Notation: I = integral sign from 0 to 1, D= partial derivative symbol.(adsbygoogle = window.adsbygoogle || []).push({});

Please help me prove that for any smooth function f:R^n -> R defined on a neighbourhood of a in R^n,

f(x) = f(a) + I{(D/Dt)f(a+t(x-a))dt}

Here's my attempt:

(D/Dt)f(a+t(x-a))dt = d[f(a+t(x-a)] (justification needed?)

so

I [(D/Dt)f(a+t(x-a))dt] = I d[f(a+t(x-a)]

= f(a+1(x-a)) - f(a+0(x-a)) (Fundamental theorem of calculus, right?)

= f(x)-f(a).

Am I right, or am I making many unjustified steps here?

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# Integration proof

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