Ngô Quốc Anh

June 11, 2021

Second-order differentiability in terms of partial derivatives

Filed under: Uncategorized — Ngô Quốc Anh @ 0:28

Let U \subset \mathbf R^n be open, a \in U is an arbitrary point, and f : U  \to \mathbf R is a function. Recall that f is called differentiable at a if there is a linear map, denoted by A, such that

\displaystyle \lim_{\|h\| \searrow 0} \frac{ \| f(a+h)-f(a) - A (h) \| }{\|h\|} = 0.

The linear map A is called derivative of f at a, denoted by f'(a). Notice that in the nominator of the above quotient, the symbol \| \cdot \| is simply the absolute value function. But this is no longer true for higher-order derivatives that we are going to define.

The following theorem is well-known.

Theorem 1 (1st order differentiability). The function f : U \to \mathbf R is differentiable at a \in U if all partial derivatives \partial_i f exist in a neighborhood of a and are continuous at a.

When f'(a) exists, we must have

\displaystyle f'(a) (h) = \sum_{i=1}^n \partial_i f(a) h_i.

In this note, we want to extend the above theorem for higher-order derivatives. To be more precise, we prove the following

Theorem 2 (2nd order differentiability). The function f : U \to \mathbf R is twice differentiable at a \in U if all partial derivatives \partial_{ij} f exist in a neighborhood of a and are continuous at a.

(more…)

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