# Administrative Information - **Class Meeting**: Wednesday, September 9, 2026, 1:00–1:50pm — **Coolbaugh 212**. *(No class next Wednesday 9/16 — Career Day; two-day weeks this week and next.)* - Previous meeting: [[MATH310F26(Day 6) - The derivative of a step, distributionally]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 7 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121181)** — 4 questions drawn from today's material, unlimited attempts, highest kept > - 🎯 **[LR] [Day 7 Reflection](https://elearning.mines.edu/courses/81632/quizzes/121183)** — one free response, no wrong answers > - 🎯 **[CA] [Yahtzee Dice Data](https://elearning.mines.edu/courses/81632/assignments/586037)** — at least 3 games, **one roll per turn**; play-sheet scan or photo due **Friday 9/11, 10:00am**. Keep the dice data coming — we'll need it. > - **Three Questions, Revisited** closed this morning (10:00am); late entries: get them in soon. **Individual feedback lands on Canvas by Friday** — recorded aloud and locally transcribed, so edit out any stray "sigh"; if anything transcribes as "eye roll," report it immediately. > - Not knowing your project yet is *fine* — the Q&A bootstraps the process; choosing now is also fine. > - 🎯 **[WB] [Workbook 6](https://elearning.mines.edu/courses/81632/assignments/586618)** — conjugate pairs and the exponential representations of $\cos(\omega t)$ and $\sin(\omega t)$; upload one PDF > - 📄 [[MATH310F26_WB6.pdf|Workbook 6 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 6 Companion) - What a conjugate pair extracts]] > - 🎯 **[WB] [Workbook 7](https://elearning.mines.edu/courses/81632/assignments/586619)** — the Fourier transforms of $\cos(\omega t)$ and $\sin(\omega t)$ from Eq.-of-the-day $\mathcal{F}\{e^{i\omega_{0}t}\}=2\pi\,\delta(\omega-\omega_{0})$, checked with the inverse; upload one PDF > - 📄 [[MATH310F26_WB7.pdf|Workbook 7 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 7 Companion) - Two tones and a constant, transformed]] > [!example]- 🗃️ COMAP Corner — three from the archive > Three from the [MCM/ICM archive](https://www.contest.comap.com/undergraduate/contests/mcm/previous-contests.php) — one data, one computational, one weird — tuned to a day spent meeting the transform: > > - **Data/stats — Predicting from A Wealth of Data (2020 MCM C).** Amazon star ratings, text reviews, and helpfulness votes for microwaves, pacifiers, and hair dryers — mine three real review datasets for patterns, including *time-based signals* that a product's reputation is rising or falling. 📄 [[2020_MCM_Problem_C_A_Wealth_of_Data.pdf|problem PDF]] · [comap.com](https://www.comap.org/component/zoo/?task=callelement&format=raw&item_id=3350&element=71992483-3d3b-47dc-a146-3c901bdea04f&method=download) > - **Computational — The Submarine Detection Problem (1996 MCM A).** The ocean hums with ambient noise at fixed frequency and amplitude; a silent submarine perturbs the field as it moves. Infer its presence, speed, size, and heading purely from the disturbance — today's mathematical prism, pointed underwater. 📄 [[1996_MCM_Problem_A_The_Submarine_Detection_Problem_Problem_B_The_Contest_Judging_Problem.html|problem page (archive)]] · [comap.com](https://www.contest.comap.com/undergraduate/contests/matrix/PDF/1996mcmProblems.htm) > - **Just weird — The Longest Lasting Sandcastle(s) (2020 MCM B).** What 3-D foundation shape survives the incoming tide longest, what's the optimal sand-to-water ratio, and what does rain do to the answer? A geometry-versus-waves showdown, at the beach. 📄 [[2020_MCM_Problem_B_The_Longest_Lasting_Sandcastles.pdf|problem PDF]] · [comap.com](https://www.comap.org/component/zoo/?task=callelement&format=raw&item_id=3349&element=71992483-3d3b-47dc-a146-3c901bdea04f&method=download) --- # Deliverables - 🎯 **[LR] [Day 7 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121181)** — 4 questions drawn from today's material, unlimited attempts, highest kept - 🎯 **[LR] [Day 7 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121183)** — one free response, no wrong answers - 🎯 **[WB] [Workbook 6](https://elearning.mines.edu/courses/81632/assignments/586618)** — what a conjugate pair extracts; upload one PDF - 🎯 **[WB] [Workbook 7](https://elearning.mines.edu/courses/81632/assignments/586619)** — Fourier transformation of sinusoids; upload one PDF - 🎯 **[CA] [Yahtzee Dice Data](https://elearning.mines.edu/courses/81632/assignments/586037)** — at least 3 games, one-roll house rules; due Friday 9/11 at 10:00 am Questions and statement are due Monday 9/14 at 11:59 pm (extended from Friday). --- # Lecture Boards + Transcript + GenAI **Seventh meeting, September 9 — the Fourier transform.** A confession opened the day: the old approach built the transform from a constructivist point of view — a nice mathematical narrative, linear-thinking-friendly — but it burned so much time that the class never got to *use* the object. This is a modeling course, and the goal is ubiquitous tools in hands. So: go for broke, straight into the Fourier transform from an intuitive perspective, with the full knowledge that copious theory lives under the hood. For those who want the hood open, the slides are here: 📄 [[DistributionsForFourier.pdf|Distributions for Fourier (slides, PDF)]]. ## Laplace, recalled and roasted First, a familiar face, written as succinctly as possible. If $f$ is of *exponential order* — $\exists\,M,c\in\mathbb{R}$, $M>0$, such that $|f(t)|\le Me^{ct}$ — then the fanciest calligraphic L you can muster (curly-brace game maintained since the C++-on-the-board years), $F(s)=\mathcal{L}\{f\}=\int_{0}^{\infty}f(t)\,e^{-st}\,dt,$ **exists and is unique to $f$.** Your Diffie-Q memories of this are mostly completing the square and partial fractions, which were there to throw you off the scent. What the thing *is*: an integral transform that trades the $t$ variable for the $s$ variable — a cracked-out coordinate change, done with calculus instead of algebra. The function $e^{-st}$ that your data gets slammed up against has a name: the **kernel** of the transform. Two genuinely nice features. It plays well with $\delta(t-t_{0})$ — delta only makes sense inside integrals, and here is an integral. And the kernel is a decaying exponential, so for suitable $s$ it squelches whatever growth $f$ attempts as $t\to\infty$: convergence by design (that is exactly what the exponential-order hypothesis is for). Now the roast, in three parts. *One:* in practice nobody uses the formula — you were handed a table and shuttled between the $t$-column and the $s$-column. *Two:* the reason you were never shown the inverse is that it is $f(t)=\mathcal{L}^{-1}\{F\}=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}F(s)\,e^{st}\,ds,$ a **contour integral in the complex plane** — a whole course of study (MATH 454, every spring) before it can be taken down. *Three,* and the real complaint: **it has no straightforward interpretation.** Read it: recover $f(t)$ by linearly combining exponentials $e^{st}$ over all those $s$, weighted by $F(s)$. Do you spend your day-to-day linearly combining exponentials in your mind? The board awarded this a sad face. (Redemption clause, for the statistics side: the Laplace transform of a probability density is its **moment generating function**, and you can learn a great deal from it. The bait-and-switch is being played knowingly. A good in-room question pushed back: think of all those exponentials, rates varying, *as a basis* — which is exactly the right instinct, and exactly the instinct Fourier will make visible.) ## The Fourier transform, stated both ways Another integral transform, with two honest advantages even before interpretation: the math is admittedly *worse* (more on that below), but **both the transform and its inverse can be stated** — no table, no dodge. The formulas, first in ordinary frequency $\nu$: $\mathcal{F}\{f\}=\hat f(\nu)=\int_{\mathbb{R}}f(t)\,e^{-2\pi i\nu t}\,dt,$ (the $\mathbb{R}$ subscript is shorthand for $\int_{-\infty}^{\infty}$ — it will come up constantly). Naming conventions: Laplace-land uses lowercase in, uppercase out; Fourier-land uses lowercase in, **hat** out. Now wrap $\omega=2\pi\nu$ and this becomes the mathematician's preferred dress: $\hat f(\omega)=\int_{\mathbb{R}}f(t)\,e^{-i\omega t}\,dt,\qquad\qquad f(t)=\mathcal{F}^{-1}\{\hat f\}=\frac{1}{2\pi}\int_{\mathbb{R}}\hat f(\omega)\,e^{i\omega t}\,d\omega.$ Here $\omega$ is an **angular frequency** (radians per time) and $\nu$ an **ordinary frequency** (complete cycles per time — hertz, for those who measure things); they differ by the $2\pi$ of one full circle, and the change of variables drags $d\omega=2\pi\,d\nu$ into the inverse — which is where that leading $\tfrac{1}{2\pi}$ comes from. > [!note]- Where the $2\pi$ lives — three conventions, one transform > A sharp question from the room: why not split it symmetrically, $\tfrac{1}{\sqrt{2\pi}}$ on each side? You can — that is a standard third formulation. The $2\pi$ is in the game because *circles* are in the game: Workbook 5's formula $e^{i\theta}=\cos\theta+i\sin\theta$ says the kernel's real part is a cosine, its imaginary part a sine, and the pair traces the **unit circle** in the complex plane ([[MATH310F26(Workbook 5 Companion) - Splitting a series by parity|the companion]] has the derivation). So a $2\pi$ must appear somewhere: all of it on the inverse (our convention), split as $\tfrac{1}{\sqrt{2\pi}}$ and $\tfrac{1}{\sqrt{2\pi}}$, or shoved into the exponent via ordinary frequency. Same object three ways; check which convention a book uses before borrowing its formulas. ## What $\hat f$ means — the payoff Concentrate your gaze on the kernel. Inside $e^{i\omega t}$ live cosines and sines — oscillations, *primitive waves*. And $\hat f(\omega)$ multiplies them, which controls not their accordion-ness but their **verticality**: if each pure frequency is a tone on an oscilloscope, $\hat f$ says **how loud each tone is**. Then read the inverse transform's elongated S as the "add" it is: *sum up all the tones at their assigned loudnesses and you reconstruct the sound.* Compare the two sentences side by side — "a linear combination of exponential functions with different amplitudes in a contour integral in the complex plane," versus "add together a bunch of notes at the right loudnesses and get the sound back." Much better interpretation. ![[day7_two_tones_spectrum.png]] Three cashings-out, straight from the board's blue bullets: - **Sound.** A speaker is a dome on an electromagnet, pushed in and out by the signal. Identify the bass frequencies — small $\omega$ — and forcibly smoosh $\hat f$ down there before transforming back: you have just quieted the bass. Equalizers are edits in $\omega$-land. - **Light.** If $f$ is an electromagnetic wave, $\hat f$ is its decomposition into spectral components, seen and unseen — a **mathematical prism**. That is how we know what stars are made of: split the starlight, read which colors appear, and match them to the atoms that alone can make them. - **Images.** If $f$ is an image, low frequency is the slowly-changing regions and high frequency the rapidly-changing ones — on the instructor's shirt, the thin stripes are high-frequency content and the blue chunks low. The transform of an image reads as a heat map of where the energy sits in frequency; you can diagnose from it, or go in and *mess with stuff in that place* to solicit behaviors. ## The problem tends to be the math If waves are the whole story, the transform ought to handle the primitive wave itself. Task: compute the Fourier transform of a pure sinusoid, $\mathcal{F}\{e^{i\omega_{0}t}\}=\int_{\mathbb{R}}e^{i\omega_{0}t}\,e^{-i\omega t}\,dt=\int_{-\infty}^{\infty}e^{i(\omega_{0}-\omega)t}\,dt,\qquad \omega_{0}\ \text{a fixed constant,}$ and busting out $e^{i\,\text{stuff}}=\cos(\text{stuff})+i\sin(\text{stuff})$, $=\int_{-\infty}^{\infty}\Big[\cos\big((\omega_{0}-\omega)t\big)+i\,\sin\big((\omega_{0}-\omega)t\big)\Big]\,dt.$ The real and imaginary parts never mix, so take them one at a time — and hit the wall: eventually you must ask a sinusoid **what it is doing at temporal infinity**, and $\cos(\pm\infty)$, $\sin(\pm\infty)$ have no answer (somewhere between $-1$ and $1$; that's all anyone can tell you). Classically, this integral does not exist. But you already know the answer. What is needed is an object that **concentrates all of the amplitude at the single frequency $\omega_{0}$** — and concentrating everything at a single point is precisely the one trick in our new toolbox. In the sense of distributions, $\mathcal{F}\{e^{i\omega_{0}t}\}=2\pi\,\delta(\omega-\omega_{0}),$ a delta spike sitting at the tone's frequency. *This* is why the last two lectures happened: the Fourier transform's most important customers — pure tones — hand back integrals that only distribution theory can cash. > [!success]- Numbers checked in one sandbox run > The pair as stated is consistent: for $f(t)=e^{-t^{2}/2}$ the forward transform returned $\sqrt{2\pi}\,e^{-\omega^{2}/2}$ to six decimals at $\omega=0,1,2.5$, and the inverse (with its $\tfrac{1}{2\pi}$) recovered $f$ exactly at test points. For the pure tone, truncating the integral at $\pm T$ gives $\int_{-T}^{T}e^{i(\omega_{0}-\omega)t}dt=\tfrac{2\sin((\omega_{0}-\omega)T)}{\omega_{0}-\omega}$: its peak height is exactly $2T$ (growing without bound as $T\to\infty$) while its area held at $6.2834\approx 2\pi$ for $T=5,50,500$ — a nascent delta of total weight $2\pi$, confirming the constant in $2\pi\,\delta(\omega-\omega_{0})$ that the board didn't chase. ## The media: a flute, and a face hidden in a song Two exhibits to close, both of which you can re-run at will. 🎬 **The flute and the frequency finder.** The live spectrum analyzer is a graph of $\hat f(\omega)$ *of the room*, updating in real time: play a low note and spikes stand up out of the noise floor, because a flute — like a guitar string — only lets certain waves live in it; blow harder without changing fingers and the *other* frequencies the tube likes appear. The mic's signal is Fourier transformed numerically, and that second perspective is the whole point: with it we can learn about things, alter things, and play fun games. ("You can just turn this on at a party or something.")  > [!quote] Attribution > **"What are harmonics?"** — World Science Festival: [youtube.com/watch?v=znbfY-tXROk](https://www.youtube.com/watch?v=znbfY-tXROk). 🎬 **The Aphex Twin spectrogram.** A **spectrogram** runs time horizontally, frequency vertically, and color-codes how much energy each frequency carries at each moment — a movie of $\hat f$. Aphex Twin (Richard D. James, a music maker from back in your instructor's day) ran the machine *backwards*: he drew pictures in frequency space and inverse-transformed them into audio. Hidden near the end of the track — the B-side of the *Windowlicker* single whose printed title is itself a mathematical equation — is his own grinning face, recoverable by anyone who looks at the song instead of listening to it.  > [!quote] Attribution > **"[Equation] — Aphex Twin Spectrogram"** — HappyQuark: [youtube.com/watch?v=M9xMuPWAZW8](https://www.youtube.com/watch?v=M9xMuPWAZW8). The parting instruction: **"We want to adopt some amount of Fourier eye."** Sound, starlight, images, El Niño — one transform, many worlds. See you Friday; feedback by then, and no class next Wednesday. > [!example]- Board scans — September 9, 2026 > ![[MATH310F26-Day7-Board-1.jpg]] > ![[MATH310F26-Day7-Board-2.jpg]] > ![[MATH310F26-Day7-Board-3.jpg]] > ![[MATH310F26-Day7-Board-4.jpg]] > ![[MATH310F26-Day7-Board-5.jpg]] > > 📄 [[MATH310F26-Day7-Boards-2026-09-09.pdf|Full board capture (5 pages, PDF)]]