6.5.E: Problems on Repeated Differentiation and Taylor Expansions
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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?
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!
Carry out the induction in Theorem 2 and Corollary 2. (Use a suitable notation for subscripts: \(k_{1} k_{2} \ldots\) instead of \(j k \ldots\).)
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 \(R_{3}\).
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) \(f(x, y)=x^{5} y-a x y^{5}-x^{3}\).
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 \(R_{2}\).
(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 \(J-Q^{\prime}, Q^{\prime}= g^{-1}[Q]\).]
(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.
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.]
\(\Rightarrow\) 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 \(\vec{p}\).]
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.]
(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 \(u=\arctan \frac{y}{x}?\)
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).
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.
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.]
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.]