I require a reference for the following theorem, which is stated without proof in Matrix Groups for Undergraduates by Kristopher Tapp. In what follows, let $p \in U \subseteq \mathbb R^m$, $v \in \mathbb R^m$, and $f: U \to \mathbb R^n$. The directional derivative of $f$ in the direction $v$ at $p$ is defined in the usual way as:

$$df_p(v) = \lim_{t \to 0} \frac{f(p+tv)-f(p)}{t}$$

if this limit exists and is finite.

The theorem, which is stated after the above definition, is as follows:

If $f$ is $C^1$ on $U$, then for all $p \in U$,

(1) $v \mapsto df_p(v)$ is a linear function from $\mathbb R^m$ to $\mathbb R^n$.

(2) $f(q) \approx f(p) + df_p(q-p)$ is a good approximation of $f$ near $p$ in the following sense: for any infinite sequence $\left\{ q_1, q_2,...\right\}$ of points in $\mathbb R^m$ converging to $p$, $$\lim_{t \to \infty} \frac{f(q_t)-f(p)-df_p(q_t-p)}{|q_t-p|} = 0.$$

Immediately before stating this theorem the author asserts 'The following is proven in any real analysis textbook.' I know what he really means is 'any advanced calculus / vector analysis' textbook (rather than real analysis), and I have checked several but am so far unsuccessful in finding the result at all, let alone the proof. If anyone could point me to a reference, either online or textbook, where it is proved I'd be very grateful.

  • $\begingroup$ Do you know the little-o notation ? Then by definition of the derivative $f(p+tv) = f(p)+t d f_p(v)+o(t)$, $df_p(tv) = t d_pf(v)$ and by continuity of the derivative $ df_{p+tv}(u) = df_p(u) +o(1)$. Thus $$f(p)+t d_p f(v+u)+o(t)= f(p+tv+tu) = (f(p)+td_pf(v)+o(t))+t(d_pf(u)+o(1))+o(t)$$ which proves the linearity. $\endgroup$ – reuns Apr 26 at 13:44

Perhaps you are having trouble finding it because exposition in calculus/analysis books is usually in somewhat different order. One usually first defines $f$ to be differentiable at $p$ if there is a linear map $df_p$ such that the approximation $f(q)\approx f(p)+df_p(q-p)$ is good. Then one shows that 1) if $f$ is $C^1$ at $p$ (in Tapp's terminology, meaning continuous partials) then it is differentiaible at $p$; and 2) if $f$ is differentiable at $p$ then $df_p(v)$ is the directional derivative of $f$ in the direction of $v$. Combining these you do obtain the theorem in your question.

You can indeed find this in many places. Statements are in most multivariable calculus books (in some with proofs of various degree of validity), and proofs are in analysis books including chapter 9 of Rudin's "Principles of Mathematical Analysis" or chapter 5 of Pugh's "Real Mathematical Analysis".


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