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Mathematical notation has been evolving for hundreds of years (really thousands, I guess, but most of the noteworthy examples seem to be only a few hundred years old). Sometimes we are stuck with old notations for things we have completely redefined. Sometimes the notations may not have been all that great to start off with, but became the standard, anyway.

What are some examples of standard mathematical notation that you think are either: 1) misleading, 2) ambiguous, or 3) just plain ugly?

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closed as primarily opinion-based by Grigory M, Claude Leibovici, William, hardmath, Arthur Fischer Jul 14 at 9:59

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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This question will probably be shut down soon (not by me) but I disagree with your complaints about calculus notation, these notations are very suggestive and we should use them to gain a deeper understanding of the subject. –  Rene Schipperus Jul 13 at 0:07
    
They all serve a very well role in physics, at least at the undergraduate level. That is why should keep it, until you find something better –  julian fernandez Jul 13 at 0:11
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The Leibniz notation is a magnificent thing. Some of the others are misleading; Leibniz's isn't. –  Michael Hardy Jul 13 at 0:12
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$\cos^kx$ and $\sin^kx$ for $(\cos x)^k$ and $(\sin x)^k$ should be abolished. It is not done for other functions, and it is ambiguous if $k=-1$. –  Per Manne Jul 13 at 7:22
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Here is a related question on mathoverflow: mathoverflow.net/questions/18593/… Some of these are notations that are actually used. Richard Stanley's answer, the leader in voting, is a notation that some mathematician once used, but I rather doubt that it's very popular. –  Michael Hardy Jul 14 at 1:13

2 Answers 2

$$\frac{\partial f}{\partial x}$$

Partial derivative notation is misleading in that it suggests "how does $f$ vary as $x$ varies?" is a meaningful question, and is an outright abomination in any context where one might consider making a change of variable.

What this notation secretly means is to ask how $f$ varies as $x$ does... while certain other quantities are held constant.

(its use in differential geometry isn't so bad, since index notation really does express which other quantities are to be held constant)

Even worse is when this notation gets abused (presumably due to a lack of familiarity with sane notation) and you see things like

$$ \frac{\partial}{\partial (x^2 + y)} f(\cos x, x^2 + y, z) $$

when one really just means to say

$$ f_2(\cos x, x^2 + y, z)$$

(where $f_2$ is notation analogous to $f'$, that means "the derivative in the second argument")

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Another example along that same line is the Jacobian notation $\frac{\partial (u, v, w)}{\partial (x,y,z)}$ –  user164076 Jul 13 at 0:23
    
@user164076: I rate the Jacobian as being similar to the differential geometry usage: it's not so bad because it does express which values are to be held constant in each partial derivative. (assuming, of course, that you only have three independent variables) –  Hurkyl Jul 13 at 0:27

Rather than directly answer the question, I prefer to address the seeming implication that the Leibniz notation is bad. Maybe a really good expository defense of that notation has yet to be written. For now I shall link to this answer I wrote a while ago: What is $dx$ in integration?

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I'm not necessarily saying that the Leibniz notation is bad -- just that students are taught that $dx$ is an infinitesimal change, when that's not what it means in standard analysis. The only point you made in your answer in that link that suggests that $dx$ is in fact anything other than a notation for which variable we're integrating, is that we can dot or cross vectors with it. But that is misleading. Really we are taking a product with either a tangent or normal vector, not the $dx$. –  user164076 Jul 13 at 0:20
    
In logically rigorous mathematics as done in the present day, $dx$ is not an infinitesimal change in $x$. But it is a serious mistake to bring logical rigor into some contexts. One of those contexts is of course the freshman calculus classroom for non-math majors. Especially when this example may be evidence that what we now understand as logical rigor may be in need of further advances in understanding. I take "logically rigorous" to mean subject to sound algorithmic proof-checking. –  Michael Hardy Jul 13 at 0:23
    
Not that I've changed my mind that Leibniz notation can be misleading, but because this post was becoming about how I'm wrong for not appreciating the almighty Leibniz, I've edited that part out of the question. –  user164076 Jul 13 at 0:50
    
I like thinking of $dx$ in terms of differential forms as linear functionals. You can induce the real algebra on them, like other functions, e.g. (f + g)(x,y) = f(x,y) + g(x,y), (f/g)(x,y) = f(x,y) / g(x,y), etc. This recovers Leibniz notation nicely. –  nomen Jul 13 at 1:08

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