Physicspupil – Page 28

October 11, 2021January 19, 2023

202110110958 Sidenote (Section 3.8.2)

Given that the binomial coefficient is so defined as

$\begin{pmatrix} N \\ n \end{pmatrix} \stackrel{\textrm{def}}{=} \displaystyle{\frac{N!}{n!(N-n)!}}$ .

Prove the identities (3.49), (3.50), and (3.51) in Section 3.8.2 Useful Identities for the Binomial Coefficients. Namely,

Eq. (3.49):

$\begin{pmatrix} N \\ 0 \end{pmatrix} = \begin{pmatrix} N \\ N \end{pmatrix} = 1$

Eq. (3.50):

$\begin{pmatrix} N - 1 \\ n \end{pmatrix} + \begin{pmatrix} N - 1 \\ n-1 \end{pmatrix} = \begin{pmatrix} N \\ n \end{pmatrix}$

Eq. (3.51):

$\begin{pmatrix} N \\ n+1 \end{pmatrix} = \displaystyle{\frac{N-n}{n+1}}\begin{pmatrix} N \\ n \end{pmatrix}$

_{R. H. Swendsen. (2012). An Introduction to Statistical Mechanics and Thermodynamics}

Proof of Eq. (3.49).

$\begin{aligned} \begin{pmatrix} N \\ 0 \end{pmatrix} & = \frac{N!}{0!(N-0)!} \\ & = \frac{N!}{1\times N!} \\ & = 1 \end{aligned}$
$\begin{aligned} \begin{pmatrix} N \\ N \end{pmatrix} & = \frac{N!}{N!(N-N)!} \\ & = \frac{N!}{N!\times 1} \\ & = 1 \end{aligned}$

Proof of Eq. (3.50).

$\begin{aligned} & \quad \begin{pmatrix} N-1 \\ n \end{pmatrix} + \begin{pmatrix} N-1 \\ n-1 \end{pmatrix} \\ & = \frac{(N-1)!}{(n)!(N-1-n)!} + \frac{(N-1)!}{(n-1)!\big((N-1)-(n-1)\big)!} \\ & = \frac{(N-1)!}{(n)!(N-1-n)!} + \frac{(N-1)!}{(n-1)!(N-n)!} \\ & = \frac{(N-n)(N-1)!+(n)(N-1)!}{(n)(n-1)!(N-n)(N-n-1)!} \\ & = \frac{N(N-1)!}{n!(N-n)!} \\ & = \frac{N!}{n!(N-n)!} \\ & = \begin{pmatrix} N \\ n \end{pmatrix} \\ \end{aligned}$

Proof of Eq. (3.51).

$\begin{aligned} &\quad \begin{pmatrix} N \\ n+1 \end{pmatrix} \\ & = \frac{N!}{(n+1)!\big(N-(n+1)\big) !} \\ & = \frac{(N!)(N-n)}{\big( (n+1)(n!)\big) \big( (N-n)(N-n-1)! \big)} \\ & = \bigg(\frac{N-n}{n+1}\bigg)\bigg(\frac{N!}{n!(N-n)!}\bigg) \\ & = \frac{N-n}{n+1}\begin{pmatrix} N \\n \end{pmatrix} \\ \end{aligned}$

QED

September 29, 2021August 11, 2022

202109291137 SQL Table 003

CREATE TABLE SQL_Table_003
	(ID int,
	Formulations_of_quantum_mechanics varchar(255),
	Proposer varchar(255),
	Time_(Yr) varchar(255)
	);
INSERT INTO SQL_Table_003
	(Formulations_of_quantum_mechanics, Proposer, Time_(Yr))
	VALUES
	('transformation theory', 'Paul Dirac', '~1927 ')
	('matrix mechanics', 'Werner Heisenberg, Max Born, Pascual Jordan', '~1925')
	('wave mechanics', 'Erwin Schrödinger', '~1925')

CREATE TABLE SQL_Table_003

(ID int,

Formulations_of_quantum_mechanics varchar(255),

Proposer varchar(255),

Time_(Yr) varchar(255)

);

INSERT INTO SQL_Table_003

(Formulations_of_quantum_mechanics, Proposer, Time_(Yr))

VALUES

('transformation theory', 'Paul Dirac', '~1927 ')

('matrix mechanics', 'Werner Heisenberg, Max Born, Pascual Jordan', '~1925')

('wave mechanics', 'Erwin Schrödinger', '~1925')

Output.
+------+-------------------------------------+--------------------------------+-------------+
|  ID  |  Formulations_of_quantum_mechanics  |  Proposer                      |  Time_(Yr)  |
+------+-------------------------------------+--------------------------------+-------------+
|  1   |  transformation theory              |  Paul Dirac                    |  ~1927      |
|  2   |  matrix mechanics                   |  Werner Heisenberg, Max Born,  |  ~1925      |
|      |                                     |    Pascual Jordan              |             |
|  3   |  wave mechanics                     |  Erwin Schrödinger             |  ~1925      |
|  4   |  path integral formulation          |  Richard Feynman               |  ~1948      |
+------+-------------------------------------+--------------------------------+-------------+

Output.

+------+-------------------------------------+--------------------------------+-------------+

+------+-------------------------------------+--------------------------------+-------------+

| 1 | transformation theory | Paul Dirac | ~1927 |

| 2 | matrix mechanics | Werner Heisenberg, Max Born, | ~1925 |

| 3 | wave mechanics | Erwin Schrödinger | ~1925 |

| 4 | path integral formulation | Richard Feynman | ~1948 |

+------+-------------------------------------+--------------------------------+-------------+

September 28, 2021November 29, 2021

202109281720 SQL Table 002

CREATE TABLE SQL_Table_002
    (ID int,
    Velocity varchar(255),
    Nonquantum varchar(255),
    Quantum varchar(255)
    );
INSERT INTO SQL_Table_002
    (Velocity, Nonquantum, Quantum)
    VALUES
    ('v << c', 'Classical Mechanics (Newton)', 'Quantum Mechanics (Bohr, Heisenberg, Schrödinger, et al.)')
    ('0 < |v-c| << 1', 'Special Relativity (Einstein)', 'Quantum Field Theory (Dirac, Pauli, Feynman, Schwinger, et al.)')

CREATE TABLE SQL_Table_002

(ID int,

Velocity varchar(255),

Nonquantum varchar(255),

Quantum varchar(255)

);

INSERT INTO SQL_Table_002

(Velocity, Nonquantum, Quantum)

VALUES

('v << c', 'Classical Mechanics (Newton)', 'Quantum Mechanics (Bohr, Heisenberg, Schrödinger, et al.)')

('0 < |v-c| << 1', 'Special Relativity (Einstein)', 'Quantum Field Theory (Dirac, Pauli, Feynman, Schwinger, et al.)')

Output.
+------+------------------+-----------------------+----------------------------+
|  ID  |  Velocity        |  Nonquantum           |  Quantum                   |
+------+------------------+-----------------------+----------------------------+
|  1   |  v << c          |  Classical Mechanics  |  Quantum Mechanics         |
|      |                  |   (Newton)            |   (Bohr, Heisenberg,       |
|      |                  |                       |   Schrödinger, et al.)     |
|  2   |  0 < |v-c| << 1  |  Special Relativity   |  Quantum Field Theory      |
|      |                  |   (Einstein)          |   (Dirac, Pauli, Feynman,  |
|      |                  |                       |    Schwinger, et al.)      |
+------+------------------+-----------------------+----------------------------+
(D. J. Griffiths on Introduction to Electrodynamics (1999), pg. xi)

Output.

+------+------------------+-----------------------+----------------------------+

+------+------------------+-----------------------+----------------------------+

| 1 | v << c | Classical Mechanics | Quantum Mechanics |

| 2 | 0 < |v-c| << 1 | Special Relativity | Quantum Field Theory |

+------+------------------+-----------------------+----------------------------+

(D. J. Griffiths on Introduction to Electrodynamics (1999), pg. xi)

September 28, 2021November 29, 2021

202109281636 SQL Table 001

CREATE TABLE SQL_Table_001
    (ID int,
    Subject varchar(255),
    Object varchar(255),
    Category varchar(255)
    );
INSERT INTO SQL_Table_001
    (Subject, Object, Category)
    VALUES
    ('algebraic geometry', 'regular (polynomial) functions', 'algebraic varieties')
    ('topology', 'continuous functions', 'topological spaces')
    ('differential topology', 'differentiable functions', 'differentiable manifolds')
    ('complex analysis', 'analytic (power series) functions', 'complex manifolds');

CREATE TABLE SQL_Table_001

(ID int,

Subject varchar(255),

Object varchar(255),

Category varchar(255)

);

INSERT INTO SQL_Table_001

(Subject, Object, Category)

VALUES

('algebraic geometry', 'regular (polynomial) functions', 'algebraic varieties')

('topology', 'continuous functions', 'topological spaces')

('differential topology', 'differentiable functions', 'differentiable manifolds')

('complex analysis', 'analytic (power series) functions', 'complex manifolds');

Output.
+------+-------------------------+-------------------------------------+----------------------------+
|  ID  |  Subject                |  Object                             |  Category                  |
+------+-------------------------+-------------------------------------+----------------------------+
|  1   |  algebraic geometry     |  regular (polynomial) functions     |  algebraic varieties       |
|  2   |  topology               |  continuous functions               |  topological spaces        |
|  3   |  differential topology  |  differentiable functions           |  differentiable manifolds  |
|  4   |  complex analysis       |  analytic (power series) functions  |  complex manifolds         |
+------+-------------------------+-------------------------------------+----------------------------+
(J.S. Milne on Algebraic Geometry (2017), pg. 7)

Output.

+------+-------------------------+-------------------------------------+----------------------------+

+------+-------------------------+-------------------------------------+----------------------------+

+------+-------------------------+-------------------------------------+----------------------------+

(J.S. Milne on Algebraic Geometry (2017), pg. 7)

September 16, 2021October 24, 2022

202109161207 Exercises 3.5.A (Q1)

Find the volume $V$ inside the paraboloid $z=x^2+y^2$ for $0\le z\le 4$ .

Background. (Triple integral in cylindrical coordinates)

$\displaystyle{\iiint\limits_{S}f(x,y,z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z=\iiint\limits_{S'}f(r\cos\theta ,r\sin\theta ,z)\, r\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}z}$ where the mapping $x=r\cos\theta$ , $y=r\sin\theta$ , $z=z$ maps the solid $S'$ in $r\theta z$ -space onto the solid $S$ in $xyz$ -space in a one-to-one manner.

_{pg.122, Chapter 3.5, Michael Corral. (2008). Vector Calculus}

Setup.

hence,

Solution.

Using vertical slices, we see that

$\begin{aligned} V & =\iint\limits_{R}(4-z)\,\mathrm{d}A \\ & = \iint\limits_{R}\big( 4-(x^2+y^2)\big)\,\mathrm{d}A\end{aligned}$

where $R=\{ (x,y):x^2+y^2\le 4\}$ is the disc in $\mathbb{R}^2$ . In polar coordinates $(r,\theta )$ we know that $x^2+y^2=r^2$ and that $R'=\{ (r,\theta ): 0\le r\le 2, 0\le\theta\le 2\pi\}$ .

Thus,

$\begin{aligned} V & = \int_{0}^{2\pi}\int_{0}^{2} (4-r^2)\, r\,\mathrm{d}r\,\mathrm{d}\theta \\ & = \int_{0}^{2\pi}\int_{0}^{2} (4r-r^3)\,\mathrm{d}r\,\mathrm{d}\theta \\ & = \int_{0}^{2\pi} \bigg[ 2r^2-\frac{r^4}{4}\bigg]\bigg|_{0}^{2}\,\mathrm{d}\theta \\ & = 4\int_{0}^{2\pi} \mathrm{d}\theta \\ & = 4(2\pi )\\ & = 8\pi\quad \textrm{(cubic units)} \end{aligned}$

September 16, 2021October 24, 2022

202109161150 Exercises 3.3.B (Q9)

Let $M$ be a constant. Show that

$\displaystyle{\int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}M\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z}=M(z_2-z_1)(y_2-y_1)(x_2-x_1)$ .

Solution.

$\begin{aligned} & \qquad \int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}M\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \\ & = M\int_{z_1}^{z_2}\int_{y_1}^{y_2}\big[ x\big]\big|_{x_1}^{x_2} \,\mathrm{d}y\,\mathrm{d}z \\ & = M(x_2-x_1)\int_{z_1}^{z_2}\int_{y_1}^{y_2} \,\mathrm{d}y\,\mathrm{d}z \\ & = M(x_2-x_1)\int_{z_1}^{z_2}\big[ y\big] \big|_{y_1}^{y_2}\,\mathrm{d}z \\ & = M(y_2-y_1)(x_2-x_1)\int_{z_1}^{z_2}\,\mathrm{d}z \\ & = M(y_2-y_1)(x_2-x_1)\big[ z\big] \big|_{z_1}^{z_2} \\ & = M(z_2-z_1)(y_2-y_1)(x_2-x_1) \\ \end{aligned}$

September 16, 2021October 24, 2022

202109161118 Exercises 3.3.A (Q3)

For Exercises 1-8, evaluate the given triple integral.

$\displaystyle{\int_{0}^{\pi}\int_{0}^{x}\int_{0}^{xy}x^2\sin z\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x}$

Solution.

$\begin{aligned} &\qquad \int_{0}^{\pi}\int_{0}^{x}\int_{0}^{xy}x^2\sin z\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{\pi}\int_{0}^{x}\big( -x^2\cos z\big)\big|_{z=0}^{z=xy}\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{\pi}\int_{0}^{x}(-x^2)(\cos xy-1)\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{\pi} (-x^2)\bigg(\frac{\sin xy}{x}-y\bigg)\bigg|_{y=0}^{y=x}\,\mathrm{d}x \\ & = \int_{0}^{\pi} (-x^2)\bigg( \frac{\sin x^2}{x}-x \bigg)\,\mathrm{d}x \\ & = \int_{0}^{\pi} (x^3 - x\sin x^2)\,\mathrm{d}x \\ & = \int_{0}^{\pi}x^3\,\mathrm{d}x - \int_{0}^{\pi} x\sin x^2\,\mathrm{d}x \\ & = \bigg( \frac{x^4}{4} \bigg)\bigg|_{0}^{\pi} - \bigg[ \cdots \bigg] \\ & = \enspace \dots \textrm{ (to be continued) }\dots \end{aligned}$

Roughwork. (Integration by parts)

$\begin{aligned} & \qquad \int x\sin x^2\,\mathrm{d}x \\ & = \int \sin x^2\,\mathrm{d}\bigg(\frac{x^2}{2}\bigg) \\ & = \frac{1}{2}\cdot\int \sin x^2\,\mathrm{d}(x^2) \\ & = - \frac{\cos x^2}{2} \\ \end{aligned}$

$\begin{aligned} & = \dots \textrm{ (to continue) }\dots \\ & = \bigg(\frac{x^4}{4}\bigg)\bigg|_{0}^{\pi} - \bigg( -\frac{\cos x^2}{2} \bigg)\bigg|_{0}^{\pi} \\ & = \bigg(\frac{\pi^4}{4}\bigg) + \bigg(\frac{\cos\pi^2}{2} - \frac{1}{2}\bigg) \\ & = -\frac{1}{2} + \frac{\pi^4}{4} + \frac{\cos\pi^2}{2} \\ \end{aligned}$

September 16, 2021October 24, 2022

202109161108 Exercises 3.3.A (Q2)

For Exercises 1-8, evaluate the given triple integral.

$\displaystyle{\int_{0}^{1}\int_{0}^{x}\int_{0}^{y}xyz\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x}$

Solution.

$\begin{aligned} &\qquad \int_{0}^{1}\int_{0}^{x}\int_{0}^{y}xyz\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1}\int_{0}^{x} \bigg( \frac{xyz^2}{2}\bigg|_{z=0}^{z=y} \bigg)\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1}\int_{0}^{x} \frac{xy^3}{2}\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1} \bigg( \frac{xy^4}{8}\bigg|_{y=0}^{y=x} \bigg) \,\mathrm{d}x \\ & = \int_{0}^{1} \frac{x^5}{8}\,\mathrm{d}x \\ & = \bigg[ \frac{x^6}{48} \bigg]_{0}^{1} \\ & = \frac{1}{48} \\ \end{aligned}$

September 16, 2021October 24, 2022

202109161055 Exercises 3.3.A (Q1)

For Exercises 1-8, evaluate the given triple integral.

$\displaystyle{\int_{0}^{3}\int_{0}^{2}\int_{0}^{1}}xyz\,\mathrm{d}x\, \,\mathrm{d}y\,\mathrm{d}z$

Solution.

$\begin{aligned} & \qquad \int_{0}^{3}\int_{0}^{2}\int_{0}^{1}xyz\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \\ & = \int_{0}^{3}\int_{0}^{2} \bigg(\frac{x^2yz}{2}\bigg|_{x=0}^{x=1}\bigg)\,\mathrm{d}y\,\mathrm{d}z \\ & = \int_{0}^{3}\int_{0}^{2}\bigg(\frac{yz}{2}\bigg)\,\mathrm{d}y\,\mathrm{d}z \\ & = \int_{0}^{3}\bigg( \frac{y^2z}{4}\bigg|_{y=0}^{y=2} \bigg) \,\mathrm{d}z \\ & = \int_{0}^{3} z\,\mathrm{d}z \\ & = \bigg[\frac{z^2}{2}\bigg]_{0}^{3} \\ & = 4\frac{1}{2} \\ \end{aligned}$

September 16, 2021October 24, 2022

202109161046 Exercises 1.1.A (Q2)

For the points $P=(1,-1,1)$ , $Q=(2,-2,2)$ , $R=(2,0,1)$ , $S=(3,-1,2)$ , does $\overrightarrow{PQ}=\overrightarrow{RS}$ ?

Solution.

$\begin{aligned} \overrightarrow{PQ} & = (2,-2,2) - (1,-1,1) \\ & = (1,-1,1) \\ \overrightarrow{RS} & = (3,-1,2) - (2,0,1) \\ & = (1,-1,1) \\ \end{aligned}$

$\therefore\enspace \overrightarrow{PQ}=\overrightarrow{RS}$