Problem 1399 (difficulty: 10/10)

Let \(\displaystyle G\subset\R^p\) be open, let \(\displaystyle f:G\to\R^p\) be differentiable and irrotational and let \(\displaystyle g,h:[0,1]\to G\) be continuously differentiable curves with the same initial and endpoints. (I.e. \(\displaystyle g(0)=h(0)\) and \(\displaystyle g(1)=h(1)\).) Assume that \(\displaystyle g\) and \(\displaystyle h\) are homotopic, \(\displaystyle \exists \phi:[0,1]^2\to\R^p\) continuous such that \(\displaystyle \phi(t,0)=g(t)\), \(\displaystyle \phi(t,1)=h(t)\), and \(\displaystyle \phi(0,u)=g(0)=h(0)\), \(\displaystyle \phi(1,u)=g(1)=h(1)\) for all \(\displaystyle u\in[0,1]\).

(a) Show from Goursat's lemma that \(\displaystyle \int_g\langle f,\mathrm{d}x\rangle =\int_h\langle f,\mathrm{d}x\rangle \).

(b) Assume in addition that \(\displaystyle \phi\) is continuously differentiable \(\displaystyle I(u) = \int_{\phi(\cdot,u)} \langle f,\mathrm{d}x\rangle \). Prove directly that \(\displaystyle I'=0\).

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