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Let C be circumference in $\mathbb{R^2}$ with centre in $(0,0)$ which we bypass counterclockwise. Compute : $$\oint_C -\frac{y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy$$ and then prove that set $\big\{(x, y) \big| 0 < x^2 + y^2 \le 1 \big\}$ is not simply connected space.

So, I computed integral and obtain $2\pi$. Expression under integral sign is total derivative and I know that line integral of total derivative by closed curve in simply connected space is zero, but our integral is not zero. But my teacher said that is not full proof. Could you complement my solution or give any tips for this?

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Expression under integral sign is total derivative and I know that line integral of total derivative by closed curve in simply connected space is zero

I think this is where your teacher has an issue: the "expression under integral sign" is not a total derivative.

However, the vector field $(f(x,y),g(x,y))$ being integrated satisfies $\frac{\partial g}{\partial x} = \frac{\partial f}{\partial y}$ over its domain, which is why you thought that it's a total derivative. If that domain were simply connected, then we could deduce that $(f,g)$ is a total derivative, which would mean that the integral should be zero. Since the integral is not zero, this means that the domain is not simply connected, which means that $(f,g)$ is not a total derivative (it also isn't really "under" the integral sign).

So in short: because $(f,g)$ is irrotational (or closed), it would be a total derivative (or exact) if the domain were simply connected. Since the integral over the loop is non-zero, $(f,g)$ cannot be exact, which means that the domain cannot be simply connected.

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