Calculus is one of the things that’s considered terrifying by most people. In fact, I’m sure a lot of people will consider me insane for trying to write a “basics” post about something like calculus. But I’m not going to try to teach you calculus – I’m just going to try to explain very roughly what it means and what it’s for.
There are actually two different things that we call calculus – but most people are only aware of one of them. There’s the standard pairing of differential and integral calculus; and then there’s what we computer science geeks call a calculus. In this post, I’m only going to talk about the standard one; the computer science kind of calculus I’ll write about some other time.
The first one – the more common one – is a branch of mathematics that uses limits
and/or infinitessimal values to analyze curves. Limits can be used in one way (the
differential calculus) to look at incredibly small sections of a curve to figure out
how it’s changing – and in particular, to find patterns in how it changes. Limits can be used in another way (the integral calculus) to compute the area under
a curve by adding up an infinitely large number of infinitely small values.
In differential calculus, what you’re usually doing is taking a curve described by an equation, and figuring out a new equation (the derivative of the curve) that describes how the first one changes. For example – look at a curve like y=x2. At any point in time, the curve has a slope – but it’s constantly changing. But what we can do is look at that curve, and say that at any point x, the slope of the curve will be 2x.
What does that mean? There are a lot of things that can be understood in
terms of rates of change. Suppose that you measured the position of a moving object,
and worked out an equation that described where it was along a line at any point in
time. Let’s say that that equation for the total distance moved in t seconds was f(t)=3t3+5t+11 meters. So:
f(t)=3t3+5t+11
Now, suppose I want to know how fast it was moving after 3 seconds, that is, at time t=3. How could I figure that out? It’s not moving at a constant speed. At any two moments, the speed is different. How can we know how fast it’s moving at a particular point in time?
The velocity that something is moving at some point in time is how much it changes its position divided by the length of the period of time; if position=p, then the
velocity v=Δp/Δt. Since the velocity is constantly changing, though, that equation isn’t too much good for us. We can’t say how fast it’s moving at t=3.
We can use it to start homing in. Between t=2 and t=3, it moved from (24+10+11)=45 to (81+15+11)=107 – so it moved 107-45=62 meters – so its average speed between t=2 and t=3 was 62 meters/1 second = 62 meters/second. Between t=2.5 and t=3, it moved from 70.375 to 107 – so its average speed for that half second was 36.625 meters/0.5 seconds = 73 1/4 meters per second. From t=2.9 to t=3, its average speed was 8.33meters/0.1 seconds=83.3 meters/second. From time t=2.9999 to time 3, it’s average speed was approximately 85.99 meters/second. To know exactly what speed it was moving at t=3, I need to know its velocity at precisely t=3 – an interval of length 0 at exactly t=3. The way that we can do that is to pull out a limit: the speed at time t=3 is limδ→0(f(3)-f(3-δ))/δ=86 meters per second.
We can do that symbolically on the original equation (I’m not going to go through the whole process), and end up with the velocity at time t=9t2+5. This second equation is called the derivative of the original equation.
In integral calculus, what you’re usually doing is taking a curve described by an
equation, and figuring out a new equation that tells you the area under the curve. (the
integral of the curve) So, again, taking the curve y=x2, we can ask
what’s the area under the curve between x=0 and x=6? The area under the curve
y=x2 is x3/3; so the area between 0 and 6 is 72. It can be used for the opposite of what we just did with the derivative – if we have an equation showing its velocity at different instants, we can figure out an equation for its position.
Differential calculus and integral calculus started out as two different (if conceptually related) fields – but they were tied together by something called the fundamental theorem of calculus. Stated very roughly and informally, what the fundamental theorem basically says is: if I start with some curve, and I take its derivative, and then I take the integral of the derivative, I’ll get back the same equation that I started with.
The history of calculus is really interesting – but to get into detail would be a whole post of its own. Basically, what we call calculus was invented roughly simultaneously by Isaac Newton and Gottfried Leibniz. Basically, Newton probably did work out the ideas of calculus first, but he didn’t publish it; Leibniz started later, but published first. The notations that we generally use for calculus are mostly those of Leibniz, as is the name calculus – Newton called it “the method of fluxions”. This conflict led to a huge feud between Leibniz and Newton, which expanded into a conflict between the mathematicians of England and the mathematicians of the European conflict.
“European continent”
Note that one place you may still see Newton-style notation is in certain physics uses – especially in classical mechanics. Leibniz notation basically rules everywhere else, and for good reason: the Leibniz notation makes more sense and is more expressive.
However, when all the derivatives you’re taking in a problem are with respect to the same variable (such as, say, time), the Newtonian notation is much more compact, which may be why it’s survived anywhere at all.
“…we computer science geeks call a calculus” You can’t say that and then not talk about it, that’s just mean.
> the fundamental theorem of calculus.
> […]: if I start with some curve, and I take its
> derivative, and then I take the integral of the
> derivative, I’ll get back the same equation that I
> started with.
Are you sure this shouldn’t be the other way round? There are a lot more functions integrable than differentiable, so it’s often only possible to start with integrating. Another point is that the derivative of x^3/3 is x^2, but an integral of x^2 could be x^3/3 + 1, i.e. the original curve translated.
“…we computer science geeks call a calculus” You can’t say that and then not talk about it, that’s just mean.
He’s probably referring to Lambda calculus.
http://goodmath.blogspot.com/2006/05/my-favorite-calculus-lambda-part-1.html
Daniel, do you know where one might look to see examples of the Newtonian notation? I don’t think I’ve ever seen it, though I haven’t studied much physics.
Dave, it think he’s probably referring to the general term calculus. That is, definition 1a here: http://www.m-w.com/dictionary/calculus
Susan, http://en.wikipedia.org/wiki/Newton%27s_notation_for_differentiation
Now I know why I used different symbols for the same thing when doing calculus in Mathematics & Engineering.
The Leibniz notation is more readable and simpler to understand, Newton’s notation is quicker to write.
I note that it is consistent if we specify boundary conditions, i.e. define where the integration or derivation took place. Then we get exactly the original curve. Otherwise we throw information away and it isn’t surprising that we no longer know (within the problem) where the original curve was situated.
In the context, for me the differential/integral picture shows how we can handle physical laws. We can conveniently specify them as laws for a general point (aha!) or helpfully describe their solutions over a specific volume (yes, please).
John Baez steps away from “the Calculus” to “a Calculus” analogously to how you went from “Algebra” to “an Algebra”:
“we’ve got differential forms, and vector fields, and so on, and a bunch of linear operations. Recently the whole lot have been formalized into a single mathematical gadget called — boldly but aptly — a ‘calculus’.”
“We can get a calculus not just from a smooth manifold but from any commutative algebra. When we use the algebra of smooth functions on a smooth manifold, we’re back where we started — but it’s really a vast generalization.”
“What if we start with a noncommutative algebra? Here it turns out we just get a ‘Calc ∞ algebra’: in other words, an infinitely categorified calculus!”
http://golem.ph.utexas.edu/category/
Is there any part of this that you want to explain, using your Categorical expertise, to audiences less technical than myself, but maybe not as bad as some Congressmen and Presidents whom I could name?
The fundamental theorem starts with a continuous function and integrates first.
However if your function is derivable you can derive first as well; but the intermediate result may not be integrable in the usual Riemann or Lebesgue senses, so you may have to use another type of integration, the Henstock-Kurzweil integral, to get back to where you started.
“… a lot more functions integrable than differentiable…”
Careful! Swept under the rug here is the question: “How many functions are there?”
Cantor proved that the number of functions (as he defined them) was a transfinite cardinal which he called F. He proved that this was a higher infinity than the number of real numbers, a transfinite cardinal which he called C. C is in turn, a higher infinity than the number of integers, a transfinite cardinal which he called Aleph-null.
He proved that for any a transfinite cardinal Aleph-k for any k, that the cardinality of the power set 2^Aleph-k is greater than Aleph-k.
So Aleph-null
Actually, Godel had already proven the consistency of CH (although there is also a forcing proof). Cohen did prove the consistency of not-CH, however.
We can dodge the counting issue by using measure theory, though. The set of differentiable functions has measure zero in the space of continuous functions. (More interestingly, the complement of the set of nowhere-differentiable functions has measure zero in the space of continuous functions.
We can also answer it in cardinality:
There are continuum many differentiable functions.
There are 2^c many integrable functions.
I’m visualizing a transfinite magician, fanning a sheaf of functions at someone from the audience.
“Pick a function, any function,” says the mathemagician.
The audience member selected, call him Zorn, picks one at random, and looks at the function almost everywhere.
“I’m guessing that…” says the wizard, “… it is NOT differentiable.”
“Wow!” exlaims Zorn. “How did you know?”
I was just thinking recently about this–that almost every function has an integral, but in practice it’s often impossible to find. Hardly any function is differentiable, but the functions that mere mortals use are almost always differentiable except for a scattered point here and there.
In arithmetic, multiplication is easier that division, but in calculus, the opposite is true.
I was also thinking about how you could view introductory calculus as a joke. The set-up is differentiation, where you show students how to differentiate all the different ways we put functions together…addition, subtraction, multiplication by a constant, multiplication, division, exponents, composition, trig functions. Everything is very straightforward, algorithmic. Then you start on integration…addition, subtraction, multiplication by a constant, multiplication (hmmm… we’ll get back to that), exponents, trig functions, multiplication…well, sometimes this works… composition–uh…hey, look over there on that blackboard! We have this cool new “log” function. Neat, huh?
“… a lot more functions integrable than differentiable…”
It’s true in a naive sense, since all the differentiable functions (say, on a closed interval) are integrable (since they are continuous), but not all the continuous functions (for which the classical Newton-Leibniz holds) are differentiable. Differentiability is a “fragile” property, it depends on the “microscopic” structure of the function that can be easily broken by adding an arbitrarily small nondifferentiable continuous function to it. The set-theoretic argument, presented in comment #12, that both sets have the same cardinality simply doesn’t capture our intuition of “a lot” and “few.” For example, both the set of points on a diagonal of a unit square and all the points in that square form the sets of the same cardinality, but everyone will agree that there are a lot of points in a unit square than on its diagomal. The probability to pick a point on the diagonal when you pick a point at random in a square is zero. The area (=the 2-dimensional Lebesque measure) of the diagonal is zero. It suggests that we use some measure when we want to make sense of how many functions among the continuous ones are differentiable. One of the popular measures on the space of continuous functions (defined for all x between 0 and 1) is the Wiener measure, used in theory of Brownian motion and other stochastic processes, which is an infinitely-dimensional analog of the Gaussian distriibution. It turnes out that the Wiener measure of the space of continuous functions is 1, while the Wiener measure of the set of all the differentiable functions is zero. And even better, the functions satisfying the Holder condition with the exponent > 1/2 form the set of the Wiener measure zero. So, if you pick a continuous function “at random” (according to Wiener measure), your chance to pick a differentiable function is 0.