---
jupytext:
  formats: ipynb,md:myst
  text_representation:
    extension: .md
    format_name: myst
    format_version: 0.13
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  display_name: Python 3 (phys-571)
  language: python
  name: phys-571
---

```{code-cell}
:tags: [hide-cell]
%pylab inline
import numpy as np, matplotlib.pyplot as plt
```

(sec:ClassLog2024)=
# Class Log - 2024

These are notes about what we did in class in the Fall 2024 offering of the
course.

## Fri 11 Oct 2024
* Finite difference and drums.
* Start complex analysis.

## Wed 9 Oct 2024
* Review midterm exam.
* Separation of variables in PDEs.
* Explicit example of Drum.

## Mon 7 Oct 2024
* Midterm I Exam

## Fri 4 Oct 2024
* PDEs.  Method of characteristics.  Traffic flow.

## Fri 4 Oct 2024
* Green's functions: 1D Oscillator
  \begin{gather*}
    \mathcal{L}u = \ddot{u} + 2\gamma \dot{u} + \omega_0^2 u = g(t).
  \end{gather*}
  Homogeneous solutions (assuming $\gamma^2 < \omega_0^2$):
  \begin{gather*}
    u(t) = e^{-\gamma t} \Bigl(a_+ e^{\I\omega t} + a_-e^{-\I\omega t}\Bigr),\qquad
    \omega = \sqrt{\omega_0^2 - \gamma^2}.
  \end{gather*}
  Green's function:
  \begin{gather*}
    \mathcal{L}G(t) = \delta(t).
  \end{gather*}
  Construct from homogeneous solutions with appropriate discontinuity at $t_0$.  Many
  ways to do this, but the following is easy:
  \begin{gather*}
    G(t) = \Theta(t)e^{-\gamma t}\frac{\sin(\omega t)}{\omega}.
  \end{gather*}

## Mon 29 Sept 2024
* Variational Method
* PDEs
  * Method of Characteristics.
  * Flow.
  * Separation of Variables.
  * Green's functions.
    * Emphasize nature of solution.  B/C.
* Wave equation.

## Fri 27 Sept 2024
* Sturm Liouville
* Self-Adjoint
* Hermite example.

## Wed 25 Sept 2024
* Series Solutions:
  * Singularities.
  * Recurrence
* Wronskian
  * Independence of solutions
* Linear Equations
  * Reduction of order
  * Homogeneous
  * Inhomogeneous (Green's functions)
* Numerical tests

## Mon 23 Sept 2024
* ODEs:
  * Separable
    \begin{gather*}
      A(x)\d{x} = B(y)\d{y}
    \end{gather*}
  * Exact
    \begin{gather*}
      P(x, y)\d{x} + Q(x, y)\d{y} = 0, \qquad
      P = \varphi_{,x}, \qquad Q = \varphi_{,y}, \qquad
      P_{,y} = Q_{,x}, \qquad
      \varphi = \int_{x_0}^{x}P(x, y)\d{y} + \int_{y_0}^{y}Q(x_0, y)\d{y} = C.
    \end{gather*}
  * First Order
    * Integrating Factors
    * Method of Variable coefficients
  * Series

  
  
## Fri 20 Sept 2024
* Tensor summary.  Note: I have made significant updates to {ref}`sec:Tensors` that I
  strongly encourage students read (especially the "Important" notes).

## Wed 18 Sept 2024
* Tensors: Followed notes {ref}`sec:Tensors` up to the definition of the metric.
* Discussed problem of computing $\norm{\op{L}^2}^2$ and the role of Christoffle
  symbols (but briefly).
* Students asked questions about matrices: especially the transpose, the difference
  between components and the actual "things" (I did not make this too explicit.)
* Mentioned curves as vectors, and derivations as vectors but no details.

## Mon 16 Sep 2024
* Tensors:
  * Vectors $\ket{A} \equiv A_{\mu}$ and co-vectors $\bra{A} \equiv A^{\mu}$.
  * Fields: $A_{\mu}$.
  * Transformations: representations of rotations etc.
    * Inverse picture of transformations that leave the inner product invariant.
    * Lorentz.
  * Connection: Example from quantum mechanics.
    \begin{gather*}
      \vect{\Psi}(\vect{x}) \rightarrow \mathcal{R}\vect{\Psi}(\vect{x})
      = \mat{R}\vect{\Psi}(\mat{R}^{-1}\vect{x}),\\
      \Psi^{a} \rightarrow [\mat{R}]^{a}{}_{b}\Psi^{b}.
    \end{gather*}


  * $\mat{1}$: $[\mat{1}]^{i}{}_{j} = \delta^{i}{}_{j} = \delta^{i}_{j}$ since $\mat{1}
    = \mat{1}^T$.
  * Metric raises indices.
  
    
  
## Fri 13 Sep 2024
* Finite differences and Hermitian vs. self-adjoint.  (Working with functional
  operators.)
* Vector Calculus:
  * Pictures.
  \begin{align*}
    \int_V \vect{\nabla}\cdot\vect{A}\d{\tau}
    & = \oint_{\partial V} \vect{A}\cdot\d{\vect{\sigma}},\\
    \int_S \vect{\nabla}\times\vect{B}\cdot\d{\vect{\sigma}}
    &= \oint_{\partial S} \vect{B}\cdot\d{\vect{r}},\\
    \oint_{C} \vect{\nabla} f \cdot \d{\vect{r}} 
    &= \int_{\partial C} \d{f}.
  \end{align*}
  * Curvilinear coordinates (keep orthonormal frame)
  * Metric

## Wed 11 Sep 2024
* Angular momentum: passive vs. active transformations.  (Translation as an example.)
* Eigenvalue problem.  Diagonalization. Simultaneous Diagonalization.
* Hermitian vs self-adjoint.  Use finite difference operator as example.
* $L_2$ (see {cite}`Hassani:2013` chapter 7, Thms. 7.2.1-7.2.3
  * Riesz-Fischer: $\mathcal{L}^2_{w}(a,b)$ is complete.
  * All complete inner product spaces with countable bases are isomorphic to
    $\mathcal{L}^2_{w}(a,b)$.
  * Stone-Weierstrass: $\bigl\{x^n \mid n \in \{0, 1, 2, \dots\}\bigr\}$ forms a basis
    for $\mathcal{L}^2_{w}(a,b)$.
* Pauli matrices.

## Mon 9 Sep 2024
* Completeness.  Fourier space.
* Angular momentum operators and function.

## Fri 6 Sep 2024
* Direct sum and tensor product:
  * $R_z = e^{\mat{\vect{\theta}\times}}$.  Is this active or passive?
  * Groups: addition of angular momentum as an example.
* Fourier transformation as change of basis.

## Wed 4 Sep 2024
* Determinants and Matrices.
* Vector spaces and Inner Product spaces.
* Gram Schmidt
  * Vectors.
  * Orthogonal Polynomials
  * Completeness.
* Projections.
* Matrix factorization.
  * $LU$: Gauss-Jordan Elimination (Solving systems)
    * $LL^T$: Cholesky decomposition (stable version for symmetric matrices)
  * $QR$: Gram-Schmidt Orthonomalization
  * $SDS^{-1}$: Diagonalization, eigenvalues.
    * $UDU^\dagger$: For Hermitian/Symmetric matrices
  * $UDV^\dagger$: Singular Value Decomposition (SVD).
    * Example: Entanglement (Schmidt decomposition)

For details, see {ref}`sec:LinearAlgebra`.

## Fri 30 Aug 2024
* Sequence acceleration.
  * Subtract a nearby sequence that you know the answer too (perhaps an integral)?
  * Partial fractions. *(did not discuss)*
  * Euler. *(mentioned - see notes)*
* Weierstrauss and Abel test example.
  * E.g. Apply Abel test for $e^{x}$ on $x \in [0, 1]$. *(did not do.)*
* Mathematical Induction.
  * Use induction to prove that $f(N) = \sum_{n=1}^{N} n = N(N+1)/2.
    1. To get answer: Consider integral $\int_{1}^{\infty}n\d{n} = (N^2-1)/2$.
    2. Guess $f(N) = a + bN + cN^2$.
    :::{margin}
      Noting that $f(N) = \sum_{n=0}^{N} n$ we can also use $f(0) = a = 0$, simplifying
      \begin{gather*}
         f(1) = b + c = 1\\
         f(2) = 2b + 4c = 3
      \end{gather*}
    :::
    3. Solve the following to get $a=0$, $b=c=1/2$:
       \begin{gather*}
         f(1) = a + b + c = 1\\
         f(2) = a + 2b + 4c = 3\\
         f(3) = a + 3b + 9c = 6
       \end{gather*}
    4. Prove by induction:
       1. The formula works for a base case $f(1) = 1$.
       2. Note that $f(N+1) = f(N) + N+1$.
       3. If $f(N) = N(N+1)/2$, then the latter implies
          \begin{gather*}
            f(N+1) = \frac{N(N+1)}{2} + N+1 
                   = \frac{N^2 + 3N + 2}{2} = 
                   = \frac{(N+2)(N+1)}{2}.
          \end{gather*}
          Hence our formula works for $N+1$.  Therefore it is correct for all integer $N\geq 1$
          by induction.
* Derivatives
  * Extrema. *(did not do)*
  * Differentiate Parameters:
    \begin{gather*}
      I_n = \int x^n e^{-x^2}.
    \end{gather*}
* Delta function. *(started)*
  * Suppose two random variables $X$ and $Y$ have PDF $p_X(x)$ and $p_Y(y)$.  What is
    the PDF for $Z = X+Y$?
  * Warmup: From intro physics lab, if $x = \bar{x} \pm \sigma_x$ and $y = \bar{y} \pm
    \sigma_y$ where $\sigma_{x,y}$ are the standard deviations ("errors"), what is the
    standard deviation ("error") $\sigma_z$ in $z = x+y$?
    * Answer: $\sigma_z = \sqrt{\sigma_x^2 + \sigma_y^2}$.  Prove this. *(Apparently
      this is no longer common knowledge.)*

## Wed 28 Aug 2024
* Power series.
  * Integration and differentiation.
    \begin{gather*}
      f(x) = \frac{1}{2!} + \frac{2}{3!} + \frac{4}{5!} + \cdots.
    \end{gather*}
    \begin{gather*}
      f(x) = \frac{1}{1\cdot2} + \frac{x}{2\cdot 3} + \frac{x^2}{3\cdot 4} +
            \frac{x^3}{4\cdot 5} + \cdots.
    \end{gather*}
* Binomial expansion.  Example: $E = \sqrt{m^2c^4 + p^2c^2} = mc^2 + \tfrac{1}{2}mv^2 +
  \cdots$.
* Weierstrauss and Abel tests.
* Mention $\delta(x)$.

## Mon 26 Aug 2024
Review of better method of series convergence: integral test + generalized ratio test.
See notes in {ref}`sec:MathematicalPreliminaries`.

## Fri 23 Aug: Series

* Convergence: Metric spaces; Cauchy convergence; $\mathbb{R} = $ Cauchy completion of
  $\mathbb{Q}$; Absolute vs. Conditional.
* Common Series:
  * Harmonic *(divergent)*:
    \begin{gather*}
      \sum_{n=1}^{\infty} \frac{1}{n} 
      = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \dots.
    \end{gather*}
  * Geometric *(convergent for $0 \leq \abs{r} < 1$)*:
  :::{margin}
  Let $G = 1 + r + r^2 + \dots$:
  \begin{gather*}
    G = 1+\underbrace{rG}_{r + r^2 + r^3 + \dots}\\
    G = \frac{1}{1-r}.
  \end{gather*}
  :::
  \begin{gather*}
    \sum_{n=1}^{\infty} \frac{1}{r^n} = 1 + r + r^2 + r^3 = \dots = \frac{1}{1-r}.
  \end{gather*}
  * [Riemann Zeta function](https://www.3blue1brown.com/lessons/zeta) shows up in
    various places, and has known values for even integers:
    \begin{gather*}
      \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \qquad
      \zeta(2) = \frac{\pi^2}{6}, \qquad
      \zeta(4) = \frac{\pi^4}{90},
    \end{gather*}
    and some other [particular values](https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function).
* Series [Convergence Tests](https://en.wikipedia.org/wiki/Convergence_tests):
  * Integral test; Comparisons; Kummer/Gauss's tests are more sensitive.
  * Proof of Ratio and Kummer tests; Telescoping Series; Enhanced Gauss's test.
* Series of Functions:
  * Uniform (sketch example 1.2.1); Independent concept from absolute convergence.
  * Weierstrass $M$ Test (only for absolutely convergent series).
  * Abel' Test.
* Taylor Series (Mclauren series are about $x=0$).
  * Power Series; $e^x$;
  * Can be integrated or differentiated (unlike Fourier series)
  * Unique; [Formal power series](https://en.wikipedia.org/wiki/Formal_power_series).
  \begin{gather*}
    e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots\\
    e^{\I x} =
    \underbrace{\left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right)}_{\cos x} +
    \I\underbrace{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\right)}_{\sin x}
  \end{gather*}

```{code-cell}
:tags: [margin, hide-input]
x = np.linspace(-5, 5.0001, 200)
fig, ax = plt.subplots(figsize=(3, 2))
ax.plot(x, np.exp(-1/x**2))
ax.set(xlabel="$x$", ylabel="$e^{-1/x^2}$");
```
  * Consider $f(x) = \exp(-1/x^2)$.  All derivatives at $x=0$ are zero: $f^{(n)}(0) =
    0$, but the function is $C^{\infty}$ -- infinitely smooth but very very flat!

## Wed 21 Aug 2024

* Introductions.
* Poll: who is interested in the mathematical theory (Hassani)?  Roughly half the class.
* Small problem:

  \begin{gather*}
    a  + b + c = 0\\
    a^2 + b^2 + c^2 = \sqrt{74}\\
    a^4 + b^4 + c^4 = ?
  \end{gather*}

  Discussed geometric picture a bit, connected (roughly) with $L_p$ norm.

* Discussed vector spaces: Most people know.
* Inner product spaces: Some confusion. (E.g. Notion of angle in complex spaces?)
* Gram-Schmidt: Not everyone knows.
* Mentioned that functions are vectors.

  \begin{gather*}
    \braket{f|g} = \int \d{x} f^*(x) g(x) \textcolor{green}{w(x)}.
  \end{gather*}

  Unifies fourier, orthogonal polynomials, many special functions, etc.
* Discussed how to learn: how I learn.
* Quick review of Syllabus, reading assignment.