We know that, f:RR to RR satisfies f′(x)=f(x) for all x in RR then f(x)=A e^x What is a function f:RR^d to RR^d, analogous to the exponential function, if x in RR^d was a d dimensional vector?

lexi13xoxla 2022-08-13 Answered
We know that, f : R R satisfies f ( x ) = f ( x ) for all x R then f ( x ) = A e x
What is a function f : R d R d , analogous to the exponential function, if x R d was a d dimensional vector?
Is there a name given to such functions whose derivatives are the same as the function itself?
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Answers (1)

Dominique Mayer
Answered 2022-08-14 Author has 10 answers
There is a problem with defining a derivative of a map as a map between the same spaces as the original map. Given a map f : R m R n its differential at a point p R m (if it exists) is a linear map D f ( p ) : T R m T R n of tangent spaces. If m=1 then f is a vector-valued function and its differential is also a vector-valued function. In this case we can define
f ( t ) = ( f 1 ( t ) , . . . , f n ( t ) )
which is in fact the differential at the point t. Note that here we identify the space R n with the space of linear maps T R T R n which is (luckily) n-dimensional. But if m>1 then this space of linear maps is no more n-dimensional and we can not define the derivative of a map in the same way we do in single variable calculus.
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