6.4.E: Further Problems on Differentiable Functions

Exercise $\PageIndex{1}$

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.]

Exercise $\PageIndex{2}$

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 $H=g \circ f, d H(\vec{p} ; \vec{t}).$

Exercise $\PageIndex{3}$

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
$$\text {(a) } k \circ f ; \quad \text {(b) } g \circ k ; \quad \text {(c) } f \circ h ; \quad \text {(d) } h \circ g.$$

Exercise $\PageIndex{4}$

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 $\vec{p}$.

Exercise $\PageIndex{5}$

From Theorem 2, with $E=E^{1}(C),$ find
$$\nabla h(\vec{p})=\sum_{k=1}^{n} D_{k} g(\vec{q}) \nabla f_{k}(\vec{p}).$$

Exercise $\PageIndex{6}$

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.]

Exercise $\PageIndex{7}$

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!]

Exercise $\PageIndex{8}$

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
$$G(\vec{y})=y_{1} y_{2}.$$

Exercise $\PageIndex{9}$

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.

Exercise $\PageIndex{10}$

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.

Exercise $\PageIndex{11}$

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) $f(\vec{x})=|\vec{x}|^{2}, \vec{x} \in E^{n}$.

Exercise $\PageIndex{12}$

(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 $\left.x>0, G=G_{0}(1).\right]$

Exercise $\PageIndex{13}$

Try Problem 12 for $f : E^{\prime} \rightarrow E,$ replacing $\vec{p} \cdot \nabla f(\vec{p})$ by $d f(\vec{p} ; \vec{p})$.

Exercise $\PageIndex{14}$

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.]

Exercise $\PageIndex{15}$

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.)