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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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?

Dominique Mayer · Accepted Answer

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&amp;gt;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.

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?

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