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.
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. Derivative is shifted-subtraction. It’s (first-)differencing, except in real calculus you would have an infinite, continuous thickness of decimals—more numbers between 1, 4, and 9 than you could possibly want. In R you can use the
diff operation on a sequence of 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
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
diff on it as many times as you like.
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.
diff verb takes you from position → velocity → acceleration, so this mathematics relates fundamental stuff in physics.
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,
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
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
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
, 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.