
# 6.5.E: Problems on Repeated Differentiation and Taylor Expansions


Exercise $$\PageIndex{1}$$

Complete all details in the proof of Theorem 1. What is the motivation for introducing the auxiliary functions $$h_{t}$$ and $$g_{s}$$ in this particular way?

Exercise $$\PageIndex{2}$$

Is symbolic "multiplication" in Note 2 always commutative? (See Example (A).) Why was it possible to collect "similar" terms
$\frac{\partial^{2} f}{\partial x \partial y} d x d y \text { and } \frac{\partial^{2} f}{\partial y \partial x} d y d x$
in Example (B)? Using (5), find the general formula for $$d^{3} f.$$ Expand it!

Exercise $$\PageIndex{3}$$

Carry out the induction in Theorem 2 and Corollary 2. (Use a suitable notation for subscripts: $$k_{1} k_{2} \ldots$$ instead of $$j k \ldots$$.)

Exercise $$\PageIndex{4}$$

Do Example(C) with $$m=3$$ (instead of $$m=2$$) and with $$\vec{p}=(0,0)$$. Show that $$R_{m} \rightarrow 0,$$ i.e., $$f$$ admits a Taylor series about $$\vec{p}.$$
Do it in the following two ways.
(i) Use Theorem 2.
(ii) Expand $$\sin y$$ as in Problem 6(a) in Chapter 5, §6, and then multiply termwise by $$x.$$
Give an estimate for $$R_{3}$$.

Exercise $$\PageIndex{5}$$

Use Theorem 2 to expand the following functions in powers of $$x-3$$ and $$y+2$$ exactly (choosing $$m$$ so that $$R_{m}=0$$).
(i) $$f(x, y)=2 x y^{2}-3 y^{3}+y x^{2}-x^{3}$$;
(ii) $$f(x, y)=x^{4}-x^{3} y^{2}+2 x y-1$$;
(iii) $$f(x, y)=x^{5} y-a x y^{5}-x^{3}$$.

Exercise $$\PageIndex{6}$$

For the functions of Problem 15 in §4, give their Taylor expansions up to $$R_{2},$$ with
$\vec{p}=\left(1, \frac{\pi}{4}, 1\right)$
in case (i) and
$\vec{p}=\left(e, \frac{\pi}{4} e\right)$
in (ii). Bound $$R_{2}$$.

Exercise $$\PageIndex{7}$$

(Generalized Taylor theorem.) Let $$\vec{u}=\vec{x}-\vec{p} \neq \overrightarrow{0}$$ in $$E^{\prime}$$($$E^{\prime}$$ need not be $$E^{n}$$ or $$C^{n}$$); let $$I=L[\vec{p}, \vec{x}].$$ Prove the following statement:
If $$f : E^{\prime} \rightarrow E$$ and the derived functions $$D_{\vec{u}}^{i} f(i \leq m)$$ are relatively continuous on $$I$$ and have $$\vec{u}$$-directed derivatives on $$I-Q$$ ($$Q$$ countable), then formula (6) and Note 3 hold, with $$d^{i} f(\vec{p} ; \vec{u})$$ replaced by $$D_{\vec{u}}^{i} f(\vec{p})$$.
[Hint: Proceed as in Theorem 2 without using the chain rule or any partials or components. Instead of (8), prove that $$h^{(i)}(t)=D_{\vec{u}}^{i} f(\vec{p}+t \vec{u})$$ on $$J-Q^{\prime}, Q^{\prime}= g^{-1}[Q]$$.]

Exercise $$\PageIndex{8}$$

(i) Modify Problem 7 by setting
$\vec{u}=\frac{\vec{x}-\vec{p}}{|\vec{x}-\vec{p}|}.$
Thus expand $$f(\vec{x})$$ in powers of $$|\vec{x}-\vec{p}|.$$
(ii) Deduce Theorem 2 from Problem 7, using Corollary 2.

Exercise $$\PageIndex{9}$$

Given $$f : E^{2}\left(C^{2}\right) \rightarrow E, f \in C D^{m}$$ on an open set $$A,$$ and $$\vec{s} \in A,$$ prove that $$\left(\forall \vec{u} \in E^{2}\left(C^{2}\right)\right)$$
$d^{i} f(\vec{s} ; \vec{u})=\sum_{j=0}^{i}\left(\begin{array}{c}{i} \\ {j}\end{array}\right) u_{1}^{j} u_{2}^{i-j} D_{k_{1} \dots k_{i}} f(\vec{s}), \quad 1 \leq i \leq m,$
where the $$\left(\begin{array}{l}{i} \\ {j}\end{array}\right)$$ are binomial coefficients, and in the $$j$$th term,
$k_{1}=k_{2}=\cdots=k_{j}=2$
and
$k_{j+1}=\cdots=k_{i}=1.$
Then restate formula (6) for $$n=2$$.
[Hint: Use induction, as in the binomial theorem.]

Exercise $$\PageIndex{10}$$

$$\Rightarrow$$ Given $$\vec{p} \in E^{\prime}=E^{n}\left(C^{n}\right)$$ and $$f : E^{\prime} \rightarrow E,$$ prove that $$f \in C D^{1}$$ at $$\vec{p}$$ iff $$f$$ is differentiable at $$\vec{p}$$ and
$(\forall \varepsilon>0)(\exists \delta>0)\left(\forall \vec{x} \in G_{\vec{p}}(\delta)\right) \quad\left\|d^{1} f(\vec{p} ; \cdot)-d^{1} f(\vec{x} ; \cdot)\right\|<\varepsilon,$
with norm $$\|$$ as in Definition 2 in §2. (Does it apply?)
[Hint: If $$f \in C D^{1},$$ use Theorem 2 in §3. For the converse, verify that
$\varepsilon \geq\left|d^{1} f(\vec{p} ; \vec{t})-d^{1} f(\vec{x} ; \vec{t})\right|=\left|\sum_{k=1}^{n}\left[D_{k} f(\vec{p})-D_{k} f(\vec{x})\right] t_{k}\right|$
if $$\vec{x} \in G_{\vec{p}}(\delta)$$ and $$|\vec{t}| \leq 1.$$ Take $$\vec{t}=\vec{e}_{k},$$ to prove continuity of $$D_{k} f$$ at $$\vec{p}$$.]

Exercise $$\PageIndex{11}$$

Prove the following.
(i) If $$\phi : E^{n} \rightarrow E^{m}$$ is linear and $$[\phi]=\left(v_{i k}\right),$$ then
$\|\phi\|^{2} \leq \sum_{i, k}\left|v_{i k}\right|^{2}.$
(ii) If $$f : E^{n} \rightarrow E^{m}$$ is differentiable at $$\vec{p},$$ then
$\|d f(\vec{p} ; \cdot)\|^{2} \leq \sum_{i, k}\left|D_{k} f_{i}(\vec{p})\right|^{2}.$
(iii) Hence find a new converse proof in Problem 10 for $$f : E^{n} \rightarrow E^{m}$$.
Consider $$f : C^{n} \rightarrow C^{m},$$ too.
[Hints: (i) By the Cauchy-Schwarz inequality, $$|\phi(\vec{x})|^{2} \leq|\vec{x}|^{2} \sum_{i, k}\left|v_{i k}\right|^{2}.$$ (Why?) (ii) Use part (i) and Theorem 4 in §3.]

Exercise $$\PageIndex{12}$$

(i) Find $$d^{2} u$$ for the functions of Problem 10 in §4, in the "variable" and "mapping" notations.
(ii) Do it also for
$u=f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-\frac{1}{2}}$
and show that $$D_{11} f+D_{22} f+D_{33} f=0$$.
(iii) Does the latter hold for $$u=\arctan \frac{y}{x}?$$

Exercise $$\PageIndex{13}$$

Let $$u=g(x, y), x=r \cos \theta, y=r \sin \theta$$ (passage to polars).
Using "variables" and then the "mappings" notation, prove that if $$g$$ is differentiable, then
(i) $$\frac{\partial u}{\partial r}=\cos \theta \frac{\partial u}{\partial x}+\sin \theta \frac{\partial u}{\partial y}$$ and
(ii) $$|\nabla g(x, y)|^{2}=\left(\frac{\partial u}{\partial r}\right)^{2}+\left(\frac{1}{r} \frac{\partial u}{\partial \theta}\right)^{2}$$.
(iii) Assuming $$g \in C D^{2},$$ express $$\frac{\partial^{2} u}{\partial r \partial \theta}, \frac{\partial^{2} u}{\partial r^{2}},$$ and $$\frac{\partial^{2} u}{\partial \theta^{2}}$$ as in (i).

Exercise $$\PageIndex{14}$$

Let $$f, g : E^{1} \rightarrow E^{1}$$ be of class $$C D^{2}$$ on $$E^{1}.$$ Verify (in "variable" notation, too) the following statements.
(i) $$D_{11} h=a^{2} D_{22} h$$ if $$a \in E^{1}$$ (fixed) and
$h(x, y)=f(a x+y)+g(y-a x).$
(ii) $$x^{2} D_{11} h(x, y)+2 x y D_{12} h(x, y)+y^{2} D_{22} h(x, y)=0$$ if
$h(x, y)=x f\left(\frac{y}{x}\right)+g\left(\frac{y}{x}\right).$
(iii) $$D_{1} h \cdot D_{21} h=D_{2} h \cdot D_{11} h$$ if
$h(x, y)=g(f(x)+y)$
Find $$D_{12} h,$$ too.

Exercise $$\PageIndex{15}$$

Assume $$E^{\prime}=E^{n}\left(C^{n}\right)$$ and $$E^{\prime \prime}=E^{m}\left(C^{m}\right).$$ Let $$f : E^{\prime} \rightarrow E^{\prime \prime}$$ and $$g : E^{\prime \prime} \rightarrow E$$ be twice differentiable at $$\vec{p} \in E^{\prime}$$ and $$\vec{q}=f(\vec{p}) \in E^{\prime \prime}$$, respectively, and set $$h=g \circ f$$.
Show that $$h$$ is twice differentiable at $$\vec{p},$$ and
$d^{2} h(\vec{p} ; \vec{t})=d^{2} g(\vec{q} ; \vec{s})+d g(\vec{q} ; \vec{v}),$
where $$\vec{t} \in E^{\prime}, \vec{s}=d f(\vec{p} ; \vec{t}),$$ and $$\vec{v}=\left(v_{1}, \ldots, v_{m}\right) \in E^{\prime \prime}$$ satisfies
$v_{i}=d^{2} f_{i}(\vec{p} ; \vec{t}), \quad i=1, \ldots, m.$
Thus the second differential is not invariant in the sense of Note 4 in §4.
[Hint: Show that
$D_{k l} h(\vec{p})=\sum_{j=1}^{m} \sum_{i=1}^{m} D_{i j} g(\vec{q}) D_{k} f_{i}(\vec{p}) D_{l} f_{j}(\vec{p})+\sum_{i=1}^{m} D_{i} g(\vec{q}) D_{k l} f_{i}(\vec{p}).$
Proceed.]

Exercise $$\PageIndex{16}$$

Continuing Problem 15, prove the invariant rule:
$d^{r} h(\vec{p} ; \vec{t})=d^{r} g(\vec{q} ; \vec{s}),$
if $$f$$ is a first-degree polynomial and $$g$$ is $$r$$ times differentiable at $$\vec{q}$$.
[Hint: Here all higher-order partials of $$f$$ vanish. Use induction.]