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The question pretty much sums it all. A few days ago when studying how to find the real part of a function knowing the imaginary part (or vice versa) I was given this formula: $$u(x, y) =\int_{x_0}^{x} \frac{\partial u} {\partial x} (x, y_0) \,dx + \int_{y_0}^{y} \frac{\partial u} {\partial y} (x, y)\, dy$$ Is this a correct notation? If not, is there a simple proof to convince someone that this is an incorrect notation? So $\int_a^x f(x) \,dx$ or $\int_a^x f(t)\, dt$?

Gibbs
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    It is a very sloppy notation if nothing else. The standard is to use a different variable name on the inside of the integral, being instead $\int\limits_{a}^xf(\tau)d\tau$ – JMoravitz May 07 '18 at 20:35
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    This was a horrible practise I encountered back in school and it has always startled me until I learned it does not make sense... Whichever book uses this - please get a better one! – mol3574710n0fN074710n May 07 '18 at 20:37
  • But, is it incorrect, or only sloppy/ horrible? I dont enjoy it either. –  May 07 '18 at 20:41

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Under the usual conventions for variable binding operators the notations $\int_c^xf(x)dx$ and $\int_c^xf(y)dy$ are both correct and mean the same thing. However, if you want to be kind to your readers, I'd suggest you avoid using $x$ as a free variable and a bound variable in one formula.

Rob Arthan
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  • So it's only a convention. thank you! –  May 07 '18 at 20:52
  • From that point of view, you are correct. That is the perspective of syntax. However in the interpretation of the integral one would typically see $x$ as varying between bounds. And that is when the notation becomes logically intractable for the human reader unfamiliar with the concept of free and bound variables. And these are most students who are first confronted with a definition of the integral. So again - Yes, you are absolutely right, but personally I would point a very strong emphasis on the last part of your answer. – mol3574710n0fN074710n May 07 '18 at 20:57
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    It's analogue to quantifiers in logic, e.g. similar confusion can arise with formulas like $(x=1)\land\forall x(x\le 1)$. And there are results - using cylindric algebras - that a first order logic with only 3 variables can express the same things as with infinitely many variables. Then it is very important to enable such formulas. – Berci May 07 '18 at 21:07
  • @mol3574710n0fN074710n: I think it should be a duty of maths educators to explain the distinctions between free and bound variables in this context. – Rob Arthan May 07 '18 at 21:07
  • @Rob Arthan: Indeed! However the problem with explaining such things is time... The institutes are expected to produce graduates in less and less time where I live and many subjects are only touched superficially. In physics they start solving ODEs by calculating with differentials as if they were variables although a proper definition of such algebra would require several years of math... It is all just done too rushed and hurried, I believe. – mol3574710n0fN074710n May 07 '18 at 21:17
  • @mol3574710n0fN074710n: you have my sympathy. – Rob Arthan May 07 '18 at 21:34
  • I concede that given enough context the intended meaning is relatively clear, but I would never call it "correct". It is not unlike $$\sum_{n=1}^n n^2,$$ which is obviously ambiguous (without extra context). Having said that, when looking for suitable examples to a calculus course from a book by a local chemistry professor, I saw him abuse notation exactly like this. – Jyrki Lahtonen May 08 '18 at 04:31
  • But, where in the WP article do they state that a it is ok to use the same variable as both free and bound in the same context? – Jyrki Lahtonen May 08 '18 at 04:36
  • @JyrkiLahtonen: your example is not ambiguous. The only possible reading makes it equivalent to $\sum_{i=1}^ni^2$. – Rob Arthan May 08 '18 at 05:24
  • No. It is also possible that $\sum_{i=1}^n n^2$ is intended. May easily occur as an intermediate step when evaluating a more complicated sum. – Jyrki Lahtonen May 08 '18 at 05:29
  • @JyrkiLahtonen: $\sum_{i=1}^{n}n^2$ is not a possible reading of $\sum_{n=1}^{n}n^2$: the $n$ on the left of the subscript $n=1$ in the latter declares the bound variable $n$ and this declaration hides the free variable $n$ in the outer scope so that the $n$ in $n^2$ refers to the bound variable and not the free variable $n$ in the outer scope. – Rob Arthan May 08 '18 at 20:33
  • But similarly the $n$ in the superscript declares it as a bounded variable. My point is that, while we often can infer the intention from the context, dual use is just plain wrong. – Jyrki Lahtonen May 08 '18 at 20:58
  • No. The $n$ in the superscript is not a declaration. The conventions here are well-defined and no "inference from the context" is required. What you describe as "dual use" is no different in kind from a $\lambda$-calculus term like $(\lambda x.x)x$, which is completely innocuous and not "plain wrong". – Rob Arthan May 08 '18 at 21:25
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I would say it is not correct as you are using the same letter for variable of integration and one of the extremes of integration. I would probably write: $$\int_{x_0}^x \frac{\partial u}{\partial x^1}(x^1,x^2_0)\,dx^1+\int_{y_0}^y \frac{\partial u}{\partial x^2}(x^1,x^2)\,dx^2.$$

Gibbs
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    The superscripts can be confused for exponents. I would have gone for different letters rather than relying on superscripts. – JMoravitz May 07 '18 at 20:36
  • Personally, I would go with $\int_{x_0}^x \frac{\partial u}{\partial x}(t,y_0),dt+\int_{y_0}^y \frac{\partial u}{\partial y}(x,t),dt$ But I would like to know a direct answer if I am allowed or not to use the first one. –  May 07 '18 at 20:38
  • Ok, here is direct: NO!!! YOU ARE NOT!! ;P - Seriously, this notation is logically inconsistent, it confuses students where no confusion need be and thus it needs to be ERADICATED!!! ONCE AND FOR ALL!! ragemode off (Btw, I mean the notation from the OPs question, what Gibbs suggested is perfectly fine!) – mol3574710n0fN074710n May 07 '18 at 20:40
  • @JMoravitz there are a lot of authors using this convention with upper indices for coordinates, especially in differential geometry and mathematical physics, as far as I know. Anyway, it is a matter of taste. When a coordinate needs to be squared I write $(x^2)^2$ to avoid confusion. Further, I guess you never write $1$ as exponent... – Gibbs May 07 '18 at 20:42
  • @Sonkun your notation looks inconsistent to me. – Gibbs May 07 '18 at 20:42
  • But the fact that it confuses students doesnt make it is incorrect, right? (but I agree with you) –  May 07 '18 at 20:43
  • @Gibbs - typically this would be a matter of context, in diff'geo exponents are rare compared to coordinate-indices and this notation is defined at the beginning of the lecture cicle, so everything is fine. Personally I often use what you suggested. For new students of math I would however suggest using a tilde or another letter altogether, as they tend to associate every superscript with an exponent and I actually witnessed a case where a (new) diff'geo-prof tried to use this in introductory calculus lectures at our institute and created a lot of confusion, even some hostility. – mol3574710n0fN074710n May 07 '18 at 20:49
  • There is nothing wrong with using the same variable name for a bound variiable and in one of the limits, but it isn't good style. – Rob Arthan May 07 '18 at 20:51
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It is quite incorrect, because the $x$ in the integrand is a dummy variable, i.e. is does not appear in the result, just like when you write $$\sum_{k=1}^5 k^2,$$ $k$ does not appear in the result, since the closed form of this sum is $55$. Here $k$ is just a temporary variable which lets you know at which step of the computation you've arrived. After the computation is finished, you throw it awway. The same is true for integrals.

Using $x$ in both places reminds me of Groucho Marx's sentence: ‘I don't want to belong to any club that will accept me as a member.’

Bernard
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  • Well, I wish this was also true in terms of syntax, but refer to Rob Arthan's answer here - x is only bound for the "inner" part of the integral operator (the f(x)dx part) and thus it can actually be used in the integral bounds without violating syntax standards... I hate to admit this because I abhor this fact. – mol3574710n0fN074710n May 07 '18 at 21:03
  • Your comment about Groucho Marx isn't relevant. The right thespian analogy is an actor coming in and out of role. – Rob Arthan May 07 '18 at 21:20
  • @RobArthan: Well, what's the difference with being a club member and not a club member? – Bernard May 07 '18 at 21:24
  • @Bernard: you introduced the analogy with Groucho Marx's quote, so it's down to you to explain how it's relevant to this question. – Rob Arthan May 10 '18 at 20:58