Line integral in parameterized curve

Question

$a,b\gt 0$ and $n \in \mathbb Z$ \ {$0$}. Parameterized curve $C: x=a\cos nt, y= b\sin nt$

where $0 \le t \le2\pi$ and n is how many times that goes around origo.

I want to calculate line integral:

$\omega(C) = \frac{1}{2\pi}\oint_C \frac{x\,dy-y\, dx}{x^2+y^2}$

I have calculated:

$dx=-a\sin( nt)n$

$dy = b\cos(nt)n$

then after that I put the values in "right" places and:

$\frac{a\,b\,n}{(a^2+b^2)(\cos^2(nt))+b^2}$

Then next step is integration and the limit values are $[0,2\pi]$.

$\omega(C) = \frac{1}{2\pi}\int_0^{2\pi}\frac{a\,b\,n}{(a^2+b^2)(\cos^2(nt))+b^2} dt$

I get some random answer for that so I know that is wrong. I think the problem is in the integration but I don't know where is the problem.

Answer what I got:

$\frac{2\tan^{-1}(\frac{b\tan(2n\pi}{a})\vert a b\vert - ab(mod(4n-1,2)-4n-1)\pi}{4\vert ab \vert\pi}$

Answer 1

So you were able to find $$ \frac{1}{2\pi}\oint_C \frac{x\,dy-y\, dx}{x^2+y^2} = \dfrac{2\tan^{-1}\left(\frac{b\tan(2n\pi)}{a}\right)\vert a b\vert - ab(\mod(4n-1,2)-4n-1)\pi}{4\vert ab \vert\pi} $$ That looks pretty complicated, but:

• $\tan(2n\pi) = 0$ for any integer $n$. Therefore $$ \tan^{-1}\left(\frac{b}{a}\tan(2n\pi)\right) = 0 $$
• $4n-1$ is odd, so $(4n-1) \bmod 2 = 1$. Therefore $$ -ab((4n-1)\bmod 2 -(4n+1))\pi = 4\pi nab $$

So the expression simplifies to: $$ \frac{1}{2\pi}\oint_C \frac{x\,dy-y\, dx}{x^2+y^2} = \frac{4\pi n ab}{4|ab|\pi} = \frac{ab}{|ab|} n = \pm n $$ where the $+$ is taken when $a$ and $b$ have the same sign, and $-$ otherwise.

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But you can approach this another way. Notice that $C$ is the same ellipse $C_0$ traversed $n$ times. So $$ \frac{1}{2\pi}\oint_C \frac{x\,dy-y\, dx}{x^2+y^2} =\frac{n}{2\pi}\oint_{C_0} \frac{x\,dy-y\, dx}{x^2+y^2} $$ Also, setting $P=\frac{-y}{x^2+y^2}$ and $Q = \frac{x}{x^2+y^2}$, we see that $P_y = Q_x$ away from the origin. The ellipse $C_0$ is homologous to either the unit circle $S^1$, or its reverse, depending on whether $ab>0$ or $ab < 0$. So by Green's Theorem, $$ \oint_{C_0} \frac{x\,dy-y\, dx}{x^2+y^2} = \pm \oint_{S^1} \frac{x\,dy-y\, dx}{x^2+y^2} $$

If you parametrize the unit circle with $x=\cos t$ and $y=\sin t$, then you'll see $$ \oint_{S^1} \frac{x\,dy-y\, dx}{x^2+y^2} = \int_0^{2\pi} 1 \,dt = 2\pi $$ and this matches your previous computation.