Problem 1409 (difficulty: 8/10)

Let \(\displaystyle G\subset\R^2\) be open and let \(\displaystyle \phi_u(t)\) \(\displaystyle [0,1]^2\to G\) be continuously differentiable family of curves. Show that for a continuously differentiable \(\displaystyle f:G\to\R^2\) irrotational vector field the \(\displaystyle I(u)=\int_{\phi_u} \langle f,\mathrm{d}x\rangle \) parametric line integral satisfies \(\displaystyle I'(u)=0\).

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