What Happens to dx in Integration?

[15adhami]

Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually.
When integrating dy/dx=x you get,
∫x dx=1/2x².
But what exactly happened to the dx, why did it disappear? I thought that dx was an infinitesimally small number, representing the width of the rectangles in a Riemann Sum.
Also, I tried solving this question geometrically, but I think that I might be doing something wrong. Starting from any point x, and using left endpoint rectangles, I got that the heights of the rectangles are x,x+dx,x+2dx,... And the width is just dx. So the area under the curve y=x is dx(x+(x+dx)+(x+2dx)+(x+3dx)...) which is equal to:
dx(x+x+x...+dx(1+2+3+4...))? Am I misinterpreting dx or can I simplify this sum even further to get 1/2x²? Thank you.

 

[andrewkirk]

Yes you can simplify it further.
Say the integral is from 0 to 1. Divide the length up into n equal segments so that $dx=\tfrac{1}{n}$.
Then what you have written as dx(x+x+x...+dx(1+2+3+4...)) is actually (substituting 0 for x)

$$dx\sum_{k=1}^n\left(0+k\,dx\right)
=\tfrac{1}{n}\left(\sum_{k=1}^n\tfrac{1}{n}k\right)
=\tfrac{1}{n^2}\left(\frac{n(n+1)}{2}\right)=\frac{1}{2}+\frac{1}{2n^2}
$$

which tends towards $\tfrac{1}{2}$ as $n\to\infty$.

 

[15adhami]

Oh ok, thank you. I was able to use the summations to also get the integral from 0 to any value b. By replacing dx with b/n I got the same Riemann sums as what I learned in high school.

 

[Mark44]

Oh ok, thank you. I was able to use the summations to also get the integral from 0 to any value b. By replacing dx with b/n I got the same Riemann sums as what I learned in high school.

That's fine, but the integral you showed, ∫x dx=1/2x², is an indefinite integral, so does not evaluate to a number.

The principal use of dx in definite or indefinite integrals is to indicate what the variable of integration is. For example, in $\int my^2 dy$, the dy tells us that the variable is y, and that for the purposes of integration, m is a constant.

Also, with indefinite integrals, be careful to include the arbitrary constant. In your example, it should be $\int x~dx = \frac 1 2 x^2 + C$.

 

[zinq]

"But what exactly happened to the dx, why did it disappear? I thought that dx was an infinitesimally small number, representing the width of the rectangles in a Riemann Sum."

For an indefinite integral, which is what you have displayed with ∫ x dx, think of this as just indicating you want the antiderivative(s) of the integrand. Since the derivative of x²/2 is in fact the integrand x, x²/2 is one antiderivative. If you slide the graph of y = x²/2 by any amount vertically (i.e., adding a constant to x²/2), this will not change the slope of the function; i.e., its derivative will still be x for any constant you add. Thus

∫ x dx = x²/2 + C

is your most general family of antiderivatives of x.

Do not think of the dx as an infinitesimally small number, but rather as an instruction in the definition of an indefinite (or definite) integral as to how to intepret the rest of the notation.

 

[Mark44]

"But what exactly happened to the dx, why did it disappear? I thought that dx was an infinitesimally small number, representing the width of the rectangles in a Riemann Sum."

For an indefinite integral, which is what you have displayed with ∫ x dx, think of this as just indicating you want the antiderivative(s) of the integrand. Since the derivative of x²/2 is in fact the integrand x, x²/2 is one antiderivative. If you slide the graph of y = x²/2 by any amount vertically (i.e., adding a constant to x²/2), this will not change the slope of the function; i.e., its derivative will still be x for any constant you add. Thus

∫ x dx = x²/2 + C

is your most general family of antiderivatives of x.

Do not think of the dx as an infinitesimally small number, but rather as an instruction in the definition of an indefinite (or definite) integral as to how to intepret the rest of the notation.

I'm glad you agree with me. See post #4.

 

[mfig]

The principal use of dx in definite or indefinite integrals is to indicate what the variable of integration is. For example, in ∫my2dy∫my2dy\int my^2 dy, the dy tells us that the variable is y, and that for the purposes of integration, m is a constant.

I disagree with this slightly. The dx is not merely an instruction or indication, it is part of the differential itself. When one sees this:

$\int x^3~dx$

It can be understood as, "find the function whose differential is $dy = x^3~dx$." So, what function has that differential? Well I think it is easy to see that if we set $y = \frac{x^4}{4}$, then the differential of y is $dy = x^3~dx$.

 

[Mark44]

It's kind of "six of one, a half-dozen of the other." In other words, you can look at $\int x^3~dx$ as a) finding that antiderivative of $x^3$, or b) finding the "antidifferential" (not sure if that's a word) of $x^3 dx$. Either way, dx doesn't play much of a role, other than to indicate with respect to which variable integration/antidifferentiation should be done.

 

[mfig]

It's kind of "six of one, a half-dozen of the other."

Yeah, maybe. I am just uncomfortable with any response that seems to indicate it has no real meaning other than as a pointer. After taking differential forms, I saw that these things have all kinds of uses and even an algebra associated with them. To say that the differential of $y = x^2$ is given by $dy = 2xdx$ , is to say something with definite meaning. We cannot dismiss, in general, the dy and dx as mere notation.

On the other hand, for someone who is just learning integration, perhaps treating them as pointers is just fine.

 

[Mark44]

Yeah, maybe. I am just uncomfortable with any response that seems to indicate it has no real meaning other than as a pointer.

I meant that only in the context of integrals, not in general.

After taking differential forms, I saw that these things have all kinds of uses and even an algebra associated with them. Saying that the differential of $y = x^2$ is given by $dy = 2xdx$ has a definite meaning. We cannot dismiss, in general, the dy and dx as mere notation.

On the other hand, for someone who is just learning integration, perhaps treating them as pointers is just fine.

I agree that they have importance other than as indicators of what the variable of integration is. Many students don't bother to write the dx (or appropriate differential) in an integral, but come to grief in integration techniques such as trig substitution or integration by parts.

 

[bcrowell]

See http://math.stackexchange.com/questions/200393/what-is-dx-in-integration .

The notion that dx is only a pointer or punctuation is misguided revisionist history. The dx represents an infinitesimal number. It doesn't matter whether you're talking about a definite integral or an indefinite integral. An indefinite integral can be thought of as a definite integral in which one of the bounds of integration is the independent variable.

 

[zinq]

bcrowell: The history of mathematics is one thing; modern mathematics is another thing. As concepts become clarified, definitions become more precise. What dx once was is not the same thing as what dx is now.

In advanced mathematics beyond elementary calculus (and which is sometimes taught in honors calculus), one learns that dx can be thought of fully rigorously as a linear function on tangent vectors. As spamanon mentions above.

There are also other ways of interpreting it that are closer to the concept of an infinitesimal, but in a modern rigorous way, far removed from the vague infinitesimals of Newton and Leibniz. Usually one must take a rigorous course in mathematical logic to fully understand these interpretations.

 

[lavinia]

As zinq has said, in modern integration theory the things that get integrated are differential forms. Differential forms are multilinear alternating tensors.

d$x$ is a tensor of degree 1. In general, a 1 form can be expressed with respect to a basis for the cotangent space as is a sum, $Σf_{i}$d$x_{i}$ .

 

[bcrowell]

bcrowell: The history of mathematics is one thing; modern mathematics is another thing. As concepts become clarified, definitions become more precise. What dx once was is not the same thing as what dx is now.

In advanced mathematics beyond elementary calculus (and which is sometimes taught in honors calculus), one learns that dx can be thought of fully rigorously as a linear function on tangent vectors. As spamanon mentions above.

There are also other ways of interpreting it that are closer to the concept of an infinitesimal, but in a modern rigorous way, far removed from the vague infinitesimals of Newton and Leibniz. Usually one must take a rigorous course in mathematical logic to fully understand these interpretations.

Your first paragraph gets the history wrong. Infinitesimals have persisted since Newton and Leibniz without interruption; scientists and engineers have been using a consistent set of practices ever since that time.

Your second paragraph gets the math wrong. Yes, one can choose to interpret a dx in certain contexts using the language of differential forms. No, that is not the only way to interpret it, and in fact it doesn't work well in many cases (e.g., second derivatives).

Your third paragraph is also wrong. No, it is not necessary to take "a rigorous course in mathematical logic to fully understand these interpretations." For a counterexample, see the freshman calculus book by Keisler (free online): http://www.math.wisc.edu/~keisler/calc.html . Your third paragraph also contradicts your first and second paragraphs, which is odd -- the first and second are written as if you were ignorant of NSA, while the third shows that you know of it.

As zinq has said, in modern integration theory the things that get integrated are differential forms.

This is wrong for the same reasons.

 

[PeroK]

Yeah, maybe. I am just uncomfortable with any response that seems to indicate it has no real meaning other than as a pointer. After taking differential forms, I saw that these things have all kinds of uses and even an algebra associated with them. To say that the differential of $y = x^2$ is given by $dy = 2xdx$ , is to say something with definite meaning. We cannot dismiss, in general, the dy and dx as mere notation.

On the other hand, for someone who is just learning integration, perhaps treating them as pointers is just fine.

Suppose, for the sake of argument, I decided to write $Int[f(x), x]$ instead of $\int f(x) dx$. Is there a theorem or result that would not hold or could not be proved using my notation, but could be proved using the standard form of the integral?

 

[bcrowell]

Suppose, for the sake of argument, I decided to write $Int[f(x), x]$ instead of $\int f(x) dx$. Is there a theorem or result that would not hold or could not be proved using my notation, but could be proved using the standard form of the integral?

The standard notation and the notation you suggest are isomorphic to each other, i.e., you can translate back and forth without ambiguity. Therefore the answer to your question is no.

However, that doesn't mean that they're both equally good notations. The standard (Leibniz) notation is good for a variety of reasons, one of the most important ones being that it makes sense of the units. For example, if I integrate velocity with respect to time, I get position. Notating it in Leibniz notation, $x=\int v dt$, we can see that the units do make sense: meters=(meters/second)(seconds).

 

[zinq]

Thanks for the link to that infinitesimal calculus book, bcrowell. As far as I can tell, however, the book merely asserts that infinitesimals exist and goes from there. Which to my way of thinking doesn't create understanding.

How would you define an infinitesimal?

 

[bcrowell]

So: How would you define an infinitesimal?

The quantity x is a positive infinitesimal if $x>0$, but $x<1$, $x<1/2$, $x<1/3$, ... (The ... means that it has to satisfy all inequalities of this form.)

The real numbers have the Archimedean property, which says that they do not include any infinitesimals. The Archimedean property is logically implied by the completeness property of the reals. The reals are the unique complete, ordered field.

 

[zinq]

You have given a defining property of a positive infinitesimal, and then stated that the real numbers do not include any infinitesimals.

So, where do we find them? What is an example of one?

 

[PeroK]

The standard notation and the notation you suggest are isomorphic to each other, i.e., you can translate back and forth without ambiguity. Therefore the answer to your question is no.

However, that doesn't mean that they're both equally good notations. The standard (Leibniz) notation is good for a variety of reasons, one of the most important ones being that it makes sense of the units. For example, if I integrate velocity with respect to time, I get position. Notating it in Leibniz notation, $x=\int v dt$, we can see that the units do make sense: meters=(meters/second)(seconds).

Yes, I agree it's good notation. But, it is "only" notation! But, then, what isn't just notation when it comes down to it?

 

[PeroK]

The quantity x is a positive infinitesimal if $x>0$, but $x<1$, $x<1/2$, $x<1/3$, ... (The ... means that it has to satisfy all inequalities of this form.)

The real numbers have the Archimedean property, which says that they do not include any infinitesimals. The Archimedean property is logically implied by the completeness property of the reals, so the reals are the unique complete, ordered field.

Which, unfortunately, rather suggests that the Riemann integral can't be defined in terms of standard Real Analysis.

 

[zinq]

The Riemann integral most assuredly can be, and is, defined in terms of standard real numbers.

 

[PeroK]

The Riemann integral most assuredly can be, and is, defined in terms of standard real numbers.

Not if it's got a "non-real" differential in it! But, if that differential is merely notation ...

 

[bcrowell]

You have given a defining property of a positive infinitesimal, and then stated that the real numbers do not include any infinitesimals.

So, where do we find them? What is an example of one?

You seem to be making some assumptions about what a number is, how it relates to the real world, and the role of the real number system. If you examine these assumptions carefully, they don't hold.

The real numbers are not "really" "real" in the sense of relating perfectly to the "real" world. For example, the real number system has a distinction between rational and irrational numbers, but such a distinction can never be empirically verified for any real-world measurement. For example, there is no way, even in principle, to determine whether the mass of a hydrogen atom is a rational number when expressed in units of kilograms.

So when you ask "where do we find them," the answer is that we don't, and in fact this applies even to the real number system, and, arguably, the integers. (If you enthusiastically assert this philosophical position for the integers, you are what's known as an ultrafinitist.)

You ask "what is an example of one?" It is possible to make up number systems in which there are concrete examples of infinitesimals. A pretty rich system of this type is the Levi-Civita numbers. Here is a calculator I wrote that allows you to play with the Levi-Civita numbers: http://lightandmatter.com/calc/inf/ . However, when you do calculus it is never necessary or desirable to define a specific, concrete infinitesimal. This is really not so different from the real numbers. The vast majority of real numbers can never be defined, because there are uncountably many real numbers, but only countably many definitions.

 

[zinq]

What "assumptions" am I making? All I did was ask questions.

 

[bcrowell]

Which, unfortunately, rather suggests that the Riemann integral can't be defined in terms of standard Real Analysis.

The Riemann integral most assuredly can be, and is, defined in terms of standard real numbers.

Both of these statements are incorrect. It is possible to give a definition of the integral in either style. See the Keisler book for a definition in the style where infinitesimals are used.

 

[bcrowell]

What "assumptions" am I making? All I did was ask questions.

Questions can imply assumptions. For example, if I ask whether dead babies go to heaven, I'm implying a bunch of assumptions about the existence of the soul and life after death. My #24 explains the assumptions that, in my opinion, are implied by your questions. If you feel that the assumptions I imputed to you were not ones that you were making, then that's fine, but #24 also answers your questions.

 

[PeroK]

Both of these statements are incorrect. It is possible to give a definition of the integral in either style. See the Keisler book for a definition in the style where infinitesimals are used.

It's just logic. If the dx in the Riemann integral is an infinitesimal and if infinitesimals are not real numbers, then the Riemann integral requires surreal analysis (I think that's the correct term). If the Riemann integral is developed using standard real analysis, then the dx cannot be an infinitesimal.

A better example, perhaps is $dy/dx$. The simplest way to handle that is to say its "just notation". It's enormously powerful and extremely useful notation. If you want to give that meaning as a ratio of infinitesimals, then you need to go beyond real analysis. I think in real anaysis "dx" is undefined.

For example, you could prove the chain rule by:

$\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt}$ simply by cancelling infinitesimals.

But, that's not a valid standard real analysis proof, which is considerably more difficult.

 

[bcrowell]

It's just logic. If the dx in the Riemann integral is an infinitesimal and if infinitesimals are not real numbers, then the Riemann integral requires surreal analysis (I think that's the correct term). If the Riemann integral is developed using standard real analysis, then the dx cannot be an infinitesimal.

I think you're saying two things here: (1) If you don't use infinitesimals, then dx isn't infinitesimal. (2) That the notation somehow constrains us to use one set of foundational principles rather than the other. #1 is trivially true, while #2 is false, since the Leibniz notation is used with a variety of foundational approaches.

BTW, surreal analysis is not the correct term here, since the surreal numbers don't have convenient properties for doing analysis. You may have had in mind non-standard analysis (NSA). There is also smooth infinitesimal analysis (SIA).

A better example, perhaps is $dy/dx$. The simplest way to handle that is to say its "just notation". It's enormously powerful and extremely useful notation. If you want to give that meaning as a ratio of infinitesimals, then you need to go beyond real analysis. I think in real anaysis "dx" is undefined.

You can define it as just notation, or you can define it using a variety of other approaches, including forms, NSA, and SIA. In treatments based on the real number system, dx may be either defined or undefined by itself. If you use forms, then it's defined by itself.

For example, you could prove the chain rule by:

$\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt}$ simply by cancelling infinitesimals.

But, that's not a valid standard real analysis proof, which is considerably more difficult.

It is possible to give a proof in this style using approaches such as NSA. Neither style is really any more or less difficult than the other. For a proof using NSA, see p. 86 of Keisler (ch. 2). A couple of complications arise, but they can be dealt with. If you use limits, you have to deal with some very similar technicalities. One thing that can cause confusion is that in NSA, the derivative is not really defined as dy/dx but rather as the standard part of dy/dx. A second issue is that we have to consider the possibility that dx/dt=0, so that if dt is a nonzero infinitesimal, the standard part of dx is zero, and then the algebra becomes invalid because we're dividing by zero.

 

[zinq]

As some may not be aware, the "Riemann integral" in mathematics is a very specific definition of how to define the integral of a function. As such it is not subject to redefinition with infinitesimals or with anything else.

The usual term for doing analysis with infinitesimals is "nonstandard analysis".

The surreal numbers are extremely convenient for doing analysis, and much has been written about how to go about this. Here is one of the first things that popped up via Google: Analysis on Surreal Numbers.

It is a famous fallacy to invent something that someone else did not say and rebut it, sometimes called the "straw man" fallacy.

 

[bcrowell]

As some may not be aware of, the "Riemann integral" in mathematics is a very specific definition of how to define the integral of a function. As such it is not subject to redefinition with infinitesimals or with anything else.

The same definition can be given in more than one way, so that they're equivalent.

The surreal numbers are extremely convenient for doing analysis, and much has been written about how to go about this. Here is one of the first things that popped up via Google: Analysis on Surreal Numbers.

No, this is incorrect. It's true that it's possible to do analysis using the surreals, but they do not have convenient properties for that purpose. One problem with the surreals is that they don't have the transfer principle. Therefore when you want to generalize objects from the reals to the surreals, you have to do work on a case-by-case basis that wouldn't be necessary with NSA. For instance, the definition of exponentiation is highly nontrivial for the surreals, and was not worked out until fairly recently. There are a lot of cases where proving the existence of things in the surreals is much, much harder than it is with NSA. There are actually some links between the surreals and NSA; see here, for example. But those links require some very high-powered math even to describe. Basically NSA is the unique system that's big enough to do all of analysis, but not so big as to be unwieldy like the surreals.

It is a famous fallacy to invent something that someone else did not say and rebut it, sometimes called the "straw man" fallacy.

If you think I've done that, feel free to point out where.

 

[zinq]

Surreal numbers were invented about 1972 and as far as I know first appeared in print in 1974 in Donald Knuth's book. Then Martin Kruskal defined exponentiation for the surreals in the mid-1970s — I was at a lecture he gave with this. An equivalent definition appeared in Harry Gonshor's book on surreal analysis published in 1986.

The transfer principle occurs in nonstandard analysis, but not in standard analysis, where it is not needed.

 

[bcrowell]

Surreal numbers were invented about 1972 and as far as I know first appeared in print in 1974 in Donald Knuth's book. Then Martin Kruskal defined exponentiation for the surreals in the mid-1970s — I was at a lecture he gave with this. An equivalent definition appeared in Harry Gonshor's book on surreal analysis published in 1986.

Yes, I'm describing 1986 as relatively recent. I guess I'm just old.

 

[micromass]

No, this is incorrect. It's true that it's possible to do analysis using the surreals, but they do not have convenient properties for that purpose. One problem with the surreals is that they don't have the transfer principle.

Your own link later on say that the surreals DO have the transfer principle. See http://www.ohio.edu/people/ehrlich/Unification.pdf

 

[micromass]

You seem to be making some assumptions about what a number is, how it relates to the real world, and the role of the real number system. If you examine these assumptions carefully, they don't hold.

The real numbers are not "really" "real" in the sense of relating perfectly to the "real" world. For example, the real number system has a distinction between rational and irrational numbers, but such a distinction can never be empirically verified for any real-world measurement. For example, there is no way, even in principle, to determine whether the mass of a hydrogen atom is a rational number when expressed in units of kilograms.

So when you ask "where do we find them," the answer is that we don't, and in fact this applies even to the real number system, and, arguably, the integers. (If you enthusiastically assert this philosophical position for the integers, you are what's known as an ultrafinitist.)

You ask "what is an example of one?" It is possible to make up number systems in which there are concrete examples of infinitesimals. A pretty rich system of this type is the Levi-Civita numbers. Here is a calculator I wrote that allows you to play with the Levi-Civita numbers: http://lightandmatter.com/calc/inf/ . However, when you do calculus it is never necessary or desirable to define a specific, concrete infinitesimal. This is really not so different from the real numbers. The vast majority of real numbers can never be defined, because there are uncountably many real numbers, but only countably many definitions.

I think you're missing zinq's post. I guess he wants to know why a system of inifinitesimals as described in Keisler is a consistent set of axioms. This is an incredibly tough question. You would first need to construct the reals rigorously, which is not easy. Then you would have to apply to the axiom of choice to define the hyperreals adequately (which has as consequence that the hyperreal number system cannot have a specific definition). And then you'll want to prove the transfer principle which is even more horrible.

Doing analysis the standard way is simply way easier since you'll only have to construct the reals and you can go from there. Sure, NSA is easy once you sweep all the annoying issues under the rug.