I am confused about differentials

[Lucien1011]

Hi everyone!
I want to ask something about differentials.
I often visit this forum and I saw people write that dx is something infinitesimal.
Well but i read some books about differentials.Some of them define dx equals [delta x] and some of them seem not to consider dx as infinitesimal thing.
Is there a contradiction? Which one is true? Can anyone give a original definition for differentials?

Any reply or materials are appreciated! Thank you very much!

 

[Kummer]

Can anyone give a original definition for differentials?

A "differential" is some crazy concept that physicists use. It really has no mathematical definition. In fact, mathematicians just avoid to use them.

 

[HallsofIvy]

As much as it hurts me, I must disagree with Kummer. The "differential" is, in fact, a perfectly good mathematical definition. It is not, however, "dx equals [delta x]"- that may well be a definition in a Physics text, which are notoriously sloppy as to mathematical concepts (according to mathematicians). If y= f(x), then "dx" is defined as a symbol representing an infinitesmal change in x (it is possible to define "differential" in specfic terms but it involves some very deep concepts of logic, most of us just accept "dx" as purely symbolic. "dy" is then defined by "If y= f(x) then dy= f'(x) dx". It is a very useful, though a bit "fishy" concept.

In differential geometry, the differential really comes into its own.

 

[matt grime]

The notions of dx etc have perfectly good formal definitions (properly as differential forms, or formal derivatives). This is not to be confused with their use in certain parts of applied mathematics where the 'notion' of 'infinitesimal' is used to great effect. Of course some parts of applied mathematics use the proper formal stuff as well.

 

[Kummer]

I was referring to how "differential" are presented in a Calculus text. A more formal approach would be a http://en.wikipedia.org/wiki/Differential_form" .

 

[Lucien1011]

OK I get you
but why do dx in physics text equals [delta x]?
I think there should be some relationships between the definition for mathematicians and then definition in physics text.
Honestly I am reading a book writing "dx equals [delta x]". It is called "Mathematics Methods in the Physical Sciences" written by Mary L.Boas. Here is the page talking about this.
Well if dx is the infinitesmal change in x, how can it equal [delta x]? I start to get more confused.

 

[jostpuur]

The definition given in that page seems to be an ok definition, although it does not agree with other definitions given in other sources. Welcome to the definition jungle, Lucien1011.

 

[animalcroc]

Equation 2 is a standard method of introducing the concept of differentials. But I don't like the paragraph where she writes "but dy is not the same as delta y". I think that should be reworded.

 

[animalcroc]

dx is an infinitesimal delta xx

 

[LorenzoMath]

In physics, you guys (I'm doing arithmetic geometry.) usually take limit after arguments in which you put physical phenomena into mathematical framework, so it is relatively OK to confuse {delta}x and dx.

In mathematics, I consider a differential d as follows:

Let A and B be rings. B is an A-algebra. Define a B-module W as a B-module generated by { dx | x is in B} with relations dx = 0 if x is in A and d(xy)=ydx+xdy. Then d can be seen as a canonical map from B to W.

Example 1:
X, a diff mfd over R.
Set B to be the R-algebra of all differentiable function on X to R, and
W to be the R-algebra of all 1-forms on X.
Then d:B->W is a differential.

Example 2:
Replace X in Example 1 with R^n. This is the situation you use in classical mechanics and thermodynamics. In physics, you integrate df right without being given sufficient explanation (if you are freshman or sophomore) why you can do such a thing, don't you? Don't worry. There is a whole theory of integration of forms worked out.

 

[LorenzoMath]

You won't need a rigorous definition of a differential until you study general theory of relativity. (In this case Example 1 is needed. Not the full definition I gave above.)

(oops. i made a mistake in Example 1. W is supposed to be the B-module of all 1-forms)

anyways, even when you study GR, there are a lot of excellent textbooks like the thick black one by wheeler and taylor and thorne, and the small green one by I-forgot-who-wrote-that, and the one by Sean Carol. Weinberg is "algebraic", though... but geometry is underlying. I think Weinberg claims in the preface of the book that his approach is heterodoxical, but it seems to me that the book is just a rewriting of the traditional GR in the lagauge of "algebra". But his chapter on cosmology is coooool! ASTEEEG!

 

[Lucien1011]

Well I feel much much better now
Thank you very much!

 

[smallphi]

A differential of a function is simply the linear term in the Taylor expansion of a function:

f(x) - f(xo) = f ' (xo) (x-xo) + f '' (xo)(x-xo)^2 / 2 + ...

The left hand side is delta_f. The first term on right hand side it df. You see that they are equal only to first order, the differential simply captures the biggest/ the linear part of delta_f and that can be seen from the graphical interpretation where delta_f is the actual change in the function while df is the change calculated by substituting the function with its tangent at point xo. They will be equal only if the higher order corrections are zero i.e. when the second and higher order derivatives of the function are zero i.e. only when the function is linear, f(x) = a x + b.

For example, if you take the function f(x) = x, the second and higher order derivatives are zero. That's why, for this function, delta_f = df i.e. delta_x = dx.

 

[mathwonk]

think of a curve. a finite piece of that curve is called deltay. if you think of the curve as made up of an infinite number of infinitely short pieces, each one of which is straight, but infinitely short, like a polygon with an infinite number of super short sides, then one of those tiny sides is called dy.now if y is linear, then not only dy, the infinitely short piece of curve, but also deltay, the finite piece of curve, are both straight. that's why, since x is a linear function of x, that deltax is hard to distinguish from dx.

now since the language above is logically nonsense, to make sense of it we try to associate in some meaningful way, a straight line to each point on a curve, and of course we use the tangent line at that point, since it supposedly has the same direction as an infinitely short piece of our curve.

so at each point of a curve, dy should somehow represent the tangent line to the curve y(x) at that point.

so here we go with the correct, but complicated meaning of dy.

if y(x) is a function of x, then deltay is a function of two variables, whose value at x,h is y(x+h)- y(x).

dy is also a function of two variables, whose value at x,h is y'(x).h.

then at the point x, deltay and dy are both functions of h, and if y is a differentiable function of x, then dy(x,h) is a good approximation to deltay(x,h) when h is small.

since x'(x) = 1, this makes deltax and dx the same function of x and h, namely both equal h for every x.

but there is a more abstract version of dy too. namely, even if we do not have coordinates x chosen in our domain space, e.g if the x-axis has no unit chosen on it, then a function y still has a graph which is a curve in the x,y plane, and this curve can have a tangent line at each point. then dy is the function of x whose value at each point x is the linear function whose graph is the tangent line to the graph of y.so for each function f, df is a family of linear functions, one for each point of the domain of f, and at each such point p, the linear function df(p) is the function whose graph is the tangent to the graph of f at the point (p,f(p)).notice that this makes sense even if there are no units on the y axis, but deltay does not.see how much nicer it was to think dy was just an infinitesimally short, hence straight, piece of the graph of y?