I'm having a little problem understanding this question on Green's Theorem. It asks to use Green’s Theorem to evaluate the line integral $\int_C F.dr$, where $F=(e^{y^2} −2y)i+(2xye^{y^2} +\sin(y^2))j$ and C goes along a straight line from $(0,0)$ to $(1,2)$ and continues along a straight line to $(3, 0)$.

I tried using the circulation form of Green's Theorem and ended up with the answer 4+2 = 6. I then tried using the divergence form and seemed to end up with a different answer. Can I check if this is the correct way of doing it? Am I supposed to get the same answer whether I'm using circulation or divergence form? The correct answer is -3.

Would really appreciate any help!


There are a few problems that I think you're running into:

  1. Green's theorem assumes counterclockwise orientation along the path. The path you've described is clockwise, so it should be $-6$ instead of $6$.
  2. Green's theorem is for closed paths. The curve you described doesn't include the bottom portion of the enclosed area, i.e., the line from $(3,0)$ back to the origin. Note that $y=0$ along this line (which I'll call $C_0$), so a quick computation shows that $F(x,0) = \mathbf{i}$, since $e^{0^2}=1$ is the only nonzero term along $C_0$. Thus $$\int_{C_0} \mathbf{F}\cdot d\mathbf{r} = \int_3^0 1\,dx = -3.$$ Subtracting this from the value from the closed path from before, we have $-6-(-3) =-3$.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.