\mathbf{F}=\langle xy,y^2\rangle$, let $C$ be the unit circle centered at the origin, and consider $\int_C \mathbf{F}\cdot d\mathbf{r}$. Which portions of $C$ contribute positively to this integral? Calculate the integral two ways, first directly and then by using Green's theorem.
For the first question. I need to maximize the line integral how do I do that?
How can I apply Maxima minima concept to this?
Best Answer
Hint: Assuming that the desired curve $C$ is positively oriented (otherwise the maximum does not exist), by Green's theorem (we can do this, since $C$ is a simple closed curve), we have that: $$\int_C \mathbf{F}\cdot d\mathbf{r}=\oint_C (x^2 y+y^3-y)~dx+(3x+2y^2 x+e^y)~dy=\iint_D (4-x^2-y^2)~dx~dy$$ Where in the $x$-$y$ plane is the integrand non-negative? So, what should you take the region $D$ to be so that the integral is maximal? And what does that imply for $C=\partial D$?