I am reading a textbook on differential forms and there is a question of the form: given a $k$-form, $\omega$, on say $\mathbb{R^3}$, find $\mu$ such that $d\mu=\omega$. For example, if $\omega= (x_{1}^2 +x_{2}^2)dx_1 \wedge dx_2$, what is $\mu$?  I am looking for a systematic method to solve these types of questions.  Thanks
Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
Should a retrieval method return 'null' or throw an exception when it can't produce the return value? I'm trying to align a legend inside a table caption, so that the colons are vertically aligned. I can do it with the align ambient outside of the caption, but if I use the same code inside the caption the pdf won't compile (I use LuaTeX to compile). I'm using the package caption.  EDIT: i want something like this:    for which i've written this code:  \begin{align} \begin{split} a : {} &amp; the~first~letter~of~the~alphabet \\ b : {} &amp; the~second~one \\ and : {} &amp; so~on \end{split} \end{align}   There might be a more elegant way... but i don't know it.  Anyway, I can't do it inside the caption, that's why i'm stuck. How do I use reflection to call a generic method? Are global variables bad? In C/C++, are global variables as bad as my professor thinks they are? Possibly two questions in here: Are these sentence constructions logical, and if they are, why are they different?     I swam across the Ocmulgee River.   I swam across the Pacific Ocean.   I swam across the Red Sea.   I swam across Lake Winnipeg.   I can't think of a way of phrasing the lake example while using an article with a proper noun. I thought at first it was because Lake Winnipeg is different; it's generic portion (the "Lake") comes first and the name of that body comes second, but I swam across Falls Lake is the same.   Yet the sentence I swam across the lake sounds right to me. ...Why? An example for a homomorphism that is not an automorphism I faced with a problem after update,   I missed 2 buttons from above panel, System menu (including shutdown, suspend &amp; …) and User (including log out &amp; …)  how can I restore them !?... How can you use optional parameters in C#? Note: This question was asked at a time when C# did not yet support optional parameters (i.e. before C# 4).  We're building a web API that's programmatically generated from a C# class. The class has method GetFooBar(int a, int b) and the API has a method GetFooBar taking query params like &amp;a=foo &amp;b=bar.   The classes needs to support optional parameters, which isn't supported in C# the language. What's the best approach? Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$  How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).  Any guidelines please? "Would" & " Used to" Does $\{f_ng_n\}\to fg$ uniformly? How to share variable between two c# projects Dictionaries on iOS 8 not working I was having a discussion with a friend of mine about some normal group properties and then came up with the question "if G is not commutative, then is there always a subgroup that is not a normal subgroup?" It's probably more easy to solve this in the following form:$$\forall H \leq G : H  \lhd G \Rightarrow \forall a,b \in G : ab=ba$$ My question is, can anybody give a proof, or a counter-example (because I don't think it holds) of this theorem? Thanks! Hide textfield blinking cursor C++ display stack trace on exception Why does this excerpt say that unbiased estimation of standard deviation usually isn't relevant? I was reading on the computation of the unbiased estimation of standard deviation and the source I read stated      (...) except in some important situations, the task has little relevance to   applications of statistics since its need is avoided by standard   procedures, such as the use of significance tests and confidence   intervals, or by using Bayesian analysis.   I was wondering if anyone could elucidate the reasoning behind this statement, for example doesn't the confidence interval use the standard deviation as part of the calculation? Therefore, wouldn't the confidence intervals be affected by a biased standard deviation?  EDIT:  Thanks for the answers so far, but I'm not quite sure I follow some of the reasoning for them so I'll add a very simple example. The point is that if the source is correct, then then something is wrong from my conclusion to the example and I would like someone to point how how the p-value doesn't depend on the standard deviation.  Suppose a researcher wished to test whether the mean score of fifth graders on a test in his or her city differed from the national mean of 76 with a significance level of 0.05. The researcher randomly sampled the scores of 20 students. The sample mean was 80.85 with a sample standard deviation of 8.87. This means: t = (80.85-76)/(8.87/sqrt(20)) = 2.44. A t-table is then used to calculate that the two-tailed probability value of a t of 2.44 with 19 df is 0.025. This is below our significance level of 0.05 so we reject the null hypothesis.  So in this example, wouldn't the p-value (and maybe your conclusion) change depending on how you estimated your sample standard deviation? Discrete and compact subset must be finite Event binding on dynamically created elements? What if MSE used a different rep model for privileges?