Homework Assignments
Homework is collected at the beginning of lecture of the specified day. No late homework will be accepted. Please write your name and the homework number on each assignment. If you would like to try writing your homework with LaTeX, here is a template that produces this output.
Homework 12, due Friday, April 27th Solutions
- 1. Let $f\colon \mathbb{R}^d \to\overline{\mathbb{R}}$. Show that $f$ is measurable if and only if $f^{-1}(\{\infty\})\in \mathcal{M}(\mathbb{R}^d)$, $f^{-1}(\{-\infty\})\in \mathcal{M}(\mathbb{R}^d)$, and $f^{-1}(U)\in \mathcal{M}(\mathbb{R}^d)$ for every open $U\subset \mathbb{R}$.
- 2. Let $f\colon \mathbb{R}^d\to\overline{R}$ be measurable. Show that the collection
\[
\mathcal{A}:=\{S\subset \overline{\mathbb{R}}\colon f^{-1}(S)\in \mathcal{M}(\mathbb{R}^d)\},
\]
is a $\sigma$-algebra:
- (i) $\emptyset \in \mathcal{A}$.
- (ii) If $E\in \mathcal{A}$, then $E^c\in \mathcal{A}$.
- (iii) If $\{E_n\}_{n\in\mathbb{N}}\subset \mathcal{A}$, then $\displaystyle \bigcup_{n=1}^\infty E_n\in \mathcal{A}$.
- 3. Let $B\subset \mathbb{R}^d$ be a box, and suppose $f\colon \mathbb{R}^d\to[0,\infty)$ is Riemann integrable over $B$. Show that the Riemann integral of $f$ over $B$ is equal to $\int_B f\ dm$, the Lebesgue integral of $f$ over $B$.
- 4. For $f\in L^1(m)$, show $m(\{x\colon |f(x)|=\infty\})=0$.
- 5.
- (a) Suppose $f\in L^1(\mathbb{R},m)$ is uniformly continuous. Show that $\displaystyle \lim_{|x|\to\infty} f(x)=0$.
- (b) Find a positive, continuous $f\in L^1(\mathbb{R},m)$ such that $\displaystyle \limsup_{x\to\infty} f(x)=\infty$.
- 6. Let $\delta=(\delta_1,\ldots, \delta_n)\in (0,\infty)^n$. For $f\colon \mathbb{R}^n\to \overline{\mathbb{R}}$, define \[ f^\delta(x_1,\ldots, x_n) = f(\delta_1x_1,\ldots, \delta_n x_n). \] If $f\in L^1(\mathbb{R}^n,m)$, show that $f^\delta\in L^1(\mathbb{R}^n,m)$ with $\int f^\delta\ dm = \frac{1}{\delta_1}\cdots \frac{1}{\delta_n} \int f\ dm$.
Homework 11, due Wednesday, April 18th (Sections 6.1 and 6.2) Solutions
- 1. Let $A\subset \mathbb{R}$ satisfy $m^*(A)>0$. Show that for every $\alpha\in (0,1)$ there exists an open interval $I$ such that \[ m^*(A\cap I) \geq \alpha m^*(I) \] [Hint: Use the definition of the outer measure to find an open set $U\supset A$ such that $m^*(A)\geq \alpha m^*(U)$. Then use the fact that every open subset of $\mathbb{R}$ is a countable disjoint union of open intervals.]
- 2. Let $E\in\mathcal{M}(\mathbb{R})$ with $m(E)>0$. Consider the difference set
\[
D(E):=\{ x-y\colon x,y\in E\}.
\]
Show that $D$ contains an open interval centered at the origin.
[Hint: Invoke the previous exercise for $\alpha\in (\frac12,1)$. Also, for any set $S$, if $D(S)$ does not contain an open interval centered at the origin, then for all $\delta > 0$ the set $(-\delta,\delta)\setminus D(S)\neq \emptyset$. Think about the relation between $S$ and its translation by an element of $(-\delta,\delta)\setminus D(S)$.] - 3. Let $A\subset [0,1]$ be the set of numbers without the digit 4 appearing in their decimal expansion. Show that $A$ is Lebesgue measurable and compute $m(A)$.
[Hint: consider for each $n\in \mathbb{N}$ the set of numbers without the digit 4 appearing in the first $n$ digits of the decimal expansion.] - 4.
- (a) Show that every closed set in $\mathbb{R}^d$ is both $G_\delta$ and $F_\sigma$.
- (b) Show that every open set in $\mathbb{R}^d$ is both $G_\delta$ and $F_\sigma$.
- (c) Show every Riemann measurable set in $\mathbb{R}^d$ is Lebesgue measurable.
- 5. [The Borel–Cantelli Lemma] Suppose $\{E_n\}_{n\in\mathbb{N}}\subset \mathcal{M}(\mathbb{R}^d)$ satisfies
\[
\sum_{n=1}^\infty m(E_n) < \infty.
\]
Consider $E:=\{x\in \mathbb{R}^d\colon x\in E_n \text{ for infinitely many }n\in \mathbb{N}\}$.
- (a) Show that \[ E=\bigcap_{N=1}^\infty \bigcup_{n\geq N} E_n. \]
- (b) Show that $E\in\mathcal{M}(\mathbb{R}^d)$ with $m(E)=0$.
- (c) Show that \[ \chi_{E}(x) = \limsup_{n\to\infty} \chi_{E_n}(x) \qquad \forall x\in \mathbb{R}^d \] [Note: for this reason, $E$ is typically denoted $\displaystyle \limsup_{n\to\infty} E_n$.]
Homework 10, due Wednesday, April 11th (Sections 5.9 and 6.1) Solutions
- 1. Let $U$ and $V$ be open subsets.
- (a) For $T\colon U\to V$ be a smooth map, show that the pullback \[ T^*\colon \Omega^k(V) \to \Omega^k(U) \] satisfies $T^*(Z^k(V))\subset Z^k(U)$ and $T^*(B^k(V))\subset B^k(U)$.
- (b) Prove that if $U$ and $V$ are $C^\infty$-diffeomorphic, then $H^k(U)\cong H^k(V)$ as vector spaces.
- 2. Let $\delta=(\delta_1,\ldots, \delta_n)\in (0,\infty)^n$. For $A\subset \mathbb{R}^n$, define \[ \delta A := \{ (\delta_1 x_1,\ldots, \delta_n x_n)\colon (x_1,\ldots, x_n)\in A\}. \] Show that $m^*(\delta A) = \delta_1\cdots \delta_n m^*(A)$.
- 3. For $A\subset \mathbb{R}^n$, the Jordan content of $A$ is the quantity
\[
J^*(A):=\inf\left\{ \sum_{k=1}^N |B_k|\colon N < \infty,\ A\subset \bigcup_{k=1}^N B_k,\ B_k\text{ open boxes}\right\}.
\]
That is, in contrast with the outer measure, here the infimum is taken over finite coverings of $A$ by open boxes.
- (a) Show that $m^*(A)\leq J^*(A)$ for all $A\subset \mathbb{R}^n$.
- (b) Show that for any subset $A\subset \mathbb{R}^n$, $J^*(A) = J^*(\overline{A})$.
- (c) Find a subset $A\subset \mathbb{R}$ such that $m^*(A) < J^*(A)$.
- 4. Let $S\subset \mathbb{R}^2$ be Riemann measurable. Show that $|S|=m^*(S)=J^*(S)$.
[Hint: first show that you can replace $S$ with $\overline{S}$, then take advantage of compactness.] - 5. [Not Collected] Consider the triangle
\[
T:=\{(x,y)\in \mathbb{R}^2\colon 0\leq y\leq x\leq 1\}.
\]
Convince yourself that computing $m^*(T)=\frac12$ directly from the definition of the outer measure is hard.
[Note: the easy way to show this is by using Exercise 4 and Fubini's theorem.]
Homework 9, due Wednesday, April 4th (Sections 5.8 and 5.9) Solutions
- 1. Let $T\colon \mathbb{R}^2\to\mathbb{R}^3$ be the smooth function defined by \[ T(z_1,z_2) = ( z_1^2+z_2^2, z_1z_2, z_1-z_2^3). \] For $\omega = y_2 d y_{(1,3)}\in \Omega^2(\mathbb{R}^3)$, compute the ascending presentation of $T^*\omega$.
- 2. Let $\varphi\in C_2(\mathbb{R}^3)$ be defined by \[ \varphi(u)= ( \cos(2\pi u_1),\sin(2\pi u_1), u_2). \] Compute the dipoles $\delta^1 \varphi$ and $\delta^2\varphi$, and the boundary $\partial \varphi$. Describe these (using words or pictures), making sure to note the proper orientations.
- 3. Let $\omega\in \Omega^1(\mathbb{R}^2)$. Assume $\omega$ is closed: $d\omega =0$.
- (a) For $p,q\in \mathbb{R}^2$, let $\varphi,\psi\in C_1(\mathbb{R}^2)$ be such that $\varphi(0)=\psi(0)=p$ and $\varphi(1)=\psi(1)=q$. Show that $\int_\varphi \omega = \int_\psi \omega$.
[Hint: consider $\sigma\in C_2(\mathbb{R}^2)$ defined by $\sigma(s,t) = (1-s) \varphi(t) + s \psi(t)$.] - (b) Fix $p\in \mathbb{R}^2$. Define $h\colon \mathbb{R}^2\to\mathbb{R}$ by $h(q):=\int_\varphi \omega$ where $\varphi$ is a $1$-cell in $\mathbb{R}^2$ satisfying $\varphi(0)=p$ and $\varphi(1)=q$. Show that $h$ is a smooth function with $dh = \omega$.
- (a) For $p,q\in \mathbb{R}^2$, let $\varphi,\psi\in C_1(\mathbb{R}^2)$ be such that $\varphi(0)=\psi(0)=p$ and $\varphi(1)=\psi(1)=q$. Show that $\int_\varphi \omega = \int_\psi \omega$.
Homework 8, due Wednesday, March 14th (Section 5.8) Solutions
- 1. Fix $(x_0, y_0)\in\mathbb{R}^2$ and $a,b>0$. Consider $\varphi\in C_1(\mathbb{R}^2)$ defined by \[ \varphi(t)=\left( a\cos(2\pi t)+x_0, b\sin(2\pi t) + y_0\right). \] For $\omega= -y_2dy_1\in\Omega^1(\mathbb{R}^2)$, compute $\int_\varphi \omega$.
- 2. Let
\[
S^2 =\{(x,y,z)\in\mathbb{R}^3 \colon x^2+y^2+z^2=1\},
\]
so that $S^2=\varphi(I^2)$ for $\varphi\in C_2(\mathbb{R}^3)$ defined by
\[
\varphi(s,t) = ( \cos(2\pi s)\sin(\pi t), \sin(2\pi s)\sin(\pi t), \cos(\pi t) ).
\]
- (a) Compute the $(2,1)$-shadow area of $\varphi$: $\int_\varphi dy_{(2,1)}$.
- (b) For $\omega = y_2 dy_{(1,3)} \in \Omega^2(\mathbb{R}^3)$, compute $\int_\varphi \omega$.
- 3. Fix $y\in\mathbb{R}^n$, $r>0$, and an ascending $A=(i_1,\ldots,i_k)\in \{1,\ldots, n\}^k$. Define $\iota \in C_k(\mathbb{R}^n)$ by \[ \iota(u_1,\ldots, u_k) = y+ r (u_1e_{i_1}+\cdots +u_ke_{i_k}), \] where $e_1,\ldots, e_n$ are the standard basis vectors in $\mathbb{R}^n$. Show that for $I\in \{1,\ldots, n\}^k$ and $u\in I^k$ \[ \frac{\partial \iota_I(u)}{\partial u} = \begin{cases} \text{sgn}(\pi) r^k & \text{if }I=\pi A\text{ for some }\pi\in S_k\\ 0 & \text{otherwise} \end{cases}. \]
- 4. Let $\omega = \sum f_A dy_A\in \Omega^k(\mathbb{R}^n)$ be in ascending presentation. Define the \text{symmetrization map} $\text{symm}\colon \Omega^k(\mathbb{R}^n)\to\Omega^k(\mathbb{R}^n)$ by \[ \text{symm}(\omega) = \frac{1}{k!} \sum_{A} \sum_{\pi \in S_k} f_A dy_{\pi A}, \] where if $A=(i_1,i_2,\ldots, i_k)$ then $\pi A = (i_{\pi(1)},i_{\pi(2)},\ldots, i_{\pi(k)})$. Show that if $k>1$, then $\text{symm}(\omega)=0$ for all $\omega\in \Omega^k(\mathbb{R}^n)$.
- 5. Let $\alpha,\beta\in \Omega^3(\mathbb{R}^6)$ be defined by
\begin{align*}
\alpha &:= a_1 dy_{(1,3,2)} + a_2 dy_{(4,2,3)} + a_3 dy_{(4,3,1)}\\
\beta&:= b_1 dy_{(3,5,6)} + b_2 dy_{(6,4,5)} + b_3 dy_{(2,5,3)},
\end{align*}
where $a_i,b_j\colon\mathbb{R}^6\to\mathbb{R}$ are smooth functions for $1\leq i,j\leq 3$.
- (a) Determine the ascending presentations of $\alpha$ and $\beta$.
- (b) Compute $\alpha\wedge \beta$ using the ascending presentations.
- (c) Using the insensitivity to presentation, compute $\alpha\wedge \beta$ with their original presentations and verify that this agrees with the answer in part (b).
Homework 7, due Wednesday, March 7th (Sections 5.7 and 5.8) Solutions
- 1. Consider $f\colon [0,1]^2\to\mathbb{R}$ defined by \[ f(x,y)=\begin{cases} 1 - \frac1q &\text{if }x,y\in\mathbb{Q}\text{ with }y=\frac{p}{q}\text{ in lowest terms}\\ 1 & \text{otherwise} \end{cases}. \] Prove that $f$ is Riemann integrable with $\int_{[0,1]^2} f=1$.
- 2. Let $0\leq \epsilon\leq 1$ and suppose $a,b>0$ satisfy \[ \frac{1}{(1+\epsilon)^n} \leq \frac{a}{b} \leq (1+\epsilon)^n. \] Show that $|a-b|\leq 2^n b\epsilon$.
- 3. Let
\[
S=\{(x,y)\in\mathbb{R}^2\colon x^2+y^2\leq a^2\},
\]
and let $f\colon S\to\mathbb{R}$ be Riemann integrable. Prove that
\[
\int_S f = \int_0^{2\pi} \int_0^a f(r\cos{\theta},r\sin{\theta}) r\ drd\theta,
\]
despite the fact that the polar coordinates do not define a $C^1$-diffeomorphism on $R=[0,a]\times [0,2\pi]$.
[Hint: approximate $S$ by keyholes.] - 4. Let $\varphi_1,\varphi_2\in C_1(\mathbb{R}^2)$ be $C^1$-diffeomorphisms such that $\varphi_1([0,1])=\varphi_2([0,1])$.
- (a) Show that either \[ (\varphi_1(0), \varphi_1(1)) = (\varphi_2(0),\varphi_2(1))\qquad\text{or}\qquad (\varphi_1(0), \varphi_1(1)) = (\varphi_2(1),\varphi_2(0)). \]
- (b) Show that for any differential $1$-form $\omega=fdx+gdy$ we have \[ \omega(\varphi_1)=\pm \omega(\varphi_2), \] where we get a `$+$' if the first case in part (a) holds and we get `$-$' otherwise.
- 5. Find $\varphi \in C_3(\mathbb{R}^3)$, a $3$-cell in $\mathbb{R}^3$, such that $\varphi(I^3)$ is the closed ball centered at $(0,0,0)$ with radius $1$.
Homework 6, due Wednesday, February 28th (Section 5.7) Solutions
- 1. Let $S\subset\mathbb{R}^2$ be a zero set. Show that its interior \[ S^\circ:=\{z\in S\colon \exists r>0 \text{ such that } B(z,r)\subset S\}, \] is empty.
- 2. Let $f\colon \mathbb{R}^2\to\mathbb{R}$ be bounded. Recall that for $z\in \mathbb{R}^2$, the oscillation of $f$ at $z$ is the quantity
\[
\text{osc}_z(f) := \lim_{r\to 0} \left[ \sup(f(B(z,r))) - \inf(f(B(z,r))) \right].
\]
- (a) Show that $f$ is continuous at $z\in\mathbb{R}^2$ if and only if $\text{osc}_z(f)=0$.
- (b) For $S\subset\mathbb{R}^2$, show that $\chi_S$ is discontinuous at $z$ if and only if $z\in \partial S$.
- 3. Let $R\subset\mathbb{R}^2$ be a rectangle. Suppose $f,g\colon R\to\mathbb{R}$ are Riemann integrable over $R$ and that
\[
P:=\{z\in R\colon |f(z)-g(z)| > 0\}
\]
is a zero set. Show that $\int_R f = \int_R g$.
[Hint: first show that for $z\in P$, $\text{osc}_z(|f-g|) \geq |f(z) - g(z)|$, then proceed as in the proof of the Riemann–Lebesgue Theorem.] - 4. Let $S\subset\mathbb{R}^2$ be bounded.
- (a) Show that if $S$ is Riemann measurable then so are $S^\circ$ (its interior) and $\overline{S}$ (its closure).
- (b) Show that if $S^\circ$ and $\overline{S}$ are Riemann measurable with $|S^\circ|=|\overline{S}|$, then $S$ is Riemann measurable with the same area.
- (c) Show that the hypothesis $|S^\circ|=|\overline{S}|$ in the previous part is necessary by considering $S=\mathbb{Q}^2\cap[0,1]^2$.
- 5. Use the volume multiplier formula to prove that the area of a parallelogram is the length of its base times its height.
Homework 5, due Friday, February 23rd (Sections 5.4 and 5.7) Solutions
- 1. Let $f\colon \mathbb{R}^3\to\mathbb{R}^2$ be of class $C^1$. Assume that $f(3,-1,2)=0$ and
\[
(Df)_{(3,-1,2)} = \left[\begin{array}{ccc} 1 & 2 & 1 \\ 1 & -1 & 1\end{array}\right].
\]
- (a) Show that there is an open neighborhood $U\subset \mathbb{R}$ of $3$ and a function $g\colon U\to\mathbb{R}^2$ of class $C^1$ such that $g(3)=(-1,2)$ and $f(x,g(x))=0$ for all $x\in U$.
- (b) Determine $(Dg)_3$.
- 2. Let $U\subset \mathbb{R}^m$ be open and let $f\colon U\to\mathbb{R}^m$ be of class $C^1$. Show that if $(Df)_x$ is invertible for all $x\in U$, then $f(U)$ is open.
- 3. Let $G\subset \mathcal{M}(n,n)$ denote the set of invertible $n\times n$ matrices, with metric given by the operator norm.
- (a)Show that $G$ is open. [Hint: use the determinant but take care with the norms.]
- (b) Show that $G$ contains $1$, is closed under multiplication, and closed under the map $\text{Inv}(A):=A^{-1}$; that is, show $G$ is a group. [Note: this group is called the general linear group and is typically denoted $GL_n(\mathbb{R})$.]
- (c) Show that $\text{Inv}\colon G\to G$ is a homeomorphism.
- (d) Show that $\text{Inv}\colon G\to G$ is a $C^1$ diffeomorphism with \[ (D \text{Inv})_A(X) = - A^{-1} X A^{-1}\qquad A\in G,\ X\in \mathcal{M}(n,n). \] [Note: this should remind you of the formula $\frac{d}{dx}\left(\frac1x\right) = -\frac{1}{x^2}$.]
- 4. Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function. Show that its graph \[ G:=\{(x,f(x))\colon x\in \mathbb{R}\}\subset\mathbb{R}^2 \] is a zero set.
Homework 4, due Wednesday, February 7th (Sections 5.2 and 5.3) Solutions
- 1. Consider $f\colon \mathbb{R}^3\to\mathbb{R}^2$ defined by
\[
f(x_1,x_2,x_3) = (x_1x_2 + x_2x_3, x_3^3).
\]
For $p\in \mathbb{R}^3$, determine the matrix representation for $(D^2 f)_p$ with respect to the ordered basis
\[
\{ (e_1,e_1), (e_1, e_2), (e_1, e_3), (e_2, e_1), (e_2, e_2), (e_2, e_3), (e_3, e_1), (e_3, e_2), (e_3,e_3)\}
\]
for $\mathbb{R}^{3^2}$. Then prove that the corresponding Taylor remainder for $Df$ at $p$ is sublinear.
[Note: you already computed $(Df)_p$ on Homework 2, so you do not need to rederive this.] - 2. Let $\beta\in \mathcal{L}^k(\mathbb{R}^n, \mathbb{R}^m)$ be a $k$-linear map. The symmetrization of $\beta$ is the $k$-linear map $\text{symm}(\beta)\in \mathcal{L}^k(\mathbb{R}^n,\mathbb{R}^m)$ defined by
\[
\text{symm}(\beta)(v_1,\ldots, v_k)= \frac{1}{k!} \sum_{\sigma \in S_k} \beta(v_{\sigma(1)},\ldots, v_{\sigma(k)}),
\]
where $S_k$ is the permutation group on $k$ elements.
- (a) Show that $\text{symm}(\beta)$ is indeed symmetric.
- (b) Show that $\beta$ is symmetric if and only if $\beta=\text{symm}(\beta)$.
- 3. Let $\beta\in\mathcal{L}^k(\mathbb{R}^n,\mathbb{R}^m)$ be a $k$-linear map. Define $f\colon \mathbb{R}^n\to\mathbb{R}^m$ by $f(x)=\beta(x,\ldots, x)$.
- (a) Show that for $p,v\in \mathbb{R}^n$ \[ (Df)_p(v) = \beta(v,p,\ldots, p)+ \beta(p,v,p,\ldots,p) + \cdots + \beta(p,\ldots, p,v). \]
- (b) Show that for $p,v_1,v_2\in \mathbb{R}^n$ \[ (D^2f)_p(v_1,w_2) = \sum_{1\leq i < j\leq k}\sum_{\sigma\in S_2} \beta(\underbrace{p,\ldots, p}_{i-1}, v_{\sigma(1)}, \underbrace{p,\ldots,p}_{j-i-1}, v_{\sigma(2)},p,\ldots, p). \]
- (c) Further assume that $\beta$ is symmetric. For $r\geq 0$, show that for $p, v_1,\ldots, v_r\in \mathbb{R}^n$
\[
(D^r f)_p(v_1,\ldots, v_r) = \begin{cases} \displaystyle \frac{k!}{(k-r)!} \beta(v_1,\ldots, v_r,p,\ldots, p) & \text{if }r\leq k\\
0 & \text{if }r>k \end{cases}.
\]
Note that this implies $(D^k f)_p = k! \text{symm}(\beta)$.
[Hint: proceed by induction and note that the base case $r=0$ is trivial.]
- 4. Consider $f\colon \mathbb{R}^2\to\mathbb{R}$ defind by \[ f(x,y) = \begin{cases} \displaystyle \frac{xy(x^2 - y^2)}{x^2+y^2} & \text{if }(x,y)\neq (0,0)\\ 0 & \text{if }(x,y)=(0,0) \end{cases}. \] Show that the second partial derivatives exists everywhere, but that $\frac{\partial^2 f(0,0)}{\partial x\partial y}\neq \frac{\partial^2 f(0,0)}{\partial y \partial x}$.
- 5. Let $f\colon U\to\mathbb{R}^m$ be $r$-times differentiable at $p\in U$ with $r\geq 3$. Use induction to show that $(D^r f)_p$ is symmetric: first show for $v_1,\ldots, v_r\in \mathbb{R}^n$ that $(D^r f)_p$ is symmetric with respect to permutations of $v_2,\ldots, v_r$, and then use the fact that $r\geq 2$ to show that it is also invariant under permutations of $v_1$ and $v_2$.
Homework 3, due Wednesday, January 31st (Section 5.2) Solutions
- 1. In this exercise, we will verify the Chain Rule. Consider the functions $f\colon \mathbb{R}^2\to\mathbb{R}^3$ and $g\colon \mathbb{R}^3\to\mathbb{R}$ defined by
\[
f(x_1,x_2) = (x_1x_2, x_1+x_2, x_1^2)\qquad g(x_1,x_2,x_3) = x_1x_2x_3.
\]
- (a) Compute $(Df)_{x}$ and $(Dg)_{y}$ for $x\in \mathbb{R}^2$ and $y\in \mathbb{R}^3$.
- (b) Determine the formula for $g\circ f\colon \mathbb{R}^2\to\mathbb{R}$ and compute $(D(g\circ f))_{x}$ for $x\in \mathbb{R}^2$.
- (c) Compute $(Dg)_{f(x)}\circ (Df)_x$ for $x\in \mathbb{R}^2$, and compare this to part (b).
- 2. In this exercise, we will verify the Product Rule (for the bilinear form determined by the standard inner product on $\mathbb{R}^2$). Consider the functions $f,g\colon \mathbb{R}^2\to\mathbb{R}^2$ defined by
\[
f(x_1,x_2) = (x_2^2, x_1^2)\qquad g(x_1,x_2) = (\cos(x_1) , \sin(x_2)).
\]
- (a) Compute $(Df)_x$ and $(Dg)_x$ for $x\in \mathbb{R}^2$.
- (b) Define $h\colon \mathbb{R}^2\to\mathbb{R}$ by \[ h(x_1,x_2) = \left\langle f(x_1,x_2), g(x_1,x_2) \right\rangle. \] Compute $(Dh)_x$ for $x\in \mathbb{R}^2$.
- (c) Show that for any $x,v\in \mathbb{R}^2$, \[ (Dh)_x(v) = \left\langle (Df)_x(v), g(x)\right\rangle + \left\langle f(x), (Dg)_x(v)\right\rangle. \] [Hint: by linearity, it suffices to check the above equality for $v=e_1,e_2$.]
- 3. [Correction: instead of assuming that $C$ below is closed, you may assume the set $B=\{(Df)_x(p-q)\colon x\in [p,q]\}$ is closed.]
Let $U\subset\mathbb{R}^n$ be open, and let $f\colon U\to\mathbb{R}^m$ be differentiable on $U$. Let $[p,q]\subset U$ be a segment. Assume the set \[ C=\{ (Df)_x\in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)\colon x\in [p,q]\} \] is closed and convex: whenever $T,S\in C$ we have $t T + (1-t)S\in C$ for $t\in [0,1]$. Show that there exists $\theta\in [p,q]$ such that \[ f(q) - f(p) = (Df)_\theta (q-p); \] that is, the direct generalization of one-dimensional the Mean Value Theorem holds. You may (and should) use the following special case of the Hahn–Banach separation theorem without proof: if $A$ and $B$ are non-empty, disjoint convex sets in $\mathbb{R}^m$ and $A$ is open, then there exists a vector $v\in \mathbb{R}^m$ and a scalar $c\in\mathbb{R}$ such that \[ \left\langle v, a\right\rangle < c \leq \left\langle v, b\right\rangle \] for all $a\in A$ and $b\in B$.
[Hint: proceed by contradiction and use the separation theorem in conjunction with the one-dimensional Mean Value Theorem.] - 4. Let $U\subset\mathbb{R}^n$ be open and connected, and let $f\colon U\to\mathbb{R}^m$ be differentiable on $U$ with $(Df)_p=0$ for all $p\in U$. Show that $f$ is constant.
- 5. Let $(E,d)$ be an arbitrary metric space and let $[a,b]\subset\mathbb{R}$. Equip $[a,b]\times E$ with the product metric:
\[
d_2( (x,y), (x',y')) = \sqrt{ |x-x'|^2 + d(y,y')^2}\qquad x,x'\in [a,b],\ y,y'\in E.
\]
Let $f\colon [a,b]\times E\to\mathbb{R}$ be a continuous function. Show that
\[
F(y)=\int_a^b f(x,y)\ dx
\]
is continuous on $E$.
[Hint: recall that $[a,b]$ is compact.]
Homework 2, due Wednesday, January 24th (Sections 5.1 and 5.2) Solutions
- 1. Two norms $|\cdot|_1$ and $|\cdot|_2$ on a vector space $V$ are equivalent if there exists positive constants $c$ and $C$ such that for all $v\in V\setminus\{0\}$
\[
c\leq \frac{|v|_1}{|v|_2} \leq C.
\]
- (a) Show that any two norms on a finite-dimensional vector space are equivalent.
- (b) Let $C([0,1])$ be the space of continuous functions from $[0,1]$ to $\mathbb{R}$. For $f\in C([0,1])$ consider the following two norms:
\[
|f|_1 := \int_0^1 |f(t)|\ dt \qquad |f|_\infty:= \max\{ |f(t)|\colon 0\leq t\leq 1\}.
\]
Show that these two norms are not equivalent.
[Note: you do not need to verify that these are in fact norms.]
- 2. Let $C([0,1])$ be as in the previous exercise, equipped with the $|\cdot|_\infty$-norm. Let $C^1([0,1])$ be the (dense) subspace of $C([0,1])$ consisting of differentiable functions whose derivatives are continuous.
- (a) Define $T\colon C([0,1]) \to C^1([0,1])$ by \[ (Tf)(t) = \int_0^t f(x)\ dx. \] Show that $T$ is continuous by determining its operator norm.
- (b) Define $S\colon C^1([0,1])\to C([0,1])$ by $S f = f'$. Show that $S$ that is not continuous by showing its operator norm is infinite.
[Hint: consider the monomials $t^n$, $n\in\mathbb{N}$.] - (c) [Not collected] Convince yourself that \[ S\circ T \colon C([0,1])\to C([0,1]) \] is the identity map, and that \[ T\circ S\colon C^1([0,1])\to C^1([0,1]) \] is the identity map minus the linear transformation that evaluates a function at $0$.
- 3. Define $f\colon \mathbb{R}^3\to\mathbb{R}^2$ by \[ f(x_1,x_2,x_3) = ( x_1x_2 + x_2x_3, x_3^3) \] For $p=(p_1,p_2,p_3)\in \mathbb{R}^3$, use the definition of the derivative to show $(Df)_p = T_A$ where \[ A = \left[\begin{array}{ccc} p_2 & p_1+ p_3 & p_2 \\ 0 & 0 & 3 p_3^2 \end{array}\right] \] (i.e. show the Taylor remainder is sublinear).
- 4. Consider the function $f\colon \mathbb{R}^2 \to \mathbb{R}$ defined by \[ f(x_1,x_2) = \left\{\begin{array}{cl} \displaystyle\frac{x_1x_2}{x_1^2 + x_2^2} & \text{if }(x_1,x_2)\neq 0\\ 0 & \text{if }(x_1,x_2)=0 \end{array}\right.. \] Show that the partial derivatives $\frac{\partial f(0)}{\partial x_1}$ and $\frac{\partial f(0)}{\partial x_2}$ exist but $(Df)_0$ does not.
Homework 1, due Friday, January 19th (Section 5.1) Solutions
- 1. Find your book from Math 110.
- 2. Compute (without proof) the following matrices:
- (a) $\displaystyle \left[\begin{array}{cc} 2 & -1 \\ -3 & 0\end{array}\right] + \left[\begin{array}{cc} 5 & 2 \\ 4 & -6\end{array}\right]$.
- (b) $\displaystyle \left[\begin{array}{ccc} 1 & 0 & 1 \\ -2 & 3 & 4\end{array}\right]\cdot\left[\begin{array}{cc} 2 & 1 \\ 0 & 3\\ -6 & 7\end{array}\right]$.
- (c) $\displaystyle \left[\begin{array}{ccc} 1 & 0 & 1 \\ -2 & 3 & 4\end{array}\right]^\text{T}$ (here 'T' denotes the transpose of the matrix).
- (d) $\displaystyle \left[\begin{array}{ccc} 1 & 2& 3 \\ 1 & 0 & 1\\ - 2 & 2 & 1\end{array}\right]^{-1}$.
- 3. Let $V$ be the vector space of degree three polynomials with real coefficients. Let $W$ be the vector space of degree two polynomials with real coefficients. Consider the linear operator $T\colon V\to W$ which sends a polynomial to its derivative. Determine (without proof) the matrix representation of $T$ with respect to the bases $\{1,x,x^2,x^3\}$ for $V$ and $\{1,x,x^2\}$ for $W$.
- 4. Denote $\displaystyle O=\left[\begin{array}{cc} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2}\end{array}\right]$.
- (a) Prove that $\left\langle Ox, Oy\right\rangle=\langle x, y\rangle$ for any vectors $x,y\in \mathbb{R}^2$.
[Note: such a matrix is called an orthogonal matrix, and its column vectors yield an orthonormal basis for $\mathbb{R}^2$.] - (b) Remind yourself what "orthonormal basis" means.
- (c) Prove that $\|O\|=1$. [Hint: use part (a).]
- (a) Prove that $\left\langle Ox, Oy\right\rangle=\langle x, y\rangle$ for any vectors $x,y\in \mathbb{R}^2$.
- 5. Let $n\in \mathbb{N}$ and let $D\in M_{n\times n}(\mathbb{R}^n)$ be a diagonal matrix with diagonal entries $d_1,d_2,\ldots, d_n\in \mathbb{R}$. Prove that $\displaystyle \|D\|=\max_{1\leq i\leq n} |d_i|$.
