
# 6.6.E: Problems on Bijective Linear Maps and Jacobians


Exercise $$\PageIndex{1}$$

(i) Can a functional determinant $$f=\operatorname{det}\left(v_{i k}\right)$$ (see Note 1) be continuous or differentiable even if the functions $$v_{i k}$$ are not?
(ii) Must a Jacobian map $$J_{f}$$ be continuous or differentiable if $$f$$ is?
Give proofs or counterexamples.

Exercise $$\PageIndex{2}$$

$$\Rightarrow$$ Prove rule (b) on determinants. More generally, show that if $$f(\vec{x})=\vec{x}$$ on an open set $$A \subseteq E^{n}\left(C^{n}\right),$$ then $$J_{f}=1$$ on $$A$$.

Exercise $$\PageIndex{3}$$

Let $$f : E^{n} \rightarrow E^{n}$$ (or $$C^{n} \rightarrow C^{n}$$), $$f=\left(f_{1}, \ldots, f_{n}\right)$$.
Suppose each $$f_{k}$$ depends on $$x_{k}$$ only, i.e.,
$f_{k}(\vec{x})=f_{k}(\vec{y}) \text{ if } x_{k}=y_{k},$
regardless of the other coordinates $$x_{i}, y_{i}.$$ Prove that $$J_{f}=\prod_{k=1}^{n} D_{k} f_{k}$$.
[Hint: Show that $$D_{k} f_{i}=0$$ if $$i \neq k$$.]

Exercise $$\PageIndex{4}$$

In Corollary 1, show that
$J_{h}(\vec{p})=\prod_{k=1}^{n} D_{k} f_{k}(\vec{p}) \cdot J_{g}(\vec{q})$
if $$f$$ also has the property specified in Problem 3. Then do all in "variables," with $$y_{k}=y_{k}\left(x_{k}\right)$$ instead of $$f_{k}$$.

Exercise $$\PageIndex{5}$$

Let $$E^{\prime}=E^{1}$$ in Note 1. Prove that if all the $$v_{i k}$$ are differentiable at $$p,$$ then $$f^{\prime}(p)$$ is the sum of $$n$$ determinants, each arising from det $$\left(v_{i k}\right),$$ by replacing the terms of one column by their derivatives.
[Hint: Use Problem 6 in Chapter 5, §1.]

Exercise $$\PageIndex{6}$$

Do Problem 5 for partials of $$f,$$ with $$E^{\prime}=E^{n}\left(C^{n}\right),$$ and for directionals $$D_{\vec{u}} f,$$ in any normed space $$E^{\prime}.$$ (First, prove formulas analogous to Problem 6 in Chapter 5, §1; use Note 3 in §1.) Finally, do it for the differential, $$d f(\vec{p} ; \cdot).$$

Exercise $$\PageIndex{7}$$

In Note 1 of §4, express the matrices in terms of partials (see Theorem 4 in §3). Invent a "variable" notation for such matrices, imitating Jacobians (Corollary 3).

Exercise $$\PageIndex{8}$$

(i) Show that
$\frac{\partial(x, y, z)}{\partial(r, \theta, \alpha)}=-r^{2} \sin \alpha$
if
$\begin{array}{l}{x=r \cos \theta,} \\ {y=r \sin \theta \sin \alpha, \text { and }} \\ {z=r \cos \alpha}\end{array}$
(This transformation is passage to polars in $$E^{3};$$ see Figure 27, where $$r=O P, \angle X O A=\theta,$$ and $$\angle A O P=\alpha.)$$
(ii) What if $$x=r \cos \theta, y=r \sin \theta,$$ and $$z=z$$ remains unchanged (passage to cylindric coordinates)?
(iii) Same for $$x=e^{r} \cos \theta, y=e^{r} \sin \theta,$$ and $$z=z$$.

Exercise $$\PageIndex{9}$$

Is $$f=\left(f_{1}, f_{2}\right) : E^{2} \rightarrow E^{2}$$ one-to-one or bijective, and is $$J_{f} \neq 0,$$ if
(i) $$f_{1}(x, y)=e^{x} \cos y$$ and $$f_{2}(x, y)=e^{x} \sin y$$;
(ii) $$f_{1}(x, y)=x^{2}-y^{2}$$ and $$f_{2}(x, y)=2 x y ?$$

Exercise $$\PageIndex{10}$$

Define $$f : E^{3} \rightarrow E^{3}$$ (or $$C^{3} \rightarrow C^{3}$$)
$f(\vec{x})=\frac{\vec{x}}{1+\sum_{k=1}^{3} x_{k}}$
on
$A=\left\{\vec{x} | \sum_{k=1}^{3} x_{k} \neq-1\right\}$
and $$f=\overrightarrow{0}$$ on $$-A.$$ Prove the following.
(i) $$f$$ is one-to-one on $$A$$ (find $$f^{-1}!$$).
(ii) $$J_{f}(\vec{x})=\frac{1}{\left(1+\sum_{k=1}^{3} x_{k}\right)^{4}}$$.
(iii) Describe $$-A$$ geometrically.

Exercise $$\PageIndex{11}$$

Given any sets $$A, B$$ and maps $$f, g : A \rightarrow E^{\prime}, h : E^{\prime} \rightarrow E,$$ and $$k : B \rightarrow A,$$ prove that
(i) $$(f \pm g) \circ k=f \circ k \pm g \circ k,$$ and
(ii) $$h \circ(f \pm g)=h \circ f \pm h \circ g$$ if $$h$$ is linear.
Use these distributive laws to verify that
$\phi^{-1} \circ(\theta-\phi) \circ \theta^{-1}=\phi^{-1}-\theta^{-1}$
In Corollary 3.
[Hint: First verify the associativity of mapping composition.]

Exercise $$\PageIndex{12}$$

Prove that if $$\phi : E^{\prime} \rightarrow E$$ is linear and one-to-one, so is $$\phi^{-1} : E^{\prime \prime} \rightarrow E^{\prime},$$ where $$E^{\prime \prime}=\phi\left[E^{\prime}\right].$$

Exercise $$\PageIndex{13}$$

Let $$\vec{v}_{1}, \ldots, \vec{v}_{n}$$ be the column vectors in $$\operatorname{det}[\phi].$$ Prove that $$\operatorname{det}[\phi]$$ turns into
(i) $$c \cdot \operatorname{det}[\phi]$$ if one of the $$\vec{v}_{k}$$ is multiplied by a scalar $$c$$;
(ii) $$-\operatorname{det}[\phi],$$ if any two of the $$\vec{v}_{k}$$ are interchanged (consider $$\lambda$$ in formula (1)).
Furthermore, show that
(iii) $$\operatorname{det}[\phi]$$ does not change if some $$\vec{v}_{k}$$ is replaced by $$\vec{v}_{k}+c \vec{v}_{i}(i \neq k)$$;
(iv) $$\operatorname{det}[\phi]=0$$ if some $$\vec{v}_{k}$$ is $$\overrightarrow{0},$$ or if two of the $$\vec{v}_{k}$$ are the same.