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The mathematician Georg Polya wrote a famous book entitled How to Solve It, in which he explored ways to approach mathematical problem solving. In the third section, "A Short Dictionary of Heuristic," he describes problem-solving by analogy, by generalization, by specialization, and by other means that are essentially the realization that this and that are related.

Let me see if I've got this - you understand the notion of the derivative and integral of a function, but you didn't understand the notion of the function itself until you saw it as the integral of the derivative?

Well, whatever works, I guess. But you also say that the introductory lecture doesn't say how to find the position function. That's because it's assumed that you already knew what it was.

It's one of your inputs. Calculus problems typically go like, "A French lady is thrown off a cliff and her height at any given time is CliffHeight - 16 * Time ^ 2. How fast is she going when she hits the ground?"

They give you the position function, H-16*t^2, and they want you to first, figure out the time it takes to hit the ground (by algebra).

Then they want you to do the calculus to get the velocity function.

Then they want you to take the time from the first step and plug it into the velocity function from the seconds step.

regards differential calculus and the real world: it is wrong! I've serious problems with Ampere's differential concept of electric current as used by Maxwell in his equations when the current is small:

He has something like: electric current I = dQ/dt, where dQ is the amount of charge flowing past any point in a circuit in time interval dt.

Problem: what happens when current is so small only say 1 electron per second passes you? Is I = dQ/dt still true?

The differential equation I = dQ/dt then becomes a statistical average, and a lie. No wonder Maxwell's equations are inconsistent with quantum field theory. Calculus is only strictly valid for continuous, classical fields. You always get discontinuties in the real world, because ultimately QFT is right. This is why QFT gives infinities and requires the renormalization of charge using UV and IR cutoffs to avoid the infinities you get from a continuous field. The continuous differential equations of QFT and their solution are wrong (before renormalization) because they (1) give infinitely strong field effects at distances approaching zero (so the number of particle creation-annihilation loops becomes infinite, and the amount of momenta they carry becomes infinite and unphysical), and (2) give finite vacuum polarization effects down to arbitrarily weak fields out to infinite distances (which again is unphysical, because the entire vacuum of the universe would then polarize until it had cancel every real charge completely!).

"When was the last time you bumped into a Calabi-Yau on a leisurely stroll through your local park?"

Maybe every time? That's what I understand string theory says, anyway (that the compactification of the extra dimensions is a Calabi-Yau manifold).

"Truth, according to Plato, is the domain of philosophy only. Consuming literature was in these days not a solitary act (people did not read books, but listened to 'performances'). Plato's main critique was that theatre was very theatrical, and therefore a rhetorican could persuade his public without necessarily telling the truth."

It is the same today: we like to learn from Professors better than from books, we like to watch performances on stage and film too.
Even writers and especially the best romantic writers are like teachers guiding you thru the plot, leaving the details to your 'imagination'

Oh, and did we forget 'operating theatres'
I just wish doctors, dentists and surgeons could just see what really bad actors they are on the stage. This is Plato's main criticism of the risks of becoming too theatrical.
And well when we watch House, Scrubs, ER and a whole plethora of hospital dramas we soon realise it is not about the patient but about the ego, vanity and drama in the doctors', nurses', consultants', specialists' lives ... and they would try and convince you they know what they are doing when they don't, and their authority rests on the inability to challenge hierarchy - in which medicine (and heart surgery) have placed themselves on top of the hierarchical pyramid, since even kings (good night al saud) and presidents need to put their lives literally in a surgeon's hand. Yes even the 'wicked' Dick Chenney has a whole retinue of docyors trying to keep him alive and make him inmortal.

The mark of the beast, may not be bowing to the $US dollar, but becoming enslaved to the medical profession and pharmaceutical industries. But het don't worry about me I'm probably just a mad poet. lol!

Maths writes the programs
by trial and error we define which work
We analise the airpor data
and we can 'predit' traffic (and the weather) too

But people shop in supermarkets what is on the shelve, you cannot buy or choose to buy what is not available on the shelve. So people do not choose what they want to buy, but people choose from what someone else chose to put on the shelf. Of course there is a dynamic interaction between the the supplier - the consumer - the supplier.

But if I had a $dollar for every coffe I've been 'served' and did not enjoy, I'd be close to a dollar millionair by now. So my guess is that the number of coffees (or even beers drunk) is not directly significant to the pleasure obtained.
I only brought this in, because on the radio there were talking about how many people get drafted or dragged into plays (theatre) even concerts (gigs) they'd really rather not be at (or pay for) but out of some false sense of loyalty to family and friends - and I'm not just on about the terrible school pantos or plays.

But the number crunching is not aware of these nuances. 5o million visited this country or that country does not really tell us why they did so.
And 50 million use this drug or that drug does not tell us it is any good - any more than the coffeee sales do. Ten million had this treatment or surgery tells us no more about the quality or efficacy of said treatment and surgery, than saying that all children's parents attended the school play, tells us about the quality of the play
And so many 'people' went to see such and such a film or read such and such a book, tells us little about the film or the book - it just helps hollywood accountants budget which films to finance

So Jen, here's wishing you a mighty fine day!

Charon makes a good point. And while you walk through that park, just think of the hundred trillion neutrinos zipping through your body each second! If you still have a TV with rabbit ears, try this: tune it to a blank station and turn down the brightness knob. A few percent of the flickering "snow" is actually cosmic microwave background radiation. The fires of creation are still burning, passing through each of us as we laugh and sleep and think.

I'm glad to see you sticking with the calculus lessons. Just wait, in no time at all you'll be defining 1-forms on differentiable manifolds!

Some excellent comments here but I wanted to respond specifically to Andy's disbelief over my ignorance of the function. It's one of those concepts that is so basic to scientists, that they forget it's just jargon to the Common Man. "Let X be a function of Y" conveys no real information to a non-scientist, even though we might be able to blithely rattle it off on a test. True understanding -- e.g., knowing the underlying connections and how it all fits together -- is not the same thing as casual recitation.

Nor am I entirely convinced that Starbird (DVD instructor) assumed his audience already knows what a function is -- the jury's still out on that. It might be that he's deliberately taking a very different approach in tailoring the course to non-scientists. And for the most part, I find his conceptualizations effective; he just moves a bit too slowly for my tastes, and left out one critical bit of information. From here on out, the lectures focus almost exclusively on the different ways to apply the fundamental theorem of calculus. And that serves to make the notion of the function crystal clear -- defining it by all the different things it can do.

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    Physics Cocktails

    • Heavy G
      The perfect pick-me-up when gravity gets you down.
      2 oz Tequila
      2 oz Triple sec
      2 oz Rose's sweetened lime juice
      7-Up or Sprite
      Mix tequila, triple sec and lime juice in a shaker and pour into a margarita glass. (Salted rim and ice are optional.) Top off with 7-Up/Sprite and let the weight of the world lift off your shoulders.
    • Listening to the Drums of Feynman
      The perfect nightcap after a long day struggling with QED equations.
      1 oz dark rum
      1/2 oz light rum
      1 oz Tia Maria
      2 oz light cream
      Crushed ice
      1/8 tsp ground nutmeg
      In a shaker half-filled with ice, combine the dark and light rum, Tia Maria, and cream. Shake well. Strain into an old fashioned glass almost filled with crushed ice. Dust with the nutmeg, and serve. Bongos optional.
    • Combustible Edison
      Electrify your friends with amazing pyrotechnics!
      2 oz brandy
      1 oz Campari
      1 oz fresh lemon juice
      Combine Campari and lemon juice in shaker filled with cracked ice. Shake and strain into chilled cocktail glass. Heat brandy in chafing dish, then ignite and pour into glass. Cocktail Go BOOM! Plus, Fire = Pretty!
    • Hiroshima Bomber
      Dr. Strangelove's drink of choice.
      3/4 Triple sec
      1/4 oz Bailey's Irish Cream
      2-3 drops Grenadine
      Fill shot glass 3/4 with Triple Sec. Layer Bailey's on top. Drop Grenadine in center of shot; it should billow up like a mushroom cloud. Remember to "duck and cover."
    • Mad Scientist
      Any mad scientist will tell you that flames make drinking more fun. What good is science if no one gets hurt?
      1 oz Midori melon liqueur
      1-1/2 oz sour mix
      1 splash soda water
      151 proof rum
      Mix melon liqueur, sour mix and soda water with ice in shaker. Shake and strain into martini glass. Top with rum and ignite. Try to take over the world.
    • Laser Beam
      Warning: may result in amplified stimulated emission.
      1 oz Southern Comfort
      1/2 oz Amaretto
      1/2 oz sloe gin
      1/2 oz vodka
      1/2 oz Triple sec
      7 oz orange juice
      Combine all liquor in a full glass of ice. Shake well. Garnish with orange and cherry. Serve to attractive target of choice.
    • Quantum Theory
      Guaranteed to collapse your wave function:
      3/4 oz Rum
      1/2 oz Strega
      1/4 oz Grand Marnier
      2 oz Pineapple juice
      Fill with Sweet and sour
      Pour rum, strega and Grand Marnier into a collins glass. Add pineapple and fill with sweet and sour. Sip until all the day's super-positioned states disappear.
    • The Black Hole
      So called because after one of these, you have already passed the event horizon of inebriation.
      1 oz. Kahlua
      1 oz. vodka
      .5 oz. Cointreau or Triple Sec
      .5 oz. dark rum
      .5 oz. Amaretto
      Pour into an old-fashioned glass over (scant) ice. Stir gently. Watch time slow.