Primer of Modern Analysis pp 278-288 | Cite as

# Higher Derivatives

Chapter

## Abstract

The partial derivatives of a function

*f:***R**^{m}→**R**^{n}are again functions from**R**^{m}to**R**^{n}which may have partial derivatives of their own. If so, the latter are called the*second partial derivatives:*$$\frac{{{\partial ^2}f\left( a \right)}}{{\partial {x_i}\partial {x_j}}} = {D_{ij}}f\left( a \right) = {D_i}\left( {{D_j}f} \right)\left( a \right)$$

(1)

The formula says that to get *D*_{ ij }*f(a)*, you take first *Dif*,
which must exist on a neighborhood of *a*, and differentiate it with respect to *xi*. It appears that this would be quite different from *D*_{ ji } *f (a)*, which is formed by differentiating first with respect to *x*_{i} and then with respect to *x*_{ j }.In fact, the two are usually the same.

## Keywords

Partial Derivative Local Maximum Linear Transformation Smooth Manifold High Derivative
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1983