Problem 1408 (difficulty: 10/10)

Let \(\displaystyle G=\R^2\setminus\{(-1,0), (1,0)\}\), and \(\displaystyle g\) be the curve shown on the figure.

(a) Show that the line integral of any differentiable irrotational vector field \(\displaystyle f:G\to\R^2\) along \(\displaystyle g\) is zero.

(b) Is \(\displaystyle g\) homotopic to a point in \(\displaystyle G\)?

(c) Is \(\displaystyle g\) homologous to \(\displaystyle 0\) in \(\displaystyle G\)?

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