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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.

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  • $\begingroup$ 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. $\endgroup$ – Doubt Jul 4 '13 at 14:04
  • $\begingroup$ While not very introductory, you could look into Differential Forms. $\endgroup$ – Sai Jul 5 '13 at 1:05
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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.

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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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