Search found 308 matches
- Sat Aug 11, 2018 10:29 am
- Forum: Multivariate Calculus
- Topic: Show that a vector field is not conservative (example)
- Replies: 4
- Views: 7426
Re: Show that a vector field is not conservative (example)
First we write down a useful theorem: If a continuously differentiable vector field $\overline{F}:U\subseteq{\mathbb{R}}^n\longrightarrow{\mathbb{R}}^n\,,$ where $U$ is open, is conservative, then, for every $\overline{x}\in U$, the Jacobian matrix ${\bf{D}}\overline{F}(\overline{x})$ of $\overline{...
- Sat May 12, 2018 1:43 pm
- Forum: Analysis
- Topic: Logrithmic Integral
- Replies: 3
- Views: 6926
Re: Logrithmic Integral
Using that the Fourier series of $\log(\sin{x})$ on $(0,\pi)$ is(*) \begin{align} \log(\sin{x})=-\log2-\mathop{\sum}\limits_{n=1}^{\infty}\frac{\cos(2nx)}{n} \end{align} then \begin{align*} \int_{0}^{\pi}x^2\log(\sin{x})\,dx&\stackrel{(1)}{=}\int_{0}^{\pi}\Big(-x^2\log2-\mathop{\sum}\limits_{n=1...
- Thu May 03, 2018 6:23 am
- Forum: Complex Analysis
- Topic: Complex Integral of a singularity function
- Replies: 1
- Views: 4473
Re: Complex Integral of a singularity function
The function $f(z)=\frac{1}{(z-2)^2(z-4)}$ is defined and is holomorphic on $\mathbb{C}\setminus\{2,4\}$. The disk $D_1=\big\{{z\in\mathbb{C}\;|\;|z|\leqslant3}\big\}$ containing the second order pole $z_1=2$, but not the simple pole $z_2=4$. By Cauchy's integral formula we have \begin{align*} \disp...
- Tue May 01, 2018 4:51 pm
- Forum: General Mathematics
- Topic: Fibonacci closed form
- Replies: 1
- Views: 5131
Re: Fibonacci closed form
Because \begin{align*} \mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{\frac{1}{F_{2^{n+1}}}}{\frac{1}{F_{2^n}}}&=\mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{F_{2^n}}{F_{2^{n+1}}}\\ &=\mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{\phi^{2^n}-(-\phi)^{-2^n}}{\phi^{2^{n+1}}-(-...
- Fri Mar 16, 2018 1:11 pm
- Forum: Real Analysis
- Topic: Convergence of a series
- Replies: 1
- Views: 5256
Re: Convergence of a series
Let $\alpha_n=\big(\frac{(2n-1)!!}{(2n)!!}\big)^2\,,\; n\in\mathbb{N}$ and $\beta_n=n\,,\; n\in\mathbb{N}$. Then $\sum_{n=1}^{\infty}\frac{1}{\beta_n}=+\infty$ and for all $n\in\mathbb{N}$ holds \begin{align*} \beta_n-\beta_{n+1}\,\frac{\alpha_{n+1}}{\alpha_n}&=-\frac{1}{4(n+1)}<0\,. \end{align*...
- Wed Feb 07, 2018 10:47 pm
- Forum: Real Analysis
- Topic: Real analysis
- Replies: 3
- Views: 7435
Re: Real analysis
what can you say about A if LUB A = GLB A ? Maybe LUB is L(owest)U(pper)B(oundary) i.e. $\sup$ and GLB is G(reatest)L(ow)B(oundary) i.e. $\inf$. The main question is what kind of set is $A$? Since we talking about $\inf{A}$ and $\sup{A}$ it must be given where lies $A$ i.e. it considered as a subse...
- Sun Jan 28, 2018 12:58 am
- Forum: Real Analysis
- Topic: Subsequences
- Replies: 1
- Views: 5216
Re: Subsequences
It seems that here we have an open problem. See
Salem numbers and uniform distribution modulo 1
Salem numbers and uniform distribution modulo 1
- Sat Jan 20, 2018 5:16 am
- Forum: Real Analysis
- Topic: Subsequences
- Replies: 1
- Views: 5216
Subsequences
Prove that the sequence $\alpha_n=\lfloor{\rm{e}}^n\rfloor\,,\; n\in\mathbb{N}$, where $\lfloor{\cdot}\rfloor$ is the floor function, has a subsequence with all its terms to being odd numbers and a subsequence with all its terms to being even numbers.
Note: I don't have a solution.
Note: I don't have a solution.
- Wed Jan 03, 2018 11:17 pm
- Forum: Linear Algebra
- Topic: Linear Algebra Book Recommendation
- Replies: 1
- Views: 5709
Re: Linear Algebra Book Recommendation
One book worthy to read is Peter D. Lax - Linear Algebra and its applications. (Wiley-Interscience)
Also there are lot of good books applying (or compining) linear algebra to (with) geometry.
Also there are lot of good books applying (or compining) linear algebra to (with) geometry.
- Sun Dec 17, 2017 3:51 pm
- Forum: Multivariate Calculus
- Topic: Calculation of the mass of solid bounded by two surfaces
- Replies: 4
- Views: 7357
Re: Calculation of the mass of solid bounded by two surfaces
Is there a general rule to solve this kind of problems, given two surfaces that bound a solid and its density function ? Yes, there is a general formula to calculate the mass of a solid $S$ with given density function $f:S\subset\mathbb{R}^3\longrightarrow\mathbb{R} $ : \[\displaystyle\mathop{\iiin...