Posts tagged with engineering

## CALCULUS: the Really Really Short Version

So, you never went to university…or you assiduously avoided all maths whilst at university…or you started but were frightened away by the epsilons and deltas…. But you know the calculus is one of the pinnacles of human thought, and it would be nice to know just a bit of what they’re talking about……

Both thorough and brief intro-to-calculus lectures can be found online. I think I can explain differentiation and integration—the two famous operations of calculus—even more briefly.

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Let’s talk about sequences of numbers. Sequences that make sense next to each other, like your child’s height at different ages

not just an unrelated assemblage of numbers which happen to be beside each other. If you have handy a sequence of numbers that’s relevant to you, that’s great.

` `

Differentiation and integration are two ways of transforming the sequence to see it differently-but-more-or-less-equivalently.

Consider the sequence 1, 2, 3, 4, 5. If I look at the differences I could rewrite this sequence as `[starting point of 1]`, +1, +1, +1, +1. All I did was look at the difference between each number in the sequence and its neighbour. If I did the same thing to the sequence 1, 4, 9, 16, 25, the differences would be `[starting point of 1]`, +3, +5, +7, +9.

That’s the derivative operation. It’s basically first-differencing, except in real calculus you would have an infinite, continuous thickness of data—as many numbers between 1, 4, and 9 as you want. In R you can use the `diff` operation on a sequence of related data to automate what I did above. For example do

• `seq <- 1:5`
• `diff(seq)`
• `seq2 <- seq*seq`
• `diff(seq2)`

A couple of things you may notice:

• I could have started at a different starting point and talked about a sequence with the same changes, changing from a different initial value. For example 5, 6, 7, 8, 9 does the same +1, +1, +1, +1 but starts at 5.
• I could second-difference the numbers, differencing the first-differences: +3, +5, +7, +9 (the differences in the sequence of square numbers) gets me ++2, ++2, ++2.
• I could third-difference the numbers, differencing the second-differences: +++0, +++0.
• Every time I `diff` I lose one of the observations. This isn’t a problem in the infinitary version although sometimes even infinitely-thick sequences can only be differentiated a few times, for other reasons.

The other famous tool for looking differently at a sequence is to look at cumulative sums: cumsum in R. This is integration. Looking at “total so far” in the sequence.

Consider again the sequence 1, 2, 3, 4, 5. If I added up the “total so far” at each point I would get 1, 3, 6, 10, 15. This is telling me the same information – just in a different way. The fundamental theorem of calculus says that if I `diff( cumsum( 1:5 ))` I will get back to +1, +2, +3, +4, +5. You can verify this without a calculator by subtracting neighbours—looking at differences—amongst 1, 3, 6, 10, 15. (Go ahead, try it; I’ll wait.)

Let’s look back at the square sequence 1, 4, 9, 25, 36. If I cumulatively sum I’d have 1, 5, 15, 40, 76. Pick any sequence of numbers that’s relevant to you and do `cumsum` and `diff` on it as many times as you like.

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Those are the basics.

##### Why are people so interested in this stuff?

Why is it useful? Why did it make such a splash and why is it considered to be in the canon of human progress? Here are a few reasons:

• If the difference in a sequence goes from +, +, +, +, … to −, −, −, −, …, then the numbers climbed a hill and started going back down. In other words the sequence reached a maximum. We like to maximize things, like efficiency, profit,
• A corresponding statement could be made for valley-bottoms. We like to minimise things like cost, waste, usage of valuable materials, etc.
• The `diff` verb takes you from position → velocity → acceleration, so this mathematics relates fundamental stuff in physics.
• The `cumsum` verb takes you from acceleration → velocity → position, which allows you to calculate stuff like work. Therefore you can pre-plan for example what would be the energy cost to do something in a large scale that’s too costly to just try it.
• What’s the difference between income and wealth? Well if you define `net income` to be what you earn less what you spend,

then `wealth = cumsum(net income)` and `net income = diff(wealth)`. Another everyday relationship made absolutely crystal clear.
• In higher-dimensional or more-abstract versions of the fundamental theorem of calculus, you find out that, sometimes, complicated questions like the sum of forces a paramecium experiences all along a sequential curved path, can be reduced to merely the start and finish (i.e., the complicatedness may be one dimension less than what you thought).
• Further-abstracted versions also allow you to optimise surfaces (including “surfaces” in phase-space) and therefore build bridges or do rocket-science.
• With the fluidity that comes with being able to `diff` and `cumsum`, you can do statistics on continuous variables like height or angle, rather than just on count variables like number of people satisfying condition X.
• At small enough scales, calculus (specifically Taylor’s theorem) tells you that "most" nonlinear functions can be linearised: i.e., approximated by repeated addition of a constant `+const+const+const+const+const+...`. That’s just about the simplest mathematical operation I can think of. It’s nice to be able to talk at least locally about a complicated phenomenon in such simple terms.
• In the infinitary version, symbolic formulae `diff` and `cumsum` to other symbolic formulae. For example `diff( x² ) = 2x` (look back at the square sequence above if you didn’t notice this the first time). This means instead of having to try (or make your computer try) a lot of stuff to see what’s going to work, you can just-plain-understand something.
• Also because of the symbolic nicety: post-calculus, if you only know how, e.g., `diff( diff( diff( x )))` relates to `x` – but don’t know a formula for `x` itself – you’re not totally up a creek. You can use calculus tools to make relationships between varying `diff` levels of a sequence, just as good as a normal formula – thus expanding the landscape of things you can mathematise and solve.
• In fact `diff( diff( x )) = − x` is the source of this, this

, this,

, and therefore the physical properties of all materials (hardness, conductivity, density, why is the sky blue, etc) – which derive from chemistry which derives from Schrödinger’s Equation, which is solved by the “harmonic” `diff( diff( x )) = − x`.

Calculus isn’t “the end” of mathematics. It’s barely even before or after other mathematical stuff you may be familiar with. For example it doesn’t come “after” trigonometry, although the two do relate to each other if you’re familiar with both. You could apply the “differencing” idea to groups, topology, imaginary numbers, or other things. Calculus is just a tool for looking at the same thing in a different way.

The Audacity of Despair

by David Simon (creator of The Wire)

• arch cynicism about the public purpose of television
• The Wire is not hyperbolic about our inability to solve our own problems.
• The news media buries and forgets relevant information.
• New Orleans was not destroyed by Hurricane Katrina. An untethered barge breached the retaining wall, destroying the Ninth Ward.
• Three years later during Hurricane Gustav, another barge was unsecured in the same canal.
• The Wire is not about sinister people doing sinister things. There’s no fun in that. There’s no drag in writing a show about bad guys and good guys. First of all, it’s not credible. And second of all, it’s not where the real evil lurks.
• As a reporter: “Every time someone dragged out a statistic, I immediately distrusted it as [probably fabricated or] dubious [method]”
• Management: No sooner does someone invent a useful measure of institutional progress, than someone else begins to game it to the point that the measure becomes useless.
• "In my city [Baltimore], every single effort to quantify progress was an effort by somebody to advance themselves.”
• People are promoted or leave to another job before anyone figures out what they got was dross.
• Cops retire with a pension despite making zero progress in 40 years in the war on drugs.

• Why? is the only of the 5W’s+1H that matters. That could have made journalism “a game for grown-ups”.
• Bulls∗∗t US government claims about progress in Vietnam.
• More profitable for Chicago Tribune Company’s shareholders to stop asking Why?—and lay off reporters.
• This was due to their monopoly: they didn’t need top-quality journalism to compete. But the drop in quality, if efficient at the time, made the papers soft targets when the Web became big half a decade later.
• He thinks Internet reporting is less magazine-like and more frothy. I contend ∃ both.
• Crime wasn’t going down anymore. So robberies became larcenies. Aggravated assaults became common assaults. Felonies were leached down to misdemeanors.” Robberies in southwest Baltimore went down 70%. The commander was promoted to head of CID. Next boss went in, crime went up 70%, he took the flack.
• "40% decline in crime, but the murder rate stayed constant. [red flag] The only thing that that says rationally is that they’ve opened up a gun range in West Baltimore and they’re better shots.”
• Any reporter who had any sense of his beat would know this was a huge red flag, would dig deeper into the data and call the complainants.
• "How is it that we’re able to talk about this in an entertainment medium—television—but not in journalism?”
• Curfew for Blacks in Baltimore (fallacious arrests). ACLU tries to sue, but by the time it wends its way through the courts the practice has stopped; the Mayor has become Governor.
• "If you walk into The Other America and ask people how they feel about certain things, you’re likely to hear how they feel.”
• "We stole facts from real life, but thematically the people we stole the most from were Euripides, Aeschylus, and Sophocles."

(Source: youtube.com)

The life forms on our planet have necessarily evolved to match the magnitude of [thermal] energy ﬂows. But while “natural man” is in balance with these heat ﬂows, “technological man” has used his mind, his back, and his will to harness and control energy ﬂows that are far more intense than those we experience naturally….

A society based on power technology teems with heat transfer problems.

John H. Lienhard IV & John H. Lienhard V, A Heat Transfer Textbook

181 Plays

BHP Billiton from The Economist

• the cut and thrust of dealmaking
• putting finance types in the C suite rather than engineers
• diversifying as mines are both very large and financially volatile

Years ago, manufacturers could build a sequence of prototypes and use these to discover and rectify any problems. But now competitive pressures [have reduced] the time to bring a vehicle to market…. Automotive manufacturers aim to … design … a new vehicle and the manufacturing facility … in an entirely virtual world.

This speeds the introduction of the new product, but it does mean that designers … aim to anticipate … problems before a physical build of the vehicle is completed or a new production facility is built. Experience [from] the past is useful, but new vehicles have new features…. For these reasons, we need models that predict how humans of different types will behave in vehicle and workplace environments.

I love Julian James Faraway's reasoning process at the beginning of his paper on ergonomic simulation. He starts out by addressing the most important question: why should I care? rather than assuming “STEM is useful” or “Mathematics is good by fiat”.

Instead of saying that some bit of maths is "important" because “important” is an adjective and he felt like putting an adjective there, Dr. Faraway explains why mathematics is relevant to this specific problem which people already care about.

• Because of the production constraints, the automobile manufacturers need to figure this out on computer before building and testing something in reality.
• Because we don’t have infinite money to build a lot of test space programmes, we have to calculate exactly the trajectories and rocket pulse timing beforehand.
• Because the Aswan dam is so hugely expensive, we need to mathematically plan how it should work before making it.

And so on. It suggests that the practical application of mathematics is in areas where prototyping is prohibitively expensive.

Or where prediction is necessary. For example, actuaries predict large-scale (i.e., central limit theorem applies) insurance losses before they happen.

Neurons are designed with a lot of listeners (the dendrite) and just one talker (the axon terminal).

If we consider the brain as a robust piece of hardware, which can

• learn across environments,
• operate independently of the rest of the organisation of the superstructure,
• and function even after sustaining physical damage,

maybe there’s a universal principle of good design here.

hi-res

[I]t is … anachronistic to apply the term artist with its modern connotation to Leonardo [da Vinci]. Artists in the sense that we understand and use the word, meaning practitioner of fine art, didn’t exist in Leonardo’s time. It would be more appropriate to use the word artisan in its meaning of craftsman or skilled hand worker.

In the historical literature ∃ a perfectly good term to describe Leonardo and his ilk, Renaissance artist-engineer, whereby one can actually drop the term Renaissance as this profession already existed in the High Middle Ages before the Renaissance is considered to have begun.

[T]he artist-engineers were … regarded as menials. An artist-engineer was expected to be a practical mathematician, surveyor, architect, cartographer, landscape gardener, designer and constructor of scientific and technical instruments, designer of war engines and supervisor of their construction, designers of masks, pageants, parades and other public entertainments oh and an artist.

The … polymath … that everybody raves about when discussing Leonardo … actually … perfectly normal … any Renaissance artist-engineer—the only difference being that Leonardo was better at nearly all of them than most of his rivals.

As far as his dissections and anatomical drawings are concerned these belong to the standard training of a Renaissance artist-engineer—the major difference here being that Leonardo appears to have carried these exercises further than his contemporaries and his anatomical sketches have survived whereas those of the other Renaissance artists have not.

Having denied Leonardo the title of artist I think it is only fair to point out that it was the generation to which Leonardo belonged who were the first to become recognised as artists rather than craftsmen and in fact it has been claimed that Raphael was the first artist in the modern sense of the word….

[an exhibition on da Vinci] emphasises the few occasions where Leonardo drew something new or unexpected whilst ignoring the vast number of scientifically normal or often incorrect drawings, thereby creating the impression that his anatomical drawings were much more revolutionary than they in reality were. Also whilst the drawings published by Vesalius in his De fabrica in 1543, i.e. a couple of decades after Leonardo’s death, are possibly not quite as good artistically, as those done by Leonardo, they are medically much more advanced.

Thony Christie (@rmathematicus)

(Source: thonyc.wordpress.com)

Diamond Mining Complex by Laura del Pino

hi-res

• soft robots, modelled after octopi, made of elastimers
• ubiquitous biological battery: a cell = a bag of hydrocarbon=lipid=water-insoluble with more potassium on the inside and more sodium on the outside