Here is a multivariable Calculus textbook, intended for undergraduates who have had only one-variable Calculus and no more advanced mathematics, that covers the Stokes Theorems using differential forms. http://matrixeditions.com/UnifiedApproach5thedSamples.html

]]>Yes, when I saw that example on differentiable map, I was at first worried that I'd made a mistake, and I tried to put that example through my proof, which made me realize that it didn't apply.

]]>Very interesting!

]]>I should probably add to differentiable function my proof that this definition of differentiability is correct. It's stronger than requiring all directional derivatives and then requiring these to depend linearly on the direction. Since the derivative of $f \circ C$ depends on $C$ only through the derivative of $C$ and yet there exist nondifferentiable functions with linear directional derivatives (example: $y^3/x$ extended as $0$, at $(0,0)$), one might think that it would be insufficient to require $(f \circ C)'$ to depend on $C$ only linearly through the derivative of $C$. However, the claim that $(f \circ C)'$ depends on $C$ only through the derivative of $C$ fails for nondifferentiable functions with linear directional derivatives! (In the example, $(f \circ C)'$ is $0$ at $(0,0)$ when $C$ is a line but not, say, when $x = y^2$ on $C$.)

]]>The current term's Mulivariable Calculus course has a revised version of the introduction to differentials and $1$-forms, split into two parts. (Although I'm not really done rewriting the second part, it's acceptable, and I needed to hand it out in class today.) There will be more handouts.

This year features the general statement that a differential form is *any* expression with differentials in it, said with the confidence that I know how to define it if pressed! But in the first handout, $\mathrm{d}x$, $\mathrm{d}^2x$, etc are treated as independent variables in a formal expression, which I never really liked. Fortunately, there is a real definition in the second handout, although what it really does is to define equality as an equivalence relation on such formal expressions.

Edit: Also, the second handout formally defines a function $f$ on a Cartesian space to be differentiable at a point $P$ if $(f \circ C)$ is not only differentiable wherever the value of $C$ is $P$ and $C$ is differentiable there, but also depends only on the derivative of $C$ there (the velocity tangent vector) and depends on that linearly (stated as the existence of an appropriate row vector $\Del{f}(P)$). That actually came out looking simpler than I had originally anticipated!

]]>Thanks!

]]>Toby, you might have something to contribute here: http://matheducators.stackexchange.com/questions/2246/practical-experience-with-teaching-differentials-in-freshman-calc

]]>Just throwing this in the mix: http://mathoverflow.net/a/7632/4177

]]>OK, yes, your idea does work!

Specifically, define $\mathrm{d}f$ as the operation (in general partially defined) on differentiable parametrized curves (in a given Cartesian space, or more generally in a $C^1$ manifold) that takes $c$ to $(f \circ c)'(0)$ (if this exists). Also define $\mathrm{d}f(p)$ to be the restriction of that operation to curves with $c(0) = p$. Then $\mathrm{d}f(p)$ might be defined on all such curves, and (if so) it might respect the equivalence of curves that defines a tangent vector at $p$, and (if so) it might be linear and so a cotangent vector at $p$. If so, then $f$ is differentiable at $p$, as desired.

The proof is that the definition of differentiability itself calls for nothing more than this cotangent vector.

The next step is to make this into a definition of generalized differentiable (rather than smooth) space, by not using the previously known structure of tangent vectors.

]]>My working definition of ‘differential form’ (of rank $1$) in this class is a formal linear combination of differentials with coefficients from the ring of appreciable quantities (not stated like that, of course, but given by examples). Then $\mathrm{d}f$ is automatically the differential form desired (well, assuming that it exists, since I actually refuse to define $\mathrm{d}f$ until $f$ is known to be differentiable, but then your proposal is circular).

But ignoring that context, and defining a differential form abstractly as an operator on differentiable curves with appropriate properties (such as linearity), then the answer is Yes if you require $\mathrm{d}f$ to be a *continuous* differential form; and in that case, we can conclude that $f$ is *continuously* differentiable.

Without that, I don't know. In the example of $x, y \mapsto y^3/(x^2 + y^2)$, not only is $\mathrm{d}f$ not continuous, it's not linear (at $(x,y) = (0,0)$); if you apply it to a line through the origin with tangent vector $[a,b]$, then the result is $b^3/(a^2 + b^2)$. So this $\mathrm{d}f$ is not a differential form.

If $\mathrm{d}f$ has the properties of a differential form, does this guarantee that $f$ is differentiable in the standard sense? That would be nice! Is additivity sufficient? That would be particularly nice! I don't know.

]]>I presume it is also not sufficient to say that $f$ is differentiable if $f\circ c$ is differentiable for all $c$ and moreover there exists a differential form $df$ such that $\langle df(p)|c\rangle = (f\circ c)'(0)$? That is, you also need to ensure that the limits defining each derivative $(f \circ c)'(0)$ happen “simultaneously in all directions”?

]]>It seems that I went to far with the strategy of pushing everything back to curves, since $x, y \mapsto y^3/(x^2 + y^2)$ (continuously extended to the origin) is not differentiable at the origin (by the usual definition), even though its composite with any differentiable curve is differentiable (indeed continuously so if the curve is continuously differentiable).

Boman's theorem says that you can push things back to curves for *smooth* maps, and this is what really matters, so I may just do that next term, leaving the fine print for merely differentiable maps to the textbook.

Re #41, the new version of the notes on differentials for my Multivariable Calclulus class are done: check them out. I now feel like the end (where I get to this bit) is a bit anticlimactic, and I wonder if I should redo the whole thing *starting* with the action of differentials on curves.

Incidentally, higher differentials such as $\mathrm{d}^2 u$ (where remember we are *not* doing the exterior differential, which would just be zero, but rather something relevant to second derivatives) cannot be understood as acting on vectors (since they act on order-$2$ jets) but can be understood as acting on curves.

With the emphasis on curves, I suppose that I'm secretly doing calculus on diffeological spaces. (I remarked in class on Thursday that there are very general notions of ‘differentiable space’ even beyond the differentiable manifolds that one is likely to meet in an advanced course, but in theory everything in *this* course is done on open subspaces of $\mathbb{R}^n$ for $n = 1, 2, 3$.)

Concerning the analysis/geometry dichotomy, in view of algebra-geometry duality, this might also be seen as the analysis/algebra dichotomy. Then we have some interesting comments by Terry Tao, which I collected here. In particular, in the section ’Tao on Buzz’ he looks to characterise analysis and algebra in terms of open and closed conditions. There’s also something there on NSA.

]]>Zoran, thanks for taking the time to provide more pointers. But I think your choice of examples – e-g- nonstandard probability spaces – confirms that the distinction between analysis and differential calculus which Mike amplified above is relevant. Probability spaces are not a topic involving differential calculus.

For me that suggestion of Mike’s is a good conclusion of this little debate here, and I’d tend to leave it at that for the moment, since I should be looking into other things. If I had more time I would maybe add a little paragraph to this effect to the nLab entry.

]]>Just a sample set-theoretic treatise in the framework of nonstandard analysis.

]]>- V. A. Lyubetskiĭ,
*Оценки и пучки. О некоторых вопросах нестандартного анализа*, Uspekhi Mat. Nauk**44**(1989), no. 4(268), 99–153, 256; translation*Valuations and sheaves. On some questions of non-standard analysis*, in Russian Math. Surveys**44**(1989), no. 4, 37–112 MR1023104 doi IOP pdf rus pdf

]]>We present some parts of a mathematical theory that is sometimes called Heyting-valued analysis (or nonstandard analysis in the broad sense). Sometimes this theory is considered as a part of general topos theory. One may surmise that this theory has some applications outside mathematical logic as well: in algebra and analysis, and even in a still wider context, for example, as in A. Robinson’s well-known work on the application of nonstandard analysis in quantum field theory.

In Chapter I we present the actual method of Heyting-valued (in particular, Boolean-valued) analysis. Chapters II–IV contain specific examples of applications of the method of Heyting-valued analysis. In Chapter II we primarily consider the problem of the existence of a model companion of a locally axiomatizable class of rings. In Chapter III we consider a conjecture of P. S. Novikov [cf. Selected works (Russian), see p. 127, “Nauka”, Moscow, 1979; MR0545907 (80i:01017)]. In this chapter we discuss the transfer from classical to intuitionistic validity in an arbitrary ring. Novikov’s paper established the possibility of such a transfer in the case of the ring Z. In Chapter IV, we construct, for some rings of continuous Y-valued functions (as algebras over the ring Y), a nonstandard representation Y˜ such that in a certain sense this algebra is similar to its ring of scalars Y. The appendix briefly describes examples of applications of Boolean-valued analysis in connection with problems of duality. Practically all the theorems and propositions are given complete proofs.

for nonstandard analysis the main statement is that elementary calculus can be phrased this way. Is there anything that goes further?

Surely, there are advanced objects like Loeb nonstandard probability spaces, nonstandard set theory, transfer at the level of certain topoi in the picture etc. Nonstandard analysis is not *only* about analysis.

Thanks, Mike, for the analysis/differential geometry dichotomy. I’ll think about that.

I have added the references that you displayed to *nonstandard analysis – References*. Incidentally, that makes the list already available there a bit longer still; and my impression is that it would be useful if some expert organized these items a little and/or added some comments as to why one would want to track down which of them.

Also perhaps interesting: http://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/

]]>If you are really interested in applications of nonstandard analysis, you could try reading some books. Here are a few on my shelf:

- Nelson,
*Radically elementary probability theory* - Diener-Diener (eds.),
*Nonstandard analysis in practice* - Imme van den Berg,
*Nonstandard asymptotic analysis* - Leob-Wolff (eds.),
*Nonstandard analysis for the working mathematician*.

I haven’t digested everything in these books, but I’ve learned a lot from them, and each of them goes *way* beyond elementary calculus. But perhaps the point of departure is that most of the applications are to *analysis*, whereas in #48 you seem to prefer applications to *geometry*. Analysis is an area of math that often doesn’t seem especially amenable to elegant category-theoretic formulations, but that doesn’t make it less important. Nonstandard analysis, essentially because it is a “synthetic” way to talk about orders of magnitude, does seem like it provides a more elegant way to do a lot of analysis.

In other words, there’s a reason we say “nonstandard *analysis*” but “synthetic differential *geometry*”. (-:

By the way, also nilpotent differentials have their transfer principle: that’s the statement that for instance the Cahier topos is a model for differential cohesion. This means in particular that there is a certain geometric morphism from the standard smooth topos to that with synthetic infinitesimals.

But i’d think these transfer principles are part of the notion of infinitesimals themselves. Saying that nonstandard analysis is good because it has a transfer principle is a bit like saying that the natural numbers are good because they have an element called zero

]]>Colin, what would be the relation of this application of the transfer principle to differential calculus eith explicit differentials?

Mike, we seem to be talking past each other. Not everything that is expressed in topos theory is therefore the natural way to do something. Maybe remember the sympathies and antipathies towards Bohr toposes for another example of just this question.

I still find that when I look around then sdg differentials play a thorough and foundational role in differential geometry both in its basic formulation but in particular in a bunch of powerful modern refinements. All of derived algebraic geometry, all of D-geometry and all the applications to pde theory, variational calculus etc that this has

In contrast, for nonstandard analysis the main statement is that elementary calculus can be phrased this way. Is there anything that goes further?

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