Problem 1389 (difficulty: 4/10)

Let \(\displaystyle *:\R^p\times \R^q\to\R^r\) be bilinear, \(\displaystyle f:\R^q\to\R^p\) continuous and \(\displaystyle g:[a,b]\to\R^q\) a continuous curve. Show that

(a) if \(\displaystyle g\) is rectifiable, then \(\displaystyle \int_g f(\mathbf{x})*\mathrm{d}\mathbf{x}\) exists;

(b) if \(\displaystyle g\) is continuously differentiable, then \(\displaystyle \int_g f(\mathbf{x})*\mathrm{d}\mathbf{x}=\int_a^b f(g(t))*g'(t)\dt.\)

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