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……
There are some good intro-to-calculus videos floating around online. I think I can explain differentiation and integration—the two famous operations of calculus—even more briefly.
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
seq2 <- seq*seq
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
diffI 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.
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 neighbors—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
diff on it as many times as you like.
Those are the basics. Why are people so interested in this stuff? 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.
diffverb takes you from position → velocity → acceleration, so this mathematics relates fundamental stuff in physics.
cumsumverb 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.
- In the infinitary version, symbolic formulae
cumsumto 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
xitself – you’re not totally up a creek. You can use calculus tools to make relationships between varying
difflevels 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 )) = − xis the source of 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.