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I'm not sure if this is the right place for this question, but here goes.

I am often inclined to treat, without knowing if this is correct, the operators of calculus, namely $d, \frac{d}{dx},\int,\int_a^b$, as if they were "constants" that you can apply the rules of multiplication to.

So for example, I would be inclined to "multiply both sides by $dx\cdot dy$" as follows: $$\frac{y^2}{dx}=\frac{x}{dy}\implies x^2dx=ydy $$

or to "apply $\int$" to both sides as follows:

$$x^2dx=ydy \implies \int x^2dx=\int ydy$$

even though I am not sure whether $\int$ is an operator by itself, or whether $\int ...dx$ together form a single operator.

So my question is, not about whether the above particular derivations are correct, but what is the topic that one should study to learn the rules of manipulation of these operators? I'd like to really understand them, not just memorize them.

(note that I've taken an introduction to real analysis, but this focused very much on general concepts such as continuity, differentiability, integrability, without touching on the implications for the rules of manipulation of these operators)

user56834
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  • "what is the topic ?" Calculus. See e.g. Lara Alcock, How to Think About Analysis, Oxford University Press (2014). – Mauro ALLEGRANZA Sep 27 '16 at 12:13
  • See this post and this one. – Mauro ALLEGRANZA Sep 27 '16 at 12:20
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    @MauroALLEGRANZA, I obviously know the topic is calculus/real analysis. It's just that my book (Elementary Analysis: the theory of calculus, by Ross) doesn't touch on when and how you are allowed to manipulate these symbols. Looking in the contents page of "How to think about analysis", I don't immediately see where to look for that. – user56834 Sep 27 '16 at 12:43
  • Because they are not "symbols" in the naive way of thinking; you can see page 167 about differentiability. The "obviuos" rules about the "operator $'$ [i.e. $f'(x)$] must be understood as results to be proved. Only after that we can apply them (the rules) in a "computational way". – Mauro ALLEGRANZA Sep 27 '16 at 13:35
  • Both displayed formulas in your question don't make any sense as stand-alone mathematical propositions with semantical content.– Of course, if $x^2 dx=y dy$ is an equality between ununderstood objects then $\int x^2 dx=\int y dy$ is an immediate consequence. – Christian Blatter Sep 27 '16 at 18:14
  • Check these articles of wikipedia: 1, 2 and 3. In each article you have some bibliography where start. In particular I dont know this topic, but I hope this will be useful for you! – Masacroso Oct 03 '16 at 19:24

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