6.4.E: Further Problems on Differentiable Functions
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For \(E=E^{r}\left(C^{r}\right)\) prove Theorem 2 directly.
[Hint: Find
\[D_{k} h_{j}(\vec{p}), \quad j=1, \ldots, r,\]
from Theorem 4 of §3, and Theorem 3 of §2. Verify that
\[D_{k} h(\vec{p})=\sum_{j=1}^{r} e_{j} D_{k} h_{j}(\vec{p}) \text { and } D_{i} g(\vec{q})=\sum_{j=1}^{r} e_{j} D_{i} g_{j}(\vec{q}),\]
where the \(e_{j}\) are the basic unit vectors in \(E^{r}.\) Proceed.]
Let \(g(x, y, z)=u, x=f_{1}(r, \theta), y=f_{2}(r, \theta), z=f_{3}(r, \theta),\) and
\[f=\left(f_{1}, f_{2}, f_{3}\right) : E^{2} \rightarrow E^{3}.\]
Assuming differentiability, verify (using "variables") that
\[d u=\frac{\partial u}{\partial x} d x+\frac{\partial u}{\partial y} d y+\frac{\partial u}{\partial z} d z=\frac{\partial u}{\partial r} d r+\frac{\partial u}{\partial \theta} d \theta\]
by computing derivatives from (8'). Then do all in the mapping notation for \(H=g \circ f, d H(\vec{p} ; \vec{t}).\)
For the specific functions \(f, g, h,\) and \(k\) of Problems 4 and 5 of §2, set up and solve problems analogous to Problem 2, using
\[\text {(a) } k \circ f ; \quad \text {(b) } g \circ k ; \quad \text {(c) } f \circ h ; \quad \text {(d) } h \circ g.\]
For the functions of Problem 5 in §1, find the formulas for \(d f(\vec{p} ; \vec{t}).\) At which \(\vec{p}\) does \(d f(\vec{p} ; \cdot)\) exist in each given case? Describe it for a chosen \(\vec{p}\).
From Theorem 2, with \(E=E^{1}(C),\) find
\[\nabla h(\vec{p})=\sum_{k=1}^{n} D_{k} g(\vec{q}) \nabla f_{k}(\vec{p}).\]
Use Theorem 1 for a new solution of Problem 7 in §3 with \(E=E^{1}(C).\)
[Hint: Define \(F\) on \(E^{\prime}\) and \(G\) on \(E^{2}\left(C^{2}\right)\) by
\[F(\vec{x})=(f(\vec{x}), g(\vec{x})) \text { and } G(\vec{y})=a y_{1}+b y_{2}.\]
Then \(h=a f+b g=G \circ F.\) (Why?) Use Problems 9 and 10(ii) of §3. Do all in "variable" notation, too.]
Use Theorem 1 for a new proof of the "only if " in Problem 9 in §3.
[Hint: Set \(f_{i}=g \circ f,\) where \(g(\vec{x})=x_{i}\) (the \(i\)th "projection map") is a monomial. Verify!]
Do Problem 8(I) in §3 for the case \(E^{\prime}=E^{2}\left(C^{2}\right),\) with
\[f(\vec{x})=x_{1} \text { and } g(\vec{x})=x_{2}.\]
(Simplify!) Then do the general case as in Problem 6 above, with
\[G(\vec{y})=y_{1} y_{2}.\]
Use Theorem 2 for a new proof of Theorem 4 in Chapter 5, §1. (Proceed as in Problems 6 and 8, with \(E^{\prime}=E^{1},\) so that \(D_{1} h=h^{\prime}\)). Do it in the "variable" notation, too.
Under proper differentiability assumptions, use formula (8') to express the partials of \(u\) if
(i) \(u=g(x, y), x=f(r) h(\theta), y=r+h(\theta)+\theta f(r)\);
(ii) \(u=g(r, \theta), r=f(x+f(y)), \theta=f(x f(y))\);
(iii) \(u=g\left(x^{y}, y^{z}, z^{x+y}\right)\).
Then redo all in the "mapping" terminology, too.
Let the map \(g : E^{1} \rightarrow E^{1}\) be differentiable on \(E^{1}.\) Find \(|\nabla h(\vec{p})|\) if
\(h=g \circ f\) and
(i) \(f(\vec{x})=\sum_{k=1}^{n} x_{k}, \vec{x} \in E^{n}\);
(ii) \(f(\vec{x})=|\vec{x}|^{2}, \vec{x} \in E^{n}\).
(Euler's theorem.) A map \(f : E^{n} \rightarrow E^{1}\) (or \(C^{n} \rightarrow C\)) is called homogeneous of degree \(m\) on \(G\) iff
\[\left(\forall t \in E^{1}(C)\right) \quad f(t \vec{x})=t^{m} f(\vec{x})\]
when \(\vec{x}, t \vec{x} \in G.\) Prove the following statements.
(i) If so, and \(f\) is differentiable at \(\vec{p} \in G\) (an open globe), then
\[\vec{p} \cdot \nabla f(\vec{p})=m f(\vec{p}).\]
*(ii) Conversely, if the latter holds for all \(\vec{p} \in G\) and if \(\overrightarrow{0} \notin G,\) then \(f\) is homogeneous of degree \(m\) on \(G.\)
(iii) What if \(\overrightarrow{0} \in G ?\)
[Hints: (i) Let \(g(t)=f(t \vec{p}) .\) Find \(g^{\prime}(1) .\) (iii) Take \(f(x, y)=x^{2} y^{2}\) if \(x \leq 0, f=0\) if \(\left.x>0, G=G_{0}(1).\right]\)
Try Problem 12 for \(f : E^{\prime} \rightarrow E,\) replacing \(\vec{p} \cdot \nabla f(\vec{p})\) by \(d f(\vec{p} ; \vec{p})\).
With all as in Theorem 1, prove the following.
(i) If \(E^{\prime}=E^{1}\) and \(\vec{s}=f^{\prime}(p) \neq \overrightarrow{0},\) then \(h^{\prime}(p)=D_{\vec{s}} g(\vec{q})\).
(ii) If \(\vec{u}\) and \(\vec{v}\) are nonzero in \(E^{\prime}\) and \(a \vec{u}+b \vec{v} \neq \overrightarrow{0}\) for some scalars \(a, b,\) then
\[D_{a \vec{u}+b \vec{v}} f(\vec{p})=a D_{\vec{u}} f(\vec{p})+b D_{\vec{v}} f(\vec{p}).\]
(iii) If \(f\) is differentiable on a globe \(G_{\vec{p}},\) and \(\vec{u} \neq \overrightarrow{0}\) in \(E^{\prime},\) then
\[D_{\vec{u}} f(\vec{p})=\lim _{\vec{x} \rightarrow \vec{u}} D_{\vec{x}}(\vec{p}).\]
[Hints: Use Theorem 2(ii) from §3 and Note 1.]
Use Theorem 2 to find the partially derived functions of \(f,\) if
(i) \(f(x, y, z)=(\sin (x y / z))^{x}\);
(ii) \(f(x, y)=\log _{x}|\tan (y / x)|\).
(Set \(f=0\) wherever undefined.)