Someone said Iain (M) Banks was* two of the best novelists we ever had and I would go with that. I've been listening to some of the Culture novels beautifully read by Peter Kenny and loving being immersed in the Culture again. I've now moved on to listening to The Bridge (again wonderfully read by Peter Kenny) which is, well, maybe a bridging novel between the two genres that Banks wrote in, certainly a bit of a homage to Lanark (Alastair Gray) but entirely, robustly, on its own two literary feet.
So far I've listened to the Player of Games and The Use of Weapons (read by Peter Kenny) and one of the great joys is hearing the names pronounced correctly out loud, and then, of course, and this is down to the skill of the reader, a series of wonderful character voices for the ships and drones. That is one of the great joys of hearing something read aloud especially with humour is that it makes you *hear* properly and pay attention to the story and the humour. (There should be a mention here of Matter read by a different reader, but just as good as kenny - in fact I think I was so impressed I reviewed it here: Matter - Amazon review
I've also just joined the Iain [M] Banks fan forum, just to read and speak and interact with people, who love these novels, just so that I can get more out of my re-readings and listenings
*Iain Banks died on the 9th June 2013 - an inconceivably bitter blow - I still feel distraught even now.
Thursday, 23 August 2018
Understanding Differentiation - Leonard Susskind & George Hrabovsky - Classical Mechanics: The Theoretical Minimum
I've up to a point dimly understood differentiation - the idea that the slope of a graph is its rate of change, and that you can differentiate an equation like x3 + 5x2 +12x -7 quite simply by following the rules - the x3 goes to 3x2, the 5x2 goes to 10x, the x goes to 12 and the 7 disappears. You can check this very easily by just cut and pasting my first formula into this brilliant derivative-calculator site Wonderful though this site is it doesn't actually explain why this is happening - it explains the individual rules, but not why the rule works.
It wasn't until I started reading Classical Mechanics: The Theoretical Minimum (Theoretical Minimum 1) by Leonard Susskind and George Hrabovsky, and then checked my answers to the differentiation question that I actually understood the marvelous mathematical sleight of hand that differentiation is.
I had to do it in my own hard way by looking at each line of the explanation, then matching up bits of equation with the next line to see what was different or missing. It was then that I really understood what that delta t, that little bit of infinitesimal time actually does. So that I never forget it I'm going to lay it out all in glorious colour here.
What we're actually doing with differentiation is taking a function like this one: f(x) = x3 + 5x2 +12x -7 and then writing out the equation again as f(x+𝛥t) - basically adding 𝛥t on every x as so:
f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7
which we then multiply out to make things easier to combine. (𝛥t or delta t as it is pronounced is just a very small bit of... time.) Of course I began to get into real grief here just sorting this out and had to visit another fantastic cheat site -Symbolab that did the expansion for me - and showed me how it got there. I had forgotten that there were formulae for multiplying out things like (x+𝛥t)3
f(x+𝛥t) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2+12x+12𝛥t-7
So there we go with the expanded f(x+𝛥t). Then we write out f(x) - f(x+𝛥t) using these expanded formulae and take one away from the other.
f(x+𝛥t)- f(x) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2-12x+12𝛥t+7- x3 - 5x2 +12x -7
The coloured pairs like x3 from both f(x) and f(x+𝛥t) nullify each other so you're left with:
f(x+𝛥t)- f(x) =3x2𝛥t+3x𝛥t2+𝛥t3+10x𝛥t+5𝛥t2+12𝛥t
Now the really clever bit - divide both sides by 𝛥t and you get 3x2+10x+12. All the multiples of delta t vanish as you go to zero - what you are doing is "taking the limit" as delta t heads for zero. I should write it out properly really but I need a better mathematical editor to do that but here it is bodged up with the space bar to line it up on two lines.
lim f(x+-𝛥t) -f(t) = 3x2+10x+12
𝛥t->0 𝛥t
Looking at the writing out of f(x) - f(x+𝛥t) using these expanded formulae above I realised I had got all the plus and minus signs in the wrong place, but the gist of it is if you take f(x) = x3 + 5x2 +12x -7 away from f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7 all you get left with is the bits that have 𝛥t in them. Well it makes sense to me now - but it's been days, days.
It wasn't until I started reading Classical Mechanics: The Theoretical Minimum (Theoretical Minimum 1) by Leonard Susskind and George Hrabovsky, and then checked my answers to the differentiation question that I actually understood the marvelous mathematical sleight of hand that differentiation is.
I had to do it in my own hard way by looking at each line of the explanation, then matching up bits of equation with the next line to see what was different or missing. It was then that I really understood what that delta t, that little bit of infinitesimal time actually does. So that I never forget it I'm going to lay it out all in glorious colour here.
What we're actually doing with differentiation is taking a function like this one: f(x) = x3 + 5x2 +12x -7 and then writing out the equation again as f(x+𝛥t) - basically adding 𝛥t on every x as so:
f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7
which we then multiply out to make things easier to combine. (𝛥t or delta t as it is pronounced is just a very small bit of... time.) Of course I began to get into real grief here just sorting this out and had to visit another fantastic cheat site -Symbolab that did the expansion for me - and showed me how it got there. I had forgotten that there were formulae for multiplying out things like (x+𝛥t)3
f(x+𝛥t) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2+12x+12𝛥t-7
So there we go with the expanded f(x+𝛥t). Then we write out f(x) - f(x+𝛥t) using these expanded formulae and take one away from the other.
f(x+𝛥t)- f(x) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2-12x+12𝛥t+7- x3 - 5x2 +12x -7
The coloured pairs like x3 from both f(x) and f(x+𝛥t) nullify each other so you're left with:
f(x+𝛥t)- f(x) =3x2𝛥t+3x𝛥t2+𝛥t3+10x𝛥t+5𝛥t2+12𝛥t
Now the really clever bit - divide both sides by 𝛥t and you get 3x2+10x+12. All the multiples of delta t vanish as you go to zero - what you are doing is "taking the limit" as delta t heads for zero. I should write it out properly really but I need a better mathematical editor to do that but here it is bodged up with the space bar to line it up on two lines.
lim f(x+-𝛥t) -f(t) = 3x2+10x+12
𝛥t->0 𝛥t
Looking at the writing out of f(x) - f(x+𝛥t) using these expanded formulae above I realised I had got all the plus and minus signs in the wrong place, but the gist of it is if you take f(x) = x3 + 5x2 +12x -7 away from f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7 all you get left with is the bits that have 𝛥t in them. Well it makes sense to me now - but it's been days, days.
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