Compute the following line integral: $\oint_C \frac{\text{d}x}{y} -\frac{\text{d}y}{x}$ Where $C$ is the circle $x^2+y^2=1$.

Question

Compute the following line integral: $$\oint_C \frac{\text{d}x}{y} -\frac{\text{d}y}{x}$$

Where $C$ is the circle $x^2+y^2=1$.

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The function $1/y$ and it's first partial derivative with respective to $y$ are not defined on the points lying in the $x$ axis, and the function $-1/x$ and it's first partial derivative with respective to $x$ are not defined on the points lying in the $y$ axis, so we can't use Green's theorem, so how to solve the problem?

Answer 1

HINT:

On $C$, parameterize $(x,y)$ as $(\cos(\phi),\sin(\phi))$ with $\phi \in [-\pi,\pi]$.

Can you finish now?