Is the vector space C∞[a,b] of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by D(f)(x)=f′(x)D(f)(x)=f′(x) and I(f)(x)=∫xaf(t)dtf(t)dt. (a) Compute (DI)(f)=D(I(f))(DI)(f)=D(I(f)). (b) Compute (ID)(f)=I(D(f))(ID)(f)=I(D(f)). (c) Do this transformations commute? That is to say, is it true that (DI)(f)=(ID)(f)(DI)(f)=(ID)(f) for all vectors f in the space?

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Transformations of functions

Amari Flowers

Answered question

2021-06-23

Is the vector space $C \infty [a, b]$ of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by $D (f) (x) = f' (x) D (f) (x) = f' (x) a n d I (f) (x) = \int x a f (t) d t f (t) d t$ . (a) Compute $(D I) (f) = D (I (f)) (D I) (f) = D (I (f))$ . (b) Compute $(I D) (f) = I (D (f)) (I D) (f) = I (D (f))$ . (c) Do this transformations commute? That is to say, is it true that $(D I) (f) = (I D) (f) (D I) (f) = (I D) (f)$ for all vectors f in the space?

Answer & Explanation

oppturf

Skilled2021-06-24Added 94 answers

a) $(D I) (f) = f (x)$
b) $(I D) (f) = f (x) - f (a)$
c) $(D I) (f) = (I D) (f)$ only when $f (a) = 0$

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