Gretings fellow mathematicians!
I'm learning multivariable calculus and i seem to be stuck in a problem which i cannot even manage to construct the intuition behind it, the problem is the following:
Using Green's theorem calculate the circulation field of $F=(x+y)i+(2y-x^2)j$ around $4x^2+y^2=4$ circled once counter clockwise.
I know it is a very newbie question but im still starting this course, any help would be appreciated!
Let $\gamma$ be a parametrization of the curve. The integral of $F$ along $\gamma$ is by definition (in differential form notation) $$\int_\gamma (x+y)dx + (2y-x^2)dy.$$ Green's theorem states that this is equal to $$\iint_E \left[\frac{\partial}{\partial x}(2y-x^2) - \frac{\partial}{\partial y}(x+y)\right] dxdy$$ Where $E$ is the area enclosed in the ellipse. Just compute this integral and you're done. The intuition i have for this value is that it's simply a measure of how much the curve "follows" the vector field (in fact it is $0$ when they're orthogonal, positive when they have the same orientation and negative when they have opposite orientations). Maybe try to plot the field and guess the sign before actually computing the integral to see if you get it.