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 Forum: Algebraic Structures   Topic: Symetry group of Tetrahedron

Posted: Fri Nov 16, 2018 11:33 am 

Replies: 2
Views: 188


We give a solution in the case of the tetrahedron: Definition: A symmetry of a (regular) tetrahedron $S$ is a linear transformation $T:\mathbb{R}^3\longrightarrow\mathbb{R}^3$ with orthogonal matrix which also leaves tetrahedron $S$ unchanged(*), i.e. $T(S)=S$. ⋅  Definition: An axis of s...

 Forum: Multivariate Calculus   Topic: Area & surface integral

 Post subject: Area & surface integral
Posted: Fri Aug 31, 2018 5:53 pm 

Replies: 0
Views: 110


Let $E$ be the surface with parametric representation \begin{align*} \overline{R}:(-3,3)&\times[0,2\pi]\longrightarrow{\mathbb{R}}^3\,; \quad \overline{R}(r,\theta)=\left({\begin{array}{c} \frac{r}{\sqrt{9-r^2}}\,\cos{\theta}\\ \frac{r}{\sqrt{9-r^2}}\,\sin{\theta}\\ \theta \end{array}}\right)\,,...

 Forum: Multivariate Calculus   Topic: Surface area of an Elliptic Paraboloid

Posted: Fri Aug 31, 2018 5:24 pm 

Replies: 1
Views: 106


... $$A_P=ab\int_0^1\int_0^{2\pi} \sqrt{1+\frac{4r^2\cos^2\theta}{a}+\frac{4r^2\sin^2\theta}{b}}\,r\,d\theta dr$$ ... The integral $\int_0^{2\pi} \sqrt{1+\frac{4r^2\cos^2\theta}{a}+\frac{4r^2\sin^2\theta}{b}}\,d\theta$ is an elliptic integral of second type. Thus, the corresponding double integral ...

 Forum: Multivariate Calculus   Topic: Volume, area & line integrals

Posted: Fri Aug 31, 2018 4:23 pm 

Replies: 0
Views: 98


Let $E$ be the surface with parametric representation \begin{align*} \overline{R}:[0,6)&\times[0,2\pi]\longrightarrow{\mathbb{R}}^3\,; \quad \overline{R}(r,\theta)=\left({\begin{array}{c} \frac{r^2}{\sqrt{36-r^2}}\,\cos{\theta}\\ \frac{r^2}{\sqrt{36-r^2}}\,\sin{\theta}\\ r \end{array}}\right)\,,...

 Forum: Multivariate Calculus   Topic: Show that a vector field is not conservative (example)

Posted: Tue Aug 14, 2018 6:41 am 

Replies: 4
Views: 258


andrew.tzeva wrote:
Thank you. The 2nd solution (with the direct counter-example) is much more helpful.
Sure, in this case! But in general, to find a suitable curve isn't easy.

 Forum: Multivariate Calculus   Topic: Show that a vector field is not conservative (example)

Posted: Sun Aug 12, 2018 11:01 am 

Replies: 4
Views: 258


...I tried using $r(t)=t\vec{i}+t\vec{j}, \space t\in[\alpha,\beta]$, but it didn't work. What curve would be a better choice for $C$ and what's the deal with $\mathrm{rot}\,F$ being zero?.. Here is a 2nd solution, choosing an appropriate (closed) curve: The line integral of $\overline{F}$ over the...

 Forum: Multivariate Calculus   Topic: Show that a vector field is not conservative (example)

Posted: Sat Aug 11, 2018 10:29 am 

Replies: 4
Views: 258


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{...

 Forum: Analysis   Topic: Logrithmic Integral

 Post subject: Re: Logrithmic Integral
Posted: Sat May 12, 2018 1:43 pm 

Replies: 3
Views: 303


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...

 Forum: Complex Analysis   Topic: Complex Integral of a singularity function

Posted: Thu May 03, 2018 6:23 am 

Replies: 1
Views: 164


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...

 Forum: General Mathematics   Topic: Fibonacci closed form

 Post subject: Re: Fibonacci closed form
Posted: Tue May 01, 2018 4:51 pm 

Replies: 1
Views: 568


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}}-(-...
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