I can not understand key difference between them, because all on the web uses mathematical notations...Can anyone explain in easy way?...please.Thanks in advance.
108272
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
91199
I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation. Here is what I understand so far from what they told me:   1)Treat the top variable as as variable when finding the derivative 2)Treat the bottom variable as a constant when finding derivative 3)It it said "Find x with respect to y", but what exactly does that mean? What does it mean for something to be in respect to something else?  It seems like $\frac{dx}{dy}$ notation changes according to values in the problem. For instance, If $y = x^3 + 2x$ and $\frac{dx}{dt} = 5$, find $\frac{dy}{dt}$ when $x=2$. Why do the values of $\frac{dx}{dy}$ change in this problem and how do I solve this?
100895
Which of the two books is suited for a student looking to learn how to write proofs? I have a working knowledge of calculus and linear algebra but I'm not good at writing proofs. My intention is to learn proofs in general, not necessarily for the two.       The reason I ask is that the latter is suggested on a highly voted question here but the former has a more apt name. The reviews aren't helping. Please don't suggest books other than these.
17258
What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when people are doing math on a board or on paper).
6937
I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=).
111297
I can't understand difference between $\varnothing$ and $\{\varnothing\}$. What is difference between $x$ and set that has only $x$ as element? I think the concept of set is meaningful when it has more than 1 element. If "$\varnothing$ and $\{\varnothing \}$ is different" is right, then I have another question. When we see the set $\{\varnothing,\{\varnothing\}\}, \varnothing$ and $\{\varnothing\}$ exist simultaneously. It means two empty sets exist. One empty set exists as $\varnothing$ itself and one empty set exists as element of $\{\varnothing\}$. I can't understand this situation. What am i misunderstanding?
84767
I know the definitions of both but can't specifically tell how they are different and if one is better than the other.  Please help.
30732
What is the difference between these two URLs:   http://example.com/ http://example.com   Should we always add the final / or avoid it? Does it make a difference?
70347
Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very curious. It seems like $\times$ is already quite sufficient as a descriptive symbol.
24251
Are these merely stylistic conventions (whether italicized or non-italicized), or are there substantive differences in the meanings of these notations?  Are there other notations meaning "the probability of" that should be considered in this question?
9579
What is the difference between HTML &lt;input type='button' /&gt; and &lt;input type='submit' /&gt;?
44253
Is there a canonical way to typeset such quantities in equations and inline math?  \documentclass{article} \begin{document} $Re$, $Nu$ are clearly not an option. $\mathit{Re}$, $\mathit{Nu}$ appear ok to me. \end{document}
18896
I've heard that you should use \[ ... \] for displayed equations instead of  $$ ... $$, but why is that?  I'd assumed that it's so that you can more easily tell which are starting and which are ending delimiters, but if I always use a syntax-highlighting text editor, I can see that easily based on the color of the symbols.  Is there any reason for this suggestion?
76571
This question is about asymptotic notation in general. For simplicity I will use examples about big-O notation for function growth as $n\to\infty$ (seen in algorithmic complexity), but the issues that arise are the same for things like $\Omega$ and $\Theta$ growth as $n\to\infty$, or for little-o as $x\to 0$ (often seen in analysis), or any other combination.  The interaction between big-O notation and equals signs can be confusing. We write things like $$\tag{1} 3n^2+4 = O(n^2)$$ $$\tag{2} 5n^2+7n = O(n^2)$$ But we're not allowed to conclude from these two statements that $3n^2+4=5n^2+7n$. Thus it seems that transitivity of equality fails when big-O is involved. Also, we never write things such as $$\tag{3} O(n^2)=3n^2+4,$$ so apparently commutativity is also at risk.  Many textbooks point this out and declare, with varying degrees of indignation, that notations such as $(1)$ and $(2)$ constitute an "abuse of notation" that students are just going to have to get used to. Very well, but then what are the rules that govern this abuse? Mathematicians seem to be able to communicate using it, so it can't be completely random.  One simple way out is to define that the notation $O(n^2)$ properly denotes the set of functions that grow at most quadratically, and that equations like $(1)$ are just conventional abbreviations for $$\tag{4} (n\mapsto 3n^2+4)\in O(n^2)$$ Some authors even insist that writing $(1)$ is plain wrong, and that $(4)$ is the only correct way to express the estimate.  However, other authors blithely write things like $$\tag{5} 5 + O(n) + O(n^2)\log(O(n^3)) = O(n^2\log n)$$ which does not seem to be easily interpretable in terms of sets of functions.  The question: How can we assign meaning to such statements in a principled way such that $(1)$, $(2)$ and $(4)$ are true but $(3)$ is not?
4340
I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated
92871
What is the difference, if there is any, between $F(x)$ and $F[x]$, where $F$ is a field and $x$ is an element of a field extension $E/F$ that is not in $F$? I have some guesses, but I was never sure about this. For example, for one specific usage in a textbook, I see $\mathbb{Q}[i]$ and $\mathbb{Q}(\pi)$. And maybe a more general question: what happens when $F$ is just a ring? For example, for the ring of Gaussian integers, the notation I usually see is $\mathbb{Z}[i]$, with brackets instead of open parentheses. I guess I never properly learned the notation.
2523
After reading it, this is not a duplicate of . The answer may be related (or even the same) but the question is different.    What is the difference and what should go in each?  If I understand the theory correctly, the query optimizer should be able to use both interchangeably.