ABSTRACT
For x = (x\, · · · , xn) and y = (j/i, · · · , yn) in Kn the inner p ro d u c t is x · y = x xyx H-------h x nyn· (See §11.3.)
A function F on a neighborhood of a point u in Rn is dif feren tiab le at u if there exist a neighborhood V of 0 in Rn and a mapping P on V into Mn such that
(1) P is continuous at 0
and
(2)
The n-dim ensional derivative (or g rad ien t) V F (u ) at u is defined by
The reader may verify that P(0) does not depend on the choice of P and V in (1) and (2). Of course P does depend on u. (Exercises 1 and 2 show the equivalence of the Caratheodory definition (3) with the usual definition of V F .)
but a particular one of its coordinates equal to 0 we get the partial derivatives as the coordinates of the gradient
If η = 1 then the gradient is just the classical derivative Using the gradient we can formulate Theorem 1 which con
cerns the differential of a function F on a curve G in Rn.
