How exactly did people come up with these theories, and how do they work to let you do things like find the area of certain objects and functions?
The derivative is the method to find the tangent line to a curve at a certain point, and the integral is the method to find the area under a curve over a given interval. Applications mainly consist of picking the right curve to solve your problem and finding either the tangent at a point or the area over an interval.
Big subject, little text areas.
The best way, I think, to approach what derivatives are is to think of them as an expression of change. Suppose that you plot a graph of distance on the vertical axis and time on the horizontal axis. Over time, the distance you’ve travelled changes and you graph some kind of curve (which may be a straight line). The question you might have with such a graph is, “How are distance and time related?” Or, “How did a change in time reflect a change in distance?” Or, most commonly, “How far did I go in such-and-such a time?” You might see I’m aiming at speed here, distance over time.
If you were travelling at a constant speed, the graph of distance versus time would be a line: in a given period of time, you travel a specific distance, and as long as your speed is constant, those portions will always be the same. So what you have is a line, which has a constant slope. But what happens when you’ve sped up and slowed down on your trip? Your graph is no longer a line, but a curve. And curves don’t have slopes in the familiar way lines do. So what is to be done?
In this case, we pick a point on the time axis, the one point where we want to know, “At this instant, how fast was I travelling?” We know that (change in distance) / (change in time) is speed (velocity), but at a single point there is no change, so how are we to go about it?
So suppose we are considering the time t1 and we want to know our velocity at precisely that point. Well, even though we have a curve, the curve is “almost” a straight line if we consider a small portion of it. So let’s look at our velocity between t1 and a small addition, t2. In this case, we’d be looking for (where d is distance):
(d2-d1) / (t2 - t1) ~ (is approximately) our velocity.
But we could make a better guess, couldn’t we? I mean, we could pick a time point that was between the two we just chose so it was a little closer to the t1 time we’re interested in. And when we calculate that, we think, “Geez, this is really close, but if I picked an even smaller interval, I’d be even closer to figuring this out.”
Because what we’re shooting for is the slope of the curve at that point. As our “little guess” gets closer and closer to the point we’re interested in, the better our “straightened” curve approximates the slope of the line that just touches our curve at exactly the point we’re interested in (this is called the tangent line).
This is, in essence, the activity of differentiation. It represents the way a free and dependent variable change with respect to each other. When we’re plotting distance and time, we get (change in distance) / (change in time) from differentiation which is velocity. When we’re plotting velocity on one axis and time on the other, we’re getting (change in velocity) / (change in time) from differentiation which is acceleration.
So think of a derivative as a rate of change, and you’ve got that down. Differential equations are the result of something that changes based on such a rate of change. Let’s take an example: cooling. Newton’s Law of Cooling states that “the rate of temperature change with respect to time” is directly proportional to “the difference in temperatures”.
We write:
(change in Temp)/(change in time) = (some constant) * (Temp1-Temp2).
In other words, how fast a glass of chilled water warms up depends on how warm the room is. The larger the temperature difference, the faster the water warms up… but as the water warms up, the difference in temperature shrinks, and it takes a longer for the water to change temperature. At first, the difference is great, and the water begins warming quickly, say a few degrees in a few seconds (not true, but this is just an example). But now the difference in temperatures has shrunk, and so must the rate of change. So now it might take twice as long to move the same few degress. And so on.
Whenever you have a situation like this, you have a differential equation.
Finally, integration. Integration comes from wanting to find the area under a curve, and is very, very similar to differentiation. Instead of trying to calculate the slope of successively smaller and smaller differences, we are using successively smaller and smaller rectangles to approximate the area under a curve where the width of the rectangles is along one axis, and the height varies depending on the curve’s height. First, we might have an average distance of 1 unit, and the rectangles aren’t a very good fit. There’s gaps in some places. So we’ll use rectangles that are a little thinner. These fit a little better… and so on.
Note that the area of a rectange is its width times its height. In the case of a velocity-versus-time graph, that means the area is given by its width=(change in time) times height=(velocity). But we know that velocity * time equals distance. So what we’ve done with integration is worked backwards from differentiation.
In general, integration is sort of the opposite of differentiation in that way. There are lots of different notations for these things, and I won’t bother with them here, but that’s is roughly how they all work together.
Differential calculus was invented/discovered (not getting into that here) by two people at roughly the same time: Newton of gravity fame, and Leibniz. There’s a more complete history here.
Draw a set of axes, x and y, and draw some arbitrary curve in the +x and +y quadrant. Place to points relatively close together on that curve. Call Dx the distance that the point furtherest from the origin is to the right of the other point and Dy the distance it is either up or down from it. Now draw a straight line through the two points on the curve.
If you then move the further point a little closer to the other and draw a new line through the two the new line will have a slightly different slope than the first unless your original curve is a straight line.
Continue moving the further point closer and when the distance gets so small you can’t distinguish the two points your line will be the tangent to the curve at that point and will be the measure of the rate of change of the curve.
What the graphical process you just went through approximates is taking the ratio of Dy/Dx as Dx gets smaller and smaller, i.e. approaches zero. The limit of Dy/Dx as Dx approaches zero (provided the limit exists) is dy/dx and is called the derivative. dy and dx are both differentials.
The derivative is the rate of change of the curve, or its slope, at any point along the curve. As has been mentioned if the curve represents distance traveled in a given time, the derivative is the velocity, if the curve represents velocity, the derivative represents acceleration.
Using the same curve, start at some distance x[sub]1[/sub] on the x axis and draw a series of vertical lines separated by the small distance Dx from the x axis up to the curve. Continue the series of vertical lines out to some distance (of your choice) x[sub]2[/sub]. Now draw a horizontal line from where the first vertical intersects the curve over to the second vertical line. Now go to where the second vertical intersects the curve and draw a horizontal to the third vertical and so on until you come to the last line. You have now filled the area under the curve with a set of rectangles each one Dx wide and y high. The area of the first rectangle is y[sub]1[/sub]*Dx, the second y[sub]2[/sub]*Dx and so on. The total area of the rectangles is the sum of the areas of all these rectangles. The area under the curve between the points x[sub]1[/sub] and x[sub]2[/sub] is obviously slightly greater than this. But if you let Dx get smaller and smaller the rectangles get narrower and narrower and the difference between the sum of their areas and the area under the curve gets smaller and smaller. The fundamental theorem of Integral Calculus states that as Dx —> zero the summation of the rectangles’ areas is equal to the area under the curve.
One example. If I have a constant acceleration, a, that can be equated to the derivative of the velocity.
dv/dt = a
I can form a differential equation by multiplying both sides by dt:
dv = a*dt
You’ll have to take my word for this but if I integrate both sides from 0 to some velocity V I get:
V = at or velocity equals accelerationtime.
going further - velocity is the derivative of distance, s, so
ds/dt = a*t
Again forming a differential equation
ds = atdt
again integrating from 0 to some distance S
S = (a*t[sup]2[/sup])/2
And those are the kinematic equations for uniform motion.
Derivatives and (definite) integrals are formally defined in most elementary calculus textbooks. However, I gather the OP is not asking for a formal definition, but the motivation behind those concepts. Since others seem to be doing an adequate job at explaing that, I think I’ll hijack this thread a little. What follows is probably more of an IMHO topic. I’ll start a new thread, if people feel that it doesn’t belong here.
Notice that the OP did not say “derivatives”, but “differentials”. I do not believe that the symbol “dx” was ever adequately defined in my elemetary calculus classes. I now know there are things called “differential forms”, though I’ve never studied them, and only have a vague idea what they are.
I usually treat the “dx” in an integral expression as a quantifier over x, the scope of which is delimited by the integral sign and “dx”. I am not exactly sure how to assign meaning to “d/dx”, however. Roughly, one treats the operand of “d/dx” as a lambda expression specifying a function, takes the derivative of that function, and then takes the value of that function at x. This covers a case like “(d/dx)x^2”, but not “dy/dx”. I suppose, in the latter case, one has to treat y as a macro. (I’m mixing math and CS terminology a bit here, but I’m not sure how else to explain it).
I’m curious what others consider “dx” to mean (at least in single-variable calculus). To me, there seems to be a gap between analysis and the traditional notation of calculus. I would agree, however, that such notation is highly useful in fields like physics.
Since others have answered the other parts of your question, I’ll take this one.
Centuries of work
-lv
You aren’t integrating both sides from 0 to V. What it looks like you are doing is taking the indefinite integral of both sides, but assuming initial velocity and distance are both zero. If you don’t assume that, you would get a slightly more complicated answer.
If you have: dv = a*dt
Taking indefinite integrals gives you: v = a*t + C where C is a constant
Since putting t=0 gives v(0) = C, C is the initial velocity, and:
v = at + v(0), or velocity equals accelerationtime plus initial velocity.
Integrating again gives s = (a*t^2)/2 + v(0)t + D where D is a constant
Since putting t=0 gives s(0) = D, D is the initial distance, and:
s = (a*t^2)/2 + v(0)t + s(0)
It’s leftover notation. Remember that Newton and Leibnitz both did all of their work using infinitesimals, which were replaced by limits in the late 19th century. By that time, the notation was widespread enough to stick.
If you use the hyperreals, you can treat dy and dx as actual numbers, and most of calculus becomes an exercise in arithmetic.
I know. I was sloppy as hell.
Wesley, I’m curious about why you ask this question. You’re a chem major, aren’t you? Didn’t they cover at least the basics of this stuff in, like, first-semester math?
For some reason it’s long bothered me that explanations of calculus always use graphs. The ideas aren’t rooted in graphs, after all.
First, the idea of functions - black boxes that take input and produce output.
Now jiggle the input of a black box a little, and see how much the output jiggles. What’s the ratio? That is, what’s the gain of the box? Note that the gain might be different for inputs of different sizes, and therefore you’d have to make the jiggles smaller and smaller to make the gain measurement more nearly correct. If you built another black box whose output was whatever the gain of the first black box was, that second box would be the derivative of the first one. And the first box would be the integral of the second one. One convention for naming these functions is if the first box is named f(x), the second box would be named f’(x).
>I’m curious what others consider “dx” to mean…
This has always bothered me - I really dislike this notation. I think it looks incorrect for second and higher derivatives because dx^2 isn’t dx * dx. But I think of dx as delta x however with a magnitude set aside for later use in normalizing everything. In other words, it’s a sort of a hot potato that you can’t use for anything else besides taking in ratio to d(somethingelse).
OK, all the people with good math skills can start jeering and making sport. I’m sure my explanation speaks volumes about why I don’t appreciate the notation.
I’m not going to do any jeering, but I kind of like Leibniz notation.
Some advantages it has for beginning calculus students are that it makes the dimensions of the derivative obvious – e.g. d[sup]2[/sup]y/dx[sup]2[/sup] has dimensions of (y-dimension)/(x-dimension^2) – and it makes the chain rule take an easy-to-remember form.
The derivative is the function assigning a linear map from the tangent space at p of a manifold M to the tangent space at f§ of a manifold N, where f:M->N. It is the “closest linear approximation”, in the sense that any linear map extends to a neighborhood of p by the exponential map and this is the closest to f in that neighborhood.
The integral is a bilinear pairing between simplicial homology classes and de Rham cohomology classes on a manifold. It defines the “n-volume” of an n-chain in general.
They’re also used as a canonical basis for the cotangent space at a point of a manifold given a local coordinate system. They’re very well-defined and not mere “leftover notation” in differential geometry.
>it makes the dimensions of the derivative obvious – e.g. d2y/dx2 has dimensions of (y-dimension)/(x-dimension^2)
Well, crap, I hadn’t noticed that. But it still doesn’t look right, does it? I mean, I see the “fraction” does wind up with one y and negative two x’s, in the dimensional sense, the way it’s written. But what’s most obvious to my crossed eyes is that it would have made more sense to say d2y/d2x because you’re applying a differential operator (bear with me here) twice to each variable. Or even to write dy2/dx2 because it’s just a notation, you’re not supposed to break the dy2 apart into smaller symbols, or the dx2. But why in the world could you write it d2y/dx2? Or why not dy2/d2x making the dimensions of the derivative obviously ydimension^2/xdimension? It still looks like some sort of trick to me.
d/dx ( d/dx (y))
Just “multiply” the “fractions” d/dx and d/dx to get d[sup]2[/sup]/dx[sup]2[/sup]
I just finished first semester math, aka calculus I, and I want to know more about the history and nature of differentials and integrals so i’m asking here. Right now I just know how to find them and use them, I don’t really know how they work or where they came from.
>Just “multiply” the “fractions” d/dx and d/dx to get d2/dx2
Ah-HEH. Are you “sure” that’s a “rigorous” and “correct” treatment? Does this have anything to do with your username? You’re not jeering and making sport, real subtle-like, are you? Wouldn’t it be easier to just “slide the letters and numbers around on the page”? I dunno…
Trust me, if you’ve just finished the first semester of calculus you have little to no chance of understanding what “exactly” are differentials and integrals. I’m still not entirely certain in all the fine details.
As for the history, Newton’s Principia Mathematica is a place to start, along with Leibniz’ works.
Really, if you want to grok it you can spend a whole career working towards it. First, you’ll need to take a proper path through calculus, what most departments call “advanced” calculus, which starts from an axiomatic presentation (and often a constructive proof of existence) of the real numbers. Then an amount of real analysis to understand the modern view of integration and functional analysis to understand function spaces. You’ll probably have to take a fair bit of abstract algebra and complex analysis to shore these subjects up, and much of the analysis doesn’t really show up until most schools’ graduate curriculum.
Anyhow, in parallel with the analysis, you’ll need a good treatment of topology, particularly one which gives a proper definition of limits from the point of view of nets. If your calculus class is like most you (no offense) have no idea what a limit is at all now. Then you’ll be ready for differential geometry, which can slip in once you’ve had advanced calculus, though knowing the Lebesgue integral will help.
This, of course, omits the integral theories (all equivalent, as it happens) of Denjoy, Perron, and Henstock, which are essential for making the fundamental theorem of calculus behave as it’s usually stated.