Understanding differentials in Calculus 1?

[Matriculator]

I'm taking a short Calculus session this summer and the teacher zooms through things. I still don't fully understand differentials. I know that derivative give you the slope of a function at any point. And I know that dy is a small change in y and dx is a small change in x and how they can be used to approximate things. But is that all there is to it? The fact that dy is a small change in y and dx is small change in y? I noticed that they're used in integrals(we just started learning about them) when referring to what we're integrating with respect to. Why did we use dy/dx is derivative but dx in integrals? Thank you.

 

[micromass]

I like to make one thing very clear. There is no such thing as $dx$. It is merely a symbol. It does not represent a "very small" number, it does not represent any number at all.

The notation $\frac{df}{dx}$ is not a fraction. It is a notation. The notation $\int f(x)dx$ is also just a notation.

Do not think of $dx$ as a number. It will confuse you a big deal. And it is not how mathematics sees it. In fact, calculus doesn't work with $dx$ at all except in things like $\frac{df}{dx}$ or $\int f(x)dx$. And even there, it is just a notation.

That said, there is a way to give meaning to $dx$ as a differential form. This is done in differential geometry. But I don't want to confuse you even more.

 

[Matriculator]

Oh wow thank you so much for this. Why don't teachers tell us this? I was initially told that dy/dx isn't a ratio, then in differentials I was told that it was and dx was a small change in x. So when I saw it in integrals I thought that we were multiplying what we were to integrate by a small change in x lol. Thank you again for the clarification.

 

[Bacle2]

The differential of a differentiable function is the linear function that (locally) approximates the change of the function at a point x. Basically, differentiability of f at x means that _the change of f_ can be locally described by a linear function. As an example, for f(x)=x², we have that f'(x)=2xdx , which means that the (linear) function 2x approximates as closely as you want ( in a precise δ-ε sense), the change of f(x)=x² near a point, and the change of f(x)=x² along a small amount dx of x can be approximated as well as you want (δ-ε) by making the change in x as small as you want--check it out by plugging-in small changes in x, and comparing the change of x² between x and x+Δx as Δx becomes small, and the change of 2x in the same interval.

The symbol dy stands for the change _along the tangent line to y=f(x)_ , while dx r
epresents the change along the x-axis. OTOH, Δy , Δx represent the _actual changes_ and not the linearized ones. If the limit exists, then the derivative, and so the local-linear approximation, also exists.

 

[micromass]

Oh wow thank you so much for this. Why don't teachers tell us this? I was initially told that dy/dx isn't a ratio, then in differentials I was told that it was and dx was a small change in x. So when I saw it in integrals I thought that we were multiplying what we were to integrate by a small change in x lol. Thank you again for the clarification.

I think it's very wrong to teach $dx$ as a small change. I understand why they do it, but I think it's a disservice to the students.

I assume you've seen limits before? Fix a point $a$. We can define $\Delta x = x-a$ and $\Delta f = f(x) - f(a)$. This is a historic notation. Then we define

\frac{df}{dx} = \lim_{x\rightarrow a} \frac{\Delta f}{\Delta x}

This explains the notation for a derivative. So the $dx$ is not a small change in itself, but should be seen a limiting value of small changes.

When doing things intuitively, it is not wrong to think of $df$ and $dx$ as very small infinitesimal changes. This is a useful thing to do in physics. But I wish to stress that this is not at all rigorous, and not at all correct. It's a very good intuition, but incorrect. (Yes, incorrect things can be good intuition)

 

[verty]

To find the slope of a curve, we must find the slope of the tangent to the curve. The slope of the tangent = $\frac{dy}{dx}$, the limit of $\frac{Δy}{Δx}$.

Now suppose we allow infinitesimals, then there is a question of what the angle is between a tangent and a curve. It may be infinitesimal, in which case there is an infinity of tangents with different infinitesimal angles. Which one is THE tangent? Which do we use to define $\frac{dy}{dx}$? Radius of curvature is problematic too, we can't say what the curvature of a curve is.

So there are major problems to be overcome with infinitesimals. Mathematics moved on without them, although in the previous century they were reborn as Non-Standard Analysis.

 

[WannabeNewton]

The notion of $dx$ being a "small change" is something that is prevalent in physics texts; it is used because it helps with calculations and derivations. It is by no means rigorous (far from it) so take it with a grain of salt. There is a mathematically precise definition of what $dx$ is from a geometric point of view but don't worry about that for now.

 

[AdkinsJr]

The differentials take on an actual value if you define or assign a value to the other differential and the derivative right?

If I have y=x^2 then \frac{dy}{dx}=2x and then dy=2x dx then you have dy in terms of the derivative and the differential.

 

[lurflurf]

^Right given y=x^2 we define two functions (R^2->R)

Δy=(x+dx)^2-x^2=2x dx+dx^2
and
dy=(x^2)'dx=2x dx
so
Δy-dy=dx^2

so dx can be a small number, it can also be a large number
often we write Δy~dy
that is they are equal in some sense
that sense is locally
if dx is large they are not close numerically
if dx=10^-6
Δy-dy=dx^2=10^-12
if dx=10^6
Δy-dy=dx^2=10^12

A geometric interpretation is that the differential defines the tangent line.
The tangent line closely matches the function for small dx.
The tangent line exists for large dx, but we do not expect a close fit.

 

[verty]

^Right given y=x^2 we define two functions (R^2->R)

Δy=(x+dx)^2-x^2=2x dx+dx^2
and
dy=(x^2)'dx=2x dx
so
Δy-dy=dx^2

so dx can be a small number, it can also be a large number
often we write Δy~dy
that is they are equal in some sense
that sense is locally
if dx is large they are not close numerically
if dx=10^-6
Δy-dy=dx^2=10^-12
if dx=10^6
Δy-dy=dx^2=10^12

A geometric interpretation is that the differential defines the tangent line.
The tangent line closely matches the function for small dx.
The tangent line exists for large dx, but we do not expect a close fit.

No one should write Δy ~ dy, this is wrong because it implies dy = 0. That is not true. And it is wrong in an irreconcilable way. DY is not an infinitesimal, forget this idea. It is a symbol that has meaning only in context, like when an integral sign precedes it.

$dy = 2x dx$ is short for $\frac{dy}{dt} = 2x \frac{dx}{dt}$ for some implicit variable $t$. It means, the rate that y changes is 2x * the rate that x changes.

I'm reporting the above post for being misleading after having been told it was so.

 

[micromass]

No one should write Δy ~ dy, this is wrong because it implies dy = 0. That is not true. And it is wrong in an irreconcilable way. DY is not an infinitesimal, forget this idea. It is a symbol that has meaning only in context, like when an integral sign precedes it.

$dy = 2x dx$ is short for $\frac{dy}{dt} = 2x \frac{dx}{dt}$ for some implicit variable $t$. It means, the rate that y changes is 2x * the rate that x changes.

I'm reporting the above post for being misleading after having been told it was so.

Verty, I absolutely agree with you. The notation is extremely misleading and wrong. It does a great disservice to the students who wish to learn calculus.

However, it appears that some calculus books, like Stewart, do teach things this way. I personally think that Stewart is a horrible book and should never be used in a classroom. But it is unfortunately a standard book.

I can only ask the OP to ignore the misleading post.

 

[lurflurf]

No one should write Δy ~ dy, this is wrong because it implies dy = 0. That is not true. And it is wrong in an irreconcilable way. DY is not an infinitesimal, forget this idea. It is a symbol that has meaning only in context, like when an integral sign precedes it.

$dy = 2x dx$ is short for $\frac{dy}{dt} = 2x \frac{dx}{dt}$ for some implicit variable $t$. It means, the rate that y changes is 2x * the rate that x changes.

I'm reporting the above post for being misleading after having been told it was so.

Verty, I absolutely disagree with you. Why are you talking about infinitesimals? dy is some number. It has a meaning without integral signs. You then go on to use it yourself. I write Δy ~ dy to avoid confusion with global equality, the equality is local. Though it would be correct to use = as others do. It is rather important to not think of dx as needing to be small, though that is often the region of interest.

You are confused. See for example
http://en.wikipedia.org/wiki/Differential_of_a_function
or
any calculus book ever

 

[micromass]

Verty, I absolutely disagree with you. Why are you talking about infinitesimals? dy is some number. It has a meaning without integral signs. You then go on to use it yourself. I write Δy ~ dy to avoid confusion with global equality, the equality is local. Though it would be correct to use = as others do. It is rather important to not think of dx as needing to be small, though that is often the region of interest.

You are confused. See for example
http://en.wikipedia.org/wiki/Differential_of_a_function
or
any calculus book ever

The mathematical rigorous definition treats $dx$ and $df$ as functions. Not as numbers. See any differential geometry book and many analysis books, for example Spivak's calculus on manifolds (or the very wiki you linked).

I propose we start referring to rigorous math books on the topic, and not wiki links. Any confused calc student can edit a wiki link.

I don't care what rubbish books like Stewart use. If they set the standard for mathematical concepts nowadays, then it's a sad world.

 

[WannabeNewton]

If you open up a book on thermodynamics, you'll see lurf's usage of the differential quite commonly. It is just a difference in the usage of the differential; it is often simply convenient to think of it in that way because it works brilliantly in calculations. The rigorous definition of the differential is quite useless at that stage and not to mention much harder to visualize than the notion of the differential as a "small quantity of such and such". As long as the student knows the difference between a non-rigorous but highly effective computational tool and a rigorous definition of the differential, I don't see immediate harm. If everyone was rigorous all the time then much of the practical stuff would never get done. As wiki puts it: the precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor.

I find it a bit contentious to report someone for something that actually exists in various texts. It's not like he/she is pulling things out of a hat.

 

[micromass]

I'm reporting the above post for being misleading after having been told it was so.

I find it a bit contentious to report someone for something that actually exists in various texts. It's not like he/she is pulling things out of a hat.

Let's keep things on-topic here and let's not resort to insults.

 

[Stephen Tashi]

I find it a bit contentious to report someone for something that actually exists in various texts. It's not like he/she is pulling things out of a hat.

It's an interesting cultural phenomenon that certain topics that don't pass muster as logical reasoning are traditionally given a pass in discussing mathematics. Many discussions about statistics illustrate this. Students get a slap on the hand for a wrong epsilon-delta proof of Lim_{x \rightarrow 1} 2x + 1 = 3, but differentials and infinitesimals slip by without similar scrutiny. Proofs of trigonometric identities are tradictionally written backwards. Many texts state definitions in an "if...then..." form when they actually mean "if and only if". (e.g. "If the group operation is commutative then we say that G is an abelian group.")

The presence of differentials and infinitesimals in textbooks is a strong argument that these topics are important from a sociological point of view - given that mathematics as practiced by humans is not a completely logical adventure.

However, this doesn't change the slightly embarassing fact that teachers expect one standard of rigor when they ask a student "What is the definition of a derivative of a function?" and a completely different standard if they ask "What is the definition of a differential"? In the math sections of the forum, I think it's appropriate to make it clear that differentials, from a calculus 1 perspective, do not have a precise definition. The fact that many people are eager to weigh-in on threads like as "Are dx and dy numbers?" "Is dy/dx a ratio?" indicates that many people wish to be helpful (and that many of them have strong private opinions), but it shouldn't be taken to mean that there is actually a precise and logical definition and development of differentials in Calculus I.

 

[lurflurf]

Widely used calculus books must be mediocre.

Lets not be needlessly pedantic. Something can be a function and a number like 5. Clearly there is a slight abuse of notation in declaring a variable is a function of another or using f(x) to denote both the value of f at x and the function f itself. It presents no problem. In fact a great advantage of differentials is that they are coordinate invariant. One can learn a lot by looking though 2^7 or so calculus books. The good ones and bad ones both define differentials. This is a frequently misunderstood topic for reasons I don't understand. I think the explanation in most books is perfectly understandable and logical. The differential is a linearization of the function determined by the function in the neighborhood of a point. This fits quite well with the notion in differential geometry, though often differential forms are used to make higher differentials coordinate invariant by making them zero. I often tell confused calculus students to try reading a book one level up, but I think in many cases reading differential geometry books might lead to more confusion. Stewart might be better book than I gave it credit for. Maybe Micro can make a ten best and ten worst calculus book thread.

 

[Stephen Tashi]

I think the explanation in most books is perfectly understandable and logical.

I don't understand the population from which the statistic is taken. Are you talking about most Calculus I books?

The differential is a linearization of the function determined by the function in the neighborhood of a point. This fits quite well with the notion in differential geometry, though often differential forms are used to make higher differentials coordinate invariant by making them zero.

I don't think anyone disputes that differentials can be given a rigouous definition in an advanced context. It isn't being excessively pedagogical to convey to a Calculus I student that differentials are not precisely defined in that course - a student shouldn't worry that a page was torn out of his book.

Can we give a Calculus student a version of the rigorous definition that provides a unified view of all the various situations where differentials appear, dy/dx , \int f(x) dx, "total differential", "area element"? I'm not posing this question as a debating point. I'm actually curious how it could be done.

 

[bolbteppa]

This video

https://www.youtube.com/watch?v=iyIVxXB5kBc

gives a overview of what's going on with the 'dx' term, as does this article, though use them merely as motivation & as a means to annoy your lecturer

No one should write Δy ~ dy, this is wrong because it implies dy = 0. That is not true. And it is wrong in an irreconcilable way. DY is not an infinitesimal, forget this idea. It is a symbol that has meaning only in context, like when an integral sign precedes it.

$dy = 2x dx$ is short for $\frac{dy}{dt} = 2x \frac{dx}{dt}$ for some implicit variable $t$. It means, the rate that y changes is 2x * the rate that x changes.

I'm reporting the above post for being misleading after having been told it was so.

If an author as esteemed as Gelfand is willing to make multiple approximations of the kind \Delta y \approx dy in, for example, his proof of Noether's theorem in a calculus of variations context then approximations of this kind are just something you'll have to live with. Constructing things in terms of differentials rigorously is methodical worker-bee stuff that merely codifies the intuition of geniuses seeing approximations like these, & acts as a means to nullify incorrect intuition, so I would say eschewing the intuitive idea is probably far more harmful than good, it's a tool one should become proficient in.

 

[WannabeNewton]

...so I would say eschewing the intuitive idea is probably far more harmful than good, it's a tool one should become proficient in.

Well said my friend. This is what I was saying as well. Perhaps physics majors just have a different mindset than math majors; I guess this relates back to Stephen Tashi's comments. Sometimes calculational prowess is more important than rigor and other times rigor is more important than calculational prowess; it would depend on the situation at hand in my opinion. If I argued that only the rigorous notion of the differential should show up in calculus contexts then I would personally be a big hypocrite because I use the idea that the differential is a "very small amount of such and such" all the time when solving say EM problems.

 

[verty]

I did not know about the Cauchy definition of dy, this is new to me. Honestly, I have always thought of $dy = 2x dx$ as being about rates of change. I think this helps with word problems like related rates problems to get over the hurdle of finding the formulas. I apologize to Lurflurf for reporting his post, given that it is recognized mathematics.

Lurflurf mentioned local equality, that one can write Δy = dy locally. Surely this locally means in an infinitesimal neighborhood of the point. The idea that a curve looks flat if we zoom in far enough was I believe the original motivation for infinitesimals, but modern intuition is that you can zoom in as far as you like and the curve is still curved. For example, the circle and it's tangent meet in a single point, there is no local region of flatness. Zooming it will leave it curved, or in the limit it will become the tangent, but at no stage does it look like a polygon. I think this is good, we should not give this up.

 

[verty]

What I mean to say is, these intuitions are good, like the intuition that curves are not infinite polygons, and as far as learning goes, we should keep them rather than follow wikipedia and give impossibly terse definitions to learners.

Let's move on.

PS. Now I can guess what was meant by the phrase "the machinery of differential forms".

 

[Mandelbroth]

Constructing things in terms of differentials rigorously is methodical worker-bee stuff that merely codifies the intuition of geniuses seeing approximations like these...

I think not. Formal proof requires far more genius than approximation. For example, one might say that Fermat had only marginal genius.

I'm sorry. I had to say that.

The key point is that math is a rigorous subject. Intuition is good, but universal acceptance is better. Rigor is by far preferable because it is always correct, while intuition can be wrong. There was another thread asking about differentials, where the OP thought that $0.\bar{9}$ was only infinitesimally close to 1. Intuition can lead you to wrong answers. Rigor makes things more accurate.

That said, I thought your video was entertaining.

Lurflurf mentioned local equality, that one can write Δy = dy locally. Surely this locally means in an infinitesimal neighborhood of the point. The idea that a curve looks flat if we zoom in far enough was I believe the original motivation for infinitesimals...

Actually, you seem to have intuitively described a 1-manifold, which is rather intrinsic to discussion of differentials in single variable calculus.

 

[verty]

Actually, you seem to have intuitively described a 1-manifold, which is rather intrinsic to discussion of differentials in single variable calculus.

I think I understand. In a more general setting, there are many different manifolds to apply analysis to, and in each case the assumptions are specific. At a higher level, intuition is not obsoleted but parametrized?

I am put at ease by this, thanks.

 

[verty]

Sorry, I should actually respond to Lurflurf's point that calculus books define the differential. Spivak's Calculus does not define the differential and MIT's online course materials do not define the differential. Spivak, the author of a series of books on differential geometry, chose not to define the differential. MIT, who surely have some of the highest regarded lecturers around, choose not to define the differential. I'm saying it is not a given that books define it. Not wanting to restate what was said before, this is all I have to say about this.

 

[WannabeNewton]

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-2-applications-of-differentiation/part-c-mean-value-theorem-antiderivatives-and-differential-equations/session-36-differentials/MIT18_01SCF10_Ses36b.pdf
http://math.berkeley.edu/~buehler/1A-Fall2012/Worksheet17-Solutions.pdf
http://isites.harvard.edu/fs/docs/icb.topic643441.files/Chapter%203/Linear.pdf
http://www.math.upenn.edu/~jbandlow/math103/2011_02_24_related_rates_and_lin_approx.pdf
http://emp.byui.edu/BrownD/Mathematics/Calculus-Rn-Rm/Increments-Differentials-and-Local-Linear-Approximations.pdf

Someone better warn the students taking these classes!

 

[verty]

What those students may be missing, if they don't know it, is that we can do differentiation like this:

$y = x^2$
$dy = 2x dx$

I have differentiated both sides against an unknown variable. Therefore, dy and dx can be seen as rates with respect to an unnamed variable.

$\int{2x}dx$
$= \int D(x^2)$
$= x^2 + C$

That second line is not strictly correct but is helpful to illustrate that a rate interpretation gives tremendous insight. Doesn't this make integration look obvious? An integral is just a starting value plus the total change.

Is this not the best way for learners to think about differentials?

 

[Stephen Tashi]

I have differentiated both sides against an unknown variable. Therefore, dy and dx can be seen as rates with respect to an unnamed variable.

To be explicit about that thought, pretend
y = y(t)
dy = y'(t)
x = x(t)
dx = x'(t)
y = x^2
Interpret
dy = 2x dx as y'(t) = 2 x(t) x'(t)

I also would like to hear comments about this way of thinking since I often use it.

 

[lurflurf]

I don't understand the population from which the statistic is taken. Are you talking about most Calculus I books?

Both calculus I books and calculus II books. Compared to other topics this seems to me to be well explained in the usual books and also commonly misunderstood. Perhaps it is because some books and lecturers make differentials sound mysterious. Then again most calculus I books explain most topics badly so it is a low bar.

I don't think anyone disputes that differentials can be given a rigouous definition in an advanced context. It isn't being excessively pedagogical to convey to a Calculus I student that differentials are not precisely defined in that course - a student shouldn't worry that a page was torn out of his book.

Differentials can, should, and sometimes are precisely defined in calculus I. The differential of a function is its linearization. Perhaps another problem is that mathematicians historically did some questionable things with differentials, and although that is all worked out now a number of different formulations are used, leading to confusion.

Can we give a Calculus student a version of the rigorous definition that provides a unified view of all the various situations where differentials appear, dy/dx , \int f(x) dx, "total differential", "area element"? I'm not posing this question as a debating point. I'm actually curious how it could be done.

The problem is the same symbol means different things in various situations. As I mentioned above, without context we might not know what dx is in a given situation for example sometimes dx^2=0 and sometimes it does not. Another problem that is really a strength is a differential is an invariant. That is it does not stand for one formula, but a family of formula. This is very useful as often we only care that it is possible to relate some variables by functions, we do not care specifically how it is done. I support covering differential forms in elementary calculus, if this is done it needs to be decided if we should take d^2x=dx^2=0 as universal or also allow other types of differentials. A few interesting examples can be given
if we consider discontinuous functions we might have
du=1 so that
$$\lim_{\mathop{dx} \rightarrow 0} \dfrac{du}{dx} \rightarrow \infty$$
That is to be expected as a function should not be differentiable at a discontinuity
if f(x,y,z)=0 we see that
$$\left( \dfrac{\partial z}{\partial x} \right)_y \left( \dfrac{\partial x}{\partial y} \right)_z \left( \dfrac{\partial y}{\partial z} \right)_x =-1$$
One problem here is the common abuse of letting the same symbol represent a variable and a function. The bigger problem is that the functions are dependent.

dy = 2x dx as y'(t) = 2 x(t) x'(t)
is surely sensible as are
$$ \dfrac{dy}{dt} = 2x \dfrac{dy}{dt} \\
\text{and} \\
\dfrac{dy}{dt} \mathop{dt}= 2x \dfrac{dy}{dt} \mathop{dt}$$

It is often said that different notations should be thought of as writing the same thing different ways. That is a missed opportunity. A good notation like differentials help ones thinking. I think differentials help by encouraging thinking about linear functions, making the chain rule seem more natural, and freeing thoughts of the functions that link variables. To others these are probably things to be avoided.

Sorry, I should actually respond to Lurflurf's point that calculus books define the differential. Spivak's Calculus does not define the differential and MIT's online course materials do not define the differential.

Quite true, I should have said it is defined in many/most any calculus books or a definition of differentials is often included. That was in response to to someone claiming ignorance of the fact that differentials are a common calculus topic. Funny that you mention Spivak as I could not remember if it was in there. It is a strange book.

Formal proof requires far more genius than approximation...
Intuition is good, but universal acceptance is better. Rigor is by far preferable because it is always correct, while intuition can be wrong.

Approximation and rigor are not mutually exclusive. Mathematics is a dynamic and nonlinear activity. Often when solving a problem less rigorous methods are used first as they are quick and powerful. Later more rigorous methods are used. Often using rigorous methods first is slower or fails. We might say "Find the answer and prove the answer is correct." works much better than "Find the answer rigorously."

 

[Mandelbroth]

$\int{2x}dx$
$= \int D(x^2)$
$= x^2 + C$

That second line is not strictly correct but is helpful to illustrate that a rate interpretation gives tremendous insight.

Your second line there isn't correct, but $\int d(x^2)=x^2+C$ for some $C\in\operatorname{ker}\left[\frac{d}{dx}\right]$ is.

Approximation and rigor are not mutually exclusive.

True, but of the two, I would pick rigor over approximation any day of the week. I prefer the foundation of my math to be logic rather than handwaving.