March 2023 – Physicspupil

March 27, 2023March 27, 2023

202303270928 Problem 1.1.1

Let $F_1$ , $F_2$ , $F_3$ , … be the Fibonacci sequence. Prove that

$F_{n+1}F_{n+2}-F_{n}F_{n+3}=(-1)^n$

for all positive integers $n$ .

_{Extracted from A. P. Hillman. (1999). Abstract Algebra A First Undergraduate Course.}

Roughwork.

Let $P(n)$ be the statement to prove. When $n=1$ ,

$P(1):\quad F_{2}F_{3}-F_{1}F_{4}\stackrel{\textrm{?}}{=}-1$

is to check.

$\begin{aligned} F_1 & = 1\\ F_2 & = 1\\ F_3 & = 2\\ F_4 & = 3\\ \dots & \dots \\ F_{n} & = F_{n-1} + F_{n-2}\quad\textrm{for integers }n\geqslant 3 \\ \end{aligned}$

First, $P(1)$ is true.

Next, be $P(n)$ true. WTS $P(n+1)$ true whenever $P(n)$ true:

$\begin{aligned} P(n+1): & \quad F_{(n+1)+1}F_{(n+1)+2}-F_{n+1}F_{(n+1)+3}\stackrel{\textrm{?}}{=}(-1)^{n+1} \\ \textrm{LHS} & = F_{n+2}F_{n+3}-F_{n+1}F_{n+4} \\ & = (F_{n+1}+F_n)F_{n+3}-F_{n+1}(F_{n+3}+F_{n+2})\\ & = F_{n}F_{n+3}-F_{n+1}F_{n+2} \\ & = -(F_{n+1}F_{n+2}-F_{n}F_{n+3}) \\ & \stackrel{.}{=} -((-1)^n) \\ & = (-1)^{n+1} \\ & = \textrm{RHS} \\ \end{aligned}$

Lastly, as $P(1)$ true and $P(n)\Rightarrow P(n+1)$ , by principle of mathematical induction, the statement is proven for all $n\in\mathbb{Z}^+$ .

March 15, 2023March 15, 2023

202303150916 Exercise 4.6.4

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be defined by

$f(x,y)=(x^3+y^3,x^3-y^3)$ for $(x,y)\in\mathbb{R}^2$ .

Prove that the Jacobian matrix $J_{f,(0,0)}$ is the zero matrix. Show that nevertheless $f$ is globally invertible on $\mathbb{R}^2$ . [Hint: prove that $f$ is $\textrm{1--1}$ on $\mathbb{R}^2$ .] Show also that the inverse function is not differentiable at $f(0,0)=(0,0)$ .

_{Extracted from P. R. Baxandall. (1986). Vector Calculus.}

Roughwork.

$\begin{aligned} f\bigg(\begin{bmatrix}x\\y\end{bmatrix}\bigg) & = \begin{bmatrix} x^3+y^3 \\ x^3-y^3 \end{bmatrix} \\ J_{f,(x,y)} & = \begin{bmatrix} \partial_x (x^3+y^3) & \partial_y(x^3+y^3) \\ \partial_x(x^3-y^3) & \partial_y(x^3-y^3) \\ \end{bmatrix} \\ & = \begin{bmatrix} 3x^2 & 3y^2 \\ 3x^2 & -3y^2 \\ \end{bmatrix} \\ J_{f,(0,0)} & = \begin{bmatrix} 3(0)^2 & 3(0)^2 \\ 3(0)^2 & -3(0)^2 \end{bmatrix} \\ & = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} \\ \end{aligned}$

The function $f:\mathbb{R}^n\to\mathbb{R}^n$ is locally invertible at $\mathbf{x}_0$ if there is an $\epsilon >0$ and a function $g:B_{\epsilon}(f(\mathbf{x}_0))\to\mathbb{R}^n$ such that

$\begin{cases} f\circ g(\mathbf{y}) \equiv \mathbf{y}\quad\textrm{for }\mathbf{y}\in B_{\epsilon}(f(\mathbf{x}_0)) \\ g\circ f(\mathbf{x}) \equiv \mathbf{x}\quad \textrm{for }\mathbf{x}\in B_{\epsilon}(\mathbf{x}_0) \\ \end{cases}$

_{Luca Rigotti. (2015). University of Pittsburgh, ECON2001 Lecture 12}

I should have felt no hesitation in applying the inverse function theorem but for some difficulties when interpreting

$\begin{aligned} &\quad\enspace\textrm{ locally invertible everywhere} \\ & \nLeftrightarrow \textrm{ globally invertible} \\ \end{aligned}$

(to be continued)

March 14, 2023March 14, 2023

202303141200 Exercise 7.7.1

Consider the coordinate transformation $(x,y)=G(u,v)$ given by

$G(u,v)=(au,bv)$ for $(u,v)\in\mathbb{R}^2$

where $a$ and $b$ are positive constants.

(a) Prove that $G$ is $\textrm{1--1}$ on $\mathbb{R}^2$ .
(b) Show that $G$ maps the circular disc

$K=\{(u,v)\in\mathbb{R}^2 : u^2+v^2\leqslant 1\}$

in the $u$ , $v$ plane onto the elliptical disc

$G(K)=\{(x,y)\in\mathbb{R}^2 : x^2/a^2+y^2/b^2\leqslant 1\}$

in the $x$ , $y$ plane.
(c) Calculate $\mathrm{det}\,J_{G,(u,v)}$ .
(d) Calculate, using Theorem 7.7.5, the area of the elliptical disc $G(K)$ . (Assume known that the area of $K$ is $\pi$ .)

_{Extracted from P. R. Baxandall. (1986). Vector Calculus.}

Roughwork.

Symbolically,

$\forall\, a,b\in X,\quad f(a)=f(b)\Rightarrow a=b$ ,

or the contrapositive

$\forall\, a,b\in X,\quad a\neq b\Rightarrow f(a)\neq f(b)$ .

_{Wikipedia on Injective function}

(a)

From $G:\mathbb{R}^2\to\mathbb{R}^2:(u,v)\mapsto (x=au,y=bv)$ ,

$\begin{aligned} & \enspace & (x_1,y_1)=G(u_1,v_1) & = G(u_2,v_2)=(x_2,y_2) \\ \Rightarrow & \enspace & (au_1,bv_1) & = (au_2,bv_2) \\ \Rightarrow & \enspace & (u_1,v_1) & = (u_2,v_2) \\ \end{aligned}$

(b) Obvious.

(c)

$\begin{aligned} J_{G}(u,v) & = \begin{bmatrix} \partial_u(au)&\partial_v(au) \\ \partial_u(bv) & \partial_v(bv) \\ \end{bmatrix} \\ & = \begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix} \\ \mathrm{det}\,J_{G,(u,v)} & = ab \\ \end{aligned}$

(d)

Theorem (Change of variables). Let $G:K\subseteq \mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ function defined on a compact set $K$ in $\mathbb{R}^2$ , and let $D\subseteq K$ be an open subset of $\mathbb{R}^2$ such that

i. $K\backslash D$ is a null set,
ii. $G$ is $\textrm{1--1}$ on $D$ ,
iii. $\mathrm{det}\,J_{G,(u,v)}\neq 0$ for all $(u,v)\in D$ .

Then, for any bounded function $f:G(K)\subseteq\mathbb{R}^2\to\mathbb{R}$ which is continuous on $G(D)$

$\displaystyle{\iint_{G(K)}f(x,y)\,\mathrm{d}x\,\mathrm{d}y=\iint_{K}f(G(u,v))|\mathrm{det}\,J_{G,(u,v)}|\,\mathrm{d}u\,\mathrm{d}v}$ .

_{See Theorem 7.7.5 on pg. 404.}

The area of the elliptical disc $G(K)$ is $\pi ab$ .

This problem is not to be attempted.

March 14, 2023March 14, 2023

202303141137 Exercise 2.7.1

Calculate the lengths of the following smooth simple arcs in $\mathbb{R}^3$ .

(a) The circular helix parametrized by

$f(t)=(t,\cos t,\sin t)$

where $t\in [a,b]$ ;
(b) the curve parametrized by

$f(t)=(e^t\cos t,e^t\sin t,e^t)$

where $t\in [0,k]$ .

_{Extracted from P. R. Baxandall. (1986). Vector Calculus.}

Roughwork.

Let $f:[a,b]\subseteq\mathbb{R}\to\mathbb{R}^n$ be a $C^1$ path in $\mathbb{R}^n$ . The length of $f$ is defined to be

$l(f)=\displaystyle{\int_{a}^{b}\|f'(t)\|\,\mathrm{d}t}$ .

_{See Definition 2.7.2 on pg.60}

(a)

$\begin{aligned} f(t) & = (t,\cos t,\sin t) \\ f'(t) & = (1,-\sin t,\cos t) \\ \|f'(t)\| & = \sqrt{(1)^2+(-\sin t)^2+(\cos t)^2} \\ & = \sqrt{2} \\ l(f) & = \int_{a}^{b} \sqrt{2}\,\mathrm{d}t \\ & = \sqrt{2}(b-a) \\ \end{aligned}$

(b) Not to be attempted.