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Forum: General Topology Topic: Continuous functions 
Grigorios Kostakos 
Posted: Sun Mar 31, 2019 8:44 am


Replies: 0 Views: 95

Let $(X,\rho)$, $(Y,d)$ two metric spaces and $f,g:X\longrightarrow Y$ two continuous functions. ⋅ Prove that the set $F=\big\{x\in X\;\; f(x)=g(x) \big\}$ is closed set of $X$. ⋅ If $D$ is a dense subset of $X$, such that $f (x) = g(x)$, for every $x\in D$, prove that $f = g$. 


Forum: General Topology Topic: Not closet set 
Grigorios Kostakos 
Posted: Sat Mar 30, 2019 2:36 pm


Replies: 0 Views: 87

Let $(X,\rho)$ a metric space and $(x_n)_{n\in{\mathbb{N}}}$ a Cauchy sequence in $X$, such that the set $\{x_n\;\; n\in{\mathbb{N}}\}$ of the terms of this sequence it isn't a closed set. Prove that exists $x\in X$, such that $x_n\stackrel{\rho}{\longrightarrow}x$. 


Forum: General Topology Topic: Two examples 
Grigorios Kostakos 
Posted: Sat Mar 30, 2019 8:43 am


Replies: 0 Views: 95

⋅ Give an example of a descending sequence $(F_n)_{n\in\mathbb{N}}$ of nonempty closed subsets of metric space $(\mathbb{R}, \cdot)$, such that $\bigcap_{n=1}^{\infty}F_n=\varnothing$. ⋅ Give an example of a descending sequence $(F_n)_{n\in\mathbb{N}}$ of nonempty closed sub... 


Forum: Algebraic Structures Topic: Symetry group of Tetrahedron 
Grigorios Kostakos 
Posted: Fri Nov 16, 2018 11:33 am


Replies: 2 Views: 314

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 
Grigorios Kostakos 
Posted: Fri Aug 31, 2018 5:53 pm


Replies: 0 Views: 191

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{9r^2}}\,\cos{\theta}\\ \frac{r}{\sqrt{9r^2}}\,\sin{\theta}\\ \theta \end{array}}\right)\,,... 


Forum: Multivariate Calculus Topic: Surface area of an Elliptic Paraboloid 
Grigorios Kostakos 
Posted: Fri Aug 31, 2018 5:24 pm


Replies: 1 Views: 194

... $$A_P=ab\int_0^1\int_0^{2\pi} \sqrt{1+\frac{4r^2\cos^2\theta}{a}+\frac{4r^2\sin^2\theta}{b}}\,r\,d\theta dr$$ ... 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 
Grigorios Kostakos 
Posted: Fri Aug 31, 2018 4:23 pm


Replies: 0 Views: 179

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{36r^2}}\,\cos{\theta}\\ \frac{r^2}{\sqrt{36r^2}}\,\sin{\theta}\\ r \end{array}}\right)\,,... 


Forum: Multivariate Calculus Topic: Show that a vector field is not conservative (example) 
Grigorios Kostakos 
Posted: Tue Aug 14, 2018 6:41 am


Replies: 4 Views: 449

andrew.tzeva wrote: Thank you. The 2nd solution (with the direct counterexample) 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) 
Grigorios Kostakos 
Posted: Sun Aug 12, 2018 11:01 am


Replies: 4 Views: 449

...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) 
Grigorios Kostakos 
Posted: Sat Aug 11, 2018 10:29 am


Replies: 4 Views: 449

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{... 


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