
# 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.)