Intellect PlusCalculus and analysis
215 cards. Each one is a 40-second story with a five-minute read behind it. Page 3 of 4.
Can boundary values dictate the behavior of an entire interior?If you know the temperature on the edge of a disk, can you predict the heat everywhere inside? This mathematical puzzle, known as the Dirichlet problem, explores how fixed boundar…
Free software is about freedom of speech, not free beerFree software can cost money and can even be sold for profit. What makes it free, in Richard Stallman's sense, is that anyone who receives it may run it, study it, change it and p…
One theorem about differential forms swallows four famous results of calculusStokes' theorem, Green's theorem, the divergence theorem and even the fundamental theorem of calculus look like separate facts in a textbook. Written in the language of differenti…
A Hiroshima regret helped inspire the idea of the digital libraryVannevar Bush had backed research behind the bomb dropped on Hiroshima. After seeing the destruction, he imagined the Memex, a desk with two screens and a keyboard giving rapid ac…
The first clickable banner ad got clicked by 44 percent of viewersIn 1994 AT&T ran the first clickable web banner, its You Will campaign, and within four months 44 percent of the people who saw it clicked. That banner belongs to the early years …
Sample a sound at twice its highest pitch and nothing is lostThe Nyquist–Shannon sampling theorem makes a startling promise: a signal can be rebuilt exactly from snapshots, provided they come faster than twice its highest frequency. That ru…
Divergence measures how much a flow swells or shrinks at a pointWarm a pocket of air and it expands in every direction, so arrows describing its motion all point outward. Mathematicians capture that outward push with a single number called div…
Count every faucet and drain inside a box to know its total outflowPicture a closed surface floating in a tank of water. The divergence theorem, also named after Gauss and Ostrogradsky, says the net amount of water crossing that surface equals th…
A beam theory from around 1750 waited for the Eiffel TowerEngineers can predict how far a beam will sag and how much it can carry thanks to Euler–Bernoulli beam theory. It was first stated around 1750, but it saw no large-scale use until…
Quake III's mysterious constant 0x5F3759DF sparked a hunt for its authorBuried in the code of Quake III Arena was a baffling line that subtracted a number from the constant 0x5F3759DF and somehow produced an inverse square root. Programmers guessed Jo…
Leibniz used the chain rule first, and fumbled a sign doing itIf a car is twice as fast as a bicycle and the bicycle four times faster than a walker, the car beats the walker eightfold. That multiplication of rates is the chain rule of calcu…
Christoffel symbols track how coordinates twist across a curved spaceOn a curved surface, the directions you call north and east quietly rotate as you move. Christoffel symbols are the bookkeeping for that drift, a three-dimensional grid of numbers…
Before microchips, computers stored bits as sound waves in mercuryEarly computers remembered things in some strange ways. One design sent pulses of sound rippling through tubes of mercury; another painted dots of charge on a cathode ray screen. …
Solving the cubic sparked secret formulas, betrayal and a maths duelRenaissance Italy settled mathematical rivalries with public contests, money on the table. The prize problem was the cubic equation. A method guarded as a secret, a broken promise…- Curl measures how much a flowing field spins at each pointDrop a tiny paddle wheel into moving water and it may start to turn. The curl of a vector field captures that spin as an arrow: its length gives the strength of the swirl, its dir…
Two vertical bars around a number trace back to Weierstrass in 1841The absolute value strips a number of its sign, so both 3 and minus 3 become 3. Think of it as distance from zero. That simple idea, written with the twin bars Karl Weierstrass in…
Some infinities are bigger than others, and Cantor labelled them with alephGeorg Cantor realised that infinite sets come in different sizes and named those sizes after the Hebrew letter aleph. The smallest, aleph-null, counts the natural numbers. Whether…
Every crystal fits one of just 14 repeating point patternsSalt, diamond and quartz look nothing alike, yet the scaffolding beneath every crystal belongs to one of only 14 point arrangements in three dimensions. Auguste Bravais catalogued…
The Cauchy–Schwarz inequality caps how much two vectors can agreeMultiply two vectors together with a dot product and the result can never exceed the product of their lengths. That modest-sounding cap, the Cauchy–Schwarz inequality, took three …
Most of modern mathematics rests on a handful of rules about setsWhen paradoxes cracked the early theory of sets, mathematicians rebuilt it from a short list of axioms. The result, Zermelo-Fraenkel set theory with the axiom of choice, is now th…
Zorn's lemma lets mathematicians finish constructions that never actually endSuppose you are building something step by step, you have not finished after infinitely many steps, and nothing seems to stop you from continuing. A single statement from set theo…
Why any two sides of a triangle always outrun the thirdIn any triangle, two sides added together are at least as long as the third. Euclid proved it in Book I of the Elements, and the same rule now helps define what it means to measur…
Type theory began as Russell's fix for a paradox and now runs proof softwareImagine a collection of every set that fails to include itself: it ends up both belonging to itself and not. Bertrand Russell's answer to that paradox was a hierarchy of types, an…
The 1872 curve that is continuous everywhere yet smooth nowhereMathematicians including Gauss had assumed a continuous curve could only have isolated sharp points. On 18 July 1872 Karl Weierstrass presented a function to the Royal Academy of …
Why slope is written m, and why vertical lines have noneSlope is just rise over run: how far a line climbs for each step sideways. Nobody is sure why mathematicians label it m, but the letter appears in English as early as 1844, and th…
Smooth is not the same as analytic: the ladder of differentiabilityIn analysis, smoothness counts how many times a function can be differentiated before something breaks. The absolute value function is continuous but has a sharp corner, so it fai…
Sobolev spaces: where equations find solutions that classical calculus cannotSome important partial differential equations have no solutions among ordinary differentiable functions, yet do have them once derivatives are understood in a looser, weak sense. …
Square numbers hide odd sums, odd divisors and a mental arithmetic trickAdd up the first few odd numbers and you always land on a perfect square: 1 plus 3 plus 5 plus 7 plus 9 makes 25. Squares are also the only positive whole numbers with an odd coun…
Stokes' theorem: add up the swirl across a surface to get the circulation at its edgePicture a vector field crossing a surface bounded by a closed loop. Stokes' theorem says the total circulation around that loop equals the sum of all the local rotation, or curl, …
From Euclid to Leibniz: the long hunt for the line that just touchesA tangent is the straight line that just touches a curve at one point, heading the same way. Pinning that down took roughly two millennia, from Euclid's circles around 300 BC thro…
Taylor's theorem: how a few derivatives let a polynomial impersonate a functionCharles Babbage's difference engine computed sines, cosines and logarithms by working with just the first 7 terms of their Taylor series. The trick rests on Taylor's theorem: near…
A test that always says positive catches every case and is uselessImagine a bogus kit that reads positive for everyone. It never misses a sick person, so its sensitivity is a perfect 100 percent. It also flags every healthy person. That thought …
Add two sine waves of the same pitch and you always get anotherMix two sine waves that share a frequency, whatever their sizes or timing, and the result is still a clean sine wave at that frequency. No other periodic wave shape behaves this w…
Matrix calculus has two rival spelling systems, and mixing them breaks answersMatrix calculus bundles hundreds of partial derivatives into tidy vectors and matrices, which is why statisticians and machine learning engineers lean on it. The catch: practition…
If your average speed was 60, at some instant you were doing exactly 60Drive smoothly from one town to another and, somewhere along the way, your speedometer must have shown precisely your average speed for the trip. That everyday certainty is the me…
Fermat's Last Theorem fell because two distant branches of mathematics turned out to matchAndrew Wiles did not attack Fermat's Last Theorem head-on. He proved a large part of a claim that elliptic curves, a kind of cubic equation, always correspond to highly symmetric …
Non-decreasing and not decreasing mean very different thingsIn mathematics, a function that never goes down is called non-decreasing, and it is allowed to sit flat for a while. A function that is merely not decreasing could zigzag up and d…
The nautilus steers its depth with a shell full of sealed chambersA nautilus lives inside a coiled shell that it keeps extending, sealing off each outgrown room behind a wall. A hatchling has about 4 chambers; an adult can have 30 or more, threa…
Why Link could never carry more than 255 rupeesIn the original Legend of Zelda, the hero's wallet capped out at 255 rupees, and Pac-Man breaks down at level 256. Neither number is arbitrary. One byte of 8 bits holds exactly 25…
Smoke alarms chase recall while courtrooms chase precisionA smoke detector that shrieks at burnt toast is doing its job: missing a real fire would be far worse than a false alarm. A court that lets ten guilty people walk rather than jail…
Leibniz found the product rule by throwing away a tiny rectangleHow fast does a product of two changing quantities change? Gottfried Leibniz pictured the product as a rectangle's area. Nudge both sides a little and the area grows by two thin s…
A theorem about trees produces a number that dwarfs Graham's numberKruskal's tree theorem sounds modest: it is about how finite branching diagrams fit inside one another. Yet a finite version of it gives rise to TREE(3), one of the largest number…- Lagrange multipliers find the best point when you must stay on a pathImagine hiking along a fixed trail and wanting its highest point. At the summit of your trail, the contour lines of the landscape just graze the path, running parallel to it. That…
The Laplacian measures how a point differs from its surroundingsTake any point, draw a tiny sphere around it, and compare the average value on that sphere with the value at the centre. That difference is what the Laplacian measures. The same s…
For most of history, mathematicians were also accountants, lawyers and doctorsDuring the Renaissance many leading mathematicians held other day jobs. Luca Pacioli founded accounting, Tartaglia worked as an engineer and bookkeeper, Robert Recorde was a physi…- One hypergeometric function quietly contains logarithms, arcsines and Bessel functionsMathematicians have published many thousands of identities for the hypergeometric function, yet nobody knows an algorithm that could generate them all, or even a scheme for organi…
Infiniti launched with car adverts that never showed the carsWhen Nissan launched its luxury brand Infiniti in North America in November 1989, the first television campaign showed rocks, trees and tranquil nature scenes in a Zen style, but …
A bracket or a parenthesis decides whether a number belongs to an intervalWrite the numbers from 0 to 1 with square brackets and both ends are included; swap one bracket for a round parenthesis and that endpoint vanishes from the set. This tiny notation…- The Helmholtz equation strips time out of waves to reveal their shapeWaves change in both space and time, which makes their equations hard to handle. A standard trick splits the problem in two: one piece describing how the wave oscillates in time, …
The Hessian matrix tells a peak from a valley or a saddleAt the top of a hill, the bottom of a bowl and the middle of a mountain pass, the ground is momentarily flat. Slope alone cannot tell these apart. The Hessian matrix, a grid of se…- The F-score folds precision and recall into a single gradeA search engine or spam filter can fail in two ways: flagging things it shouldn't, or missing things it should catch. The F-score blends both kinds of error into one number betwee…
In Göttingen, new doctors must kiss a bronze goose girlNewly minted doctors in Göttingen are pulled through town in handcarts to the Gänseliesel fountain, where they climb up and kiss the statue of a goose girl. It is officially banne…
Gabriel's horn holds a finite volume but has endless surfaceSpin the curve y equals one over x around an axis and you get an infinitely long, ever-narrowing horn. Its inside holds a finite amount, never more than pi. Yet its outer surface …
Adding halves forever: how geometric series tame infinityAdd a half, then a quarter, then an eighth, and keep going without end. Despite infinitely many terms, the total settles on a definite value. That is a geometric series, where eac…
One plus a half plus a third, forever, never stops growingAdd one, then a half, then a third, a quarter, and so on through every unit fraction. The pieces keep shrinking toward nothing, so surely the total levels off. It does not. The ha…- Vector calculus tracks fields with gradients, curls, and fluxThe subject differentiates and integrates vector fields mainly in three-dimensional Euclidean space, and is sometimes equated with broader multivariable calculus. Physics and engi…
Real analysis rebuilds calculus on a gapless number lineUndergraduate and graduate courses use real analysis to make limits, continuity, derivatives, integrals, and series rigorous over the reals and Euclidean space. Completeness—no ho…- Riemann surfaces are complex planes with wild global shapesNamed for Bernhard Riemann, these objects are connected one-complex-dimensional manifolds—locally like open disks in the complex plane, yet globally able to look like spheres, tor…
- Sequences order terms so repetition and position both matterUnlike a set, a mathematical sequence keeps order and allows the same object to appear more than once. Finite lists, infinite streams, Fibonacci numbers, primes, and decimal appro…
Infinite series add forever yet may still sum finitelyA series stacks terms of a sequence one after another without end. Greeks found the idea paradoxical in Zeno's puzzles, yet Archimedes used infinite sums practically. Modern analy…