Class Log - 2024

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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 4. Tensors and Manifolds that I strongly encourage students read (especially the “Important” notes).

Wed 18 Sept 2024#

  • Tensors: Followed notes 4. Tensors and Manifolds 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 [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 Linear Algebra.

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 \(f(N) = a + bN + cN^2\).

      1. Solve the following to get \(a=0\), \(b=c=1/2\):

        \[\begin{gather*} f(1) = a + b + c = 1\\ f(2) = a + 2b + 4c = 3\\ f(3) = a + 3b + 9c = 6 \end{gather*}\]

      2. 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 1. Mathematical Preliminaries.

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 \(0 \leq \abs{r} < 1\)):

    \[\begin{gather*} \sum_{n=1}^{\infty} \frac{1}{r^n} = 1 + r + r^2 + r^3 = \dots = \frac{1}{1-r}. \end{gather*}\]

    • Riemann Zeta function 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.

  • Series 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. \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*}

Hide code cell source

 
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="$e^{-1/x^2}$");
_images/2b84acabd5da2fb962bf7fed2024128054eff50cf3236911fd599b192375683f.png

  • 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.

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