# Differential equations book that explains differential forms

All books that I have read so far say something like this:

From your calculus course you probably know differential forms:

$$M(x,y)dx + N(x,y)dy = 0$$

This is really bothering me. Even though I am fine with intuition that $$y' = \frac{dy}{dx}$$ can be thought of as a ratio, all analysis and calculus books kept telling me that we can't do that. Thats why differential equation above has no meaning for me right now.

So my question is, are there books on DEs that give justification for using $$dx$$ and $$dy$$ separately?

• In undergraduate courses on differential equations, whenever $dx$ and $dy$ are treated as individual quantities, there is always an easy way to rephrase the argument to avoid doing so. For example, the equation you mentioned can be written as $M(x,y) + N(x,y) y' = 0$. So you don't need differential forms to understand this material. – littleO Aug 14 at 6:17

In addition to subrosar’s answer, I personally really like this introduction to differential forms: https://www.math.purdue.edu/~dvb/preprints/diffforms.pdf

It is quite elementary and offers quite a bit of material. I think just by skimming through it you will find what you are looking for.

Link to Terry Tao's explanation of the differential form:

https://www.math.ucla.edu/~tao/preprints/forms.pdf

I'd also like to take a shot at lessening your mathematical angsts myself, so I have two explanations for the equation brought up in your post. The first is more of a formal explanation and the second is more of an informal explanation.

My formal understanding of differential forms is they are maps taking vectors and outputting scalars. So the equation you mention is a statement about the orientation of the vectors in a certain vector field (the tangent vectors of solutions to a differential equation most likely). You might make a similar statement about the tangent vectors to a closed curve because the x and y components of tangent vectors are related by such a a formula. Differential forms can be used in for example a line integral in a kind of generalized Riemann sum. Divvy up the curve into small vectors and then sum over your integrand multiplied by the scalar gotten when the differential form acts on these vectors.

My other, less formal explanation of the equation $$M(x,y)dx+N(x,y)dy=0$$ is that you can think of it as being approximately/almost true for small change in $$x$$ and $$y$$. If on some solution to a diff eq, $$f(x_0)=y_0$$ and $$f(x_0+\Delta x)=y_0+\Delta y$$ then $$M(x,y) \Delta x + N(x,y) \Delta y \approx 0.$$ Put more rigorously, $$\frac{M(x,y) \Delta x + N(x,y) \Delta y} {\sqrt{\Delta x^2+\Delta y^2}}$$ converges to $$0$$ as $$\Delta x$$ and $$\Delta y$$ get small.

Also you should know that infinitesimal calculus can be made completely rigorous, something mathematicians discovered in the mid-20th century. Angst not!

It's unfortunate that that's the case in most contemporary undergraduate textbooks. But also one can't blame the book makers much -- that's the dominant approach to all things infinitesimal these days. And while so-called nonstandard analysis has been justified, many safely stick to the mainstream (although I personally think the differential approach is more natural). By the way, I trust many of those books who tell you not to treat differentials as objects in their own right go on to calculate with differentials when they treat the calculation of integrals; what do you say to that, eh?

Well, having said that, how may one think of these things? These differentials. Well, intuitively, differentials are a type of entity called an infinitesimal quantity. And, what is that? You may think of it as a way to capture our notion of such things as an instant of time, a point of a line, a plane section of a solid, etc. One sometimes (again informally) says it's an infinitely small quantity. This is usually formalised by saying that an infinitesimal quantity is one that is approaching zero, is vanishing, or is evanescent -- note the present continuous tenses here. Infinitesimals are not zero, as many 18th century mathematicians thought (even Euler championed this view in his Foundations of the Differential Calculus) -- this got them into a lot of conceptual (usually not computational) trouble and led to the crises in the foundations of analysis that led to the thorough examination of these in the 19th century and the basing of calculus by Cauchy on the notion of limit. One way of semi-formally saying this is that an infinitesimal is a quantity that approaches or tends to zero. However, it need never be zero. That is to say, although the limits of all infinitesimals are zero, they themselves are not zero. Also, infinitesimals are not numbers or functions tending to zero -- they are new objects that simply help us talk about our intuition of an infinitely small quantity in a continuum, just like the vectors of early physics helped them to talk about directed quantities in general (indeed you'll see that infinitesimals may be though of as an example of vectors in the sense in which mathematicians use that term today).

Now, having tried to explain what the notion of infinitesimal tries to capture, what do we do with these things? It's easy to then see that we may combine infinitesimals using something very like addition, and that this always gives us another infinitesimal. Also, we may multiply an infinitesimal by a quantity (a number or function), and this gives us another infinitesimal. Thus, the infinitesimals form a vector space over the field of functions of a real variable. However, we can do more. We may multiply infinitesimals, and the product is also an infinitesimal. You might think that we may divide infinitesimals too, but here we have to be careful -- division of infinitesimals doesn't give us another infinitesimal, but a (normal) quantity, that is, a function whenever the infinitesimal in the denominator is not the zero infinitesimal. Thus, the infinitesimals stop short of forming a field. But they do form a ring; indeed an integral domain. But all that is just jargon (or a summary term) for the operations on infinitesimals I've been talking about. Finally, we may also consider an infinitely small part of an infinitesimal, and that leads to the notion of higher order infinitesimals -- infinitesimals of infinitesimals, etc.

Fine, you might say. But what's all this got to do with differentials? Everything, because differentials are just a special type of infinitesimal. What type? A differential is an infinitesimal difference or change or perturbation, etc. That's all, yes. Indeed, our most important examples of infinitesimals are differentials, from an instant of time to an ordinate of a plane region, etc. This is just like the fact that the most common examples of power series are Taylor series. Thus, of course, the differential calculus is about the investigation of the division of infinitesimal differences (differentials), while the integral calculus is about summing infinitely many differentials, as it were. Finally, a differential equation is just that -- an equation relating the differentials of (related) quantities. That's it, really. Nothing mysterious. It's only recently with the dominance of the limit approach that everything has been cleaned up and talk of differentials all but banished from the beginning -- although they can't always escape it in the computation of integrals, LOL! Finally, a form is just an expression; historically, this has been used to expressions that have been of geometric or some other significance (as an invariant, for example), so that the term is used majorly for homogeneous expressions. Thus we say that $$ax+by$$ is a linear form, $$ax^2+bxy+cy^2$$ a quadratic form, and so on. But a form is just any expression, as it were. A differential form, then, is just an expression formed by differentials. The particular example you give comes when one talks about the differential of a function of two variables (in particular, about implicit differentiation) $$f(x,y).$$ The differential of this quantity is given by $$f'_x\mathrm d x+f'_y\mathrm d y.$$ These days, they talk about gradients and directional derivatives instead, but it's all the same thing but in a language trying to avoid talking about differentials at all costs.

Finally, to see more of this approach, you will have to look to the older books (but you will have to be a bit careful since there may be some error or inclarity) like the ones by Leonhard Euler or Isaac Todhunter; or read about nonstandard analysis. All the best.