Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# Derivatives and Differentials §6.1 Differential Coefficients from the Gradient

DOI link for Derivatives and Differentials §6.1 Differential Coefficients from the Gradient

Derivatives and Differentials §6.1 Differential Coefficients from the Gradient book

# Derivatives and Differentials §6.1 Differential Coefficients from the Gradient

DOI link for Derivatives and Differentials §6.1 Differential Coefficients from the Gradient

Derivatives and Differentials §6.1 Differential Coefficients from the Gradient book

## 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.