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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).

Solution

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).