please answer all of it, I am so appreciated! Show that partial differential/par
ID: 3403046 • Letter: P
Question
please answer all of it, I am so appreciated! Show that partial differential/partial differential x is a derivation on C^infinity(R, R). Show that Der(A) is an "A-module," which consists of showing: (add) if delta and esplion are derivations, then delta + epsilon is a derivation. (null) if delta is a derivation and f C^infinity (R, R) then f middot delta is also a derivation. Show that every derivation on C^infinity (R, R) is of the form f partial differential/partial differential x for some f C^infinity(R, R). Show that if delta and epsilon are two derivations, then delta epsilon is not a derivation (unless one of them is zero).Explanation / Answer
A linear endomorphism d:A-> A is called derivation if the following Leibnitz rule holds,
d(a,b)=d(a).b+a.d(b)
the set of all derivatives of A is denoted by Der(A). This is clearly a vector subspace of End(A).
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