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What is a good reference to learn about differentials and related topics. Some of my questions are:

  • Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$?
  • In a separated differential equation such as $F(x)dx + G(y)dy = 0$, what is the physical intuition behind "$F(x)dx$"?
  • When integrating the latter equation, what variable is integrated over? Usually, if $f(x) = 0$, then we can integrate over $x$, like $\int f(x)\,dx=c$. But how de we arrive at $\int F(x)dx + \int G(y)dy = c$?

A detailed, but introductory-level reference is sought. As a bonus, feel free to shed light on any of the above questions.

Doubt
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  • I considered naming this question "Understanding exact differentials", but did not want to necessarily exclude inexact differentials and infinitesimals from the discussion. Thanks for the book link, but I am also looking for some background/theory, rather than just methods of solving ODEs. – Doubt Jul 04 '13 at 14:04
  • While not very introductory, you could look into Differential Forms. – Sai Jul 05 '13 at 01:05

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It's really not possible to write it as a fraction. The notation $\frac{dy}{dx}$ is a limiting process, not an actual fraction so it doesn't make sense to split them. It's just a notational convenience because it is reminiscent of difference quotients and slope formula. Really what is going on in a differential equations course is that you're doing chain rule without the "inconvenience" of having to think about doing the chain rule. Take for instance the following differential equation:

$$\frac{dy}{dx} = y.$$

In a differential equations course they tell you to move the $y$ over and then multiply by $dx$ but since $\frac{dy}{dx}$ isn't really a fraction, instead what you should really be doing is the following:

$$\frac{1}{y}\frac{dy}{dx} = 1.$$

We can recognize the left hand side as the derivative of $C\log|y(x)|$ (you can check this by differentiating via chain rule). If you then write $y$ in terms of $x$, you'll get the same expression as you would if you "separated" the differentials.

In the language of differential forms in differential geometry, expressions like $F(x)dx+G(y)dy$ mean something very specific and you can actually make sense of them however even then $dx$ and $dy$ are (co)vectors so it doesn't make sense to divide them and again they're not differentials. It's just suggestive notation that happens to unify some seemingly disjoint theorems in mathematics.

Cameron Williams
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As a start I'd recommend having a quick read through Arturo Magidin's brilliant answer to this question. Then, I'd check out an introductory real analysis course--like this one--and follow it up with a simple text on the subject--for example, the first 5 chapters of this book. Good luck.

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jkn
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