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by Grigorios Kostakos
Sun Mar 31, 2019 8:44 am
Forum: General Topology
Topic: Continuous functions
Replies: 0
Views: 2916

Continuous functions

Let $(X,\rho)$, $(Y,d)$ two metric spaces and $f,g:X\longrightarrow Y$ two continuous functions.
  1. Prove that the set $F=\big\{x\in X\;|\; f(x)=g(x) \big\}$ is closed set of $X$.
  2. If $D$ is a dense subset of $X$, such that $f (x) = g(x)$, for every $x\in D$, prove that $f = g$.
by Grigorios Kostakos
Sat Mar 30, 2019 2:36 pm
Forum: General Topology
Topic: Not closet set
Replies: 0
Views: 2383

Not closet set

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 $x_n\stackrel{\rho}{\longrightarrow}x$.
by Grigorios Kostakos
Sat Mar 30, 2019 8:43 am
Forum: General Topology
Topic: Two examples
Replies: 0
Views: 2414

Two examples

Give an example of a descending sequence $(F_n)_{n\in\mathbb{N}}$ of non-empty 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 non-empty closed subsets of metric space $(\...
by Grigorios Kostakos
Fri Nov 16, 2018 11:33 am
Forum: Algebraic Structures
Topic: Symetry group of Tetrahedron
Replies: 2
Views: 997

Re: Symetry group of Tetrahedron

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 symmetry of a ...
by Grigorios Kostakos
Fri Aug 31, 2018 5:53 pm
Forum: Multivariate Calculus
Topic: Area & surface integral
Replies: 0
Views: 739

Area & surface integral

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)\,, \en...
by Grigorios Kostakos
Fri Aug 31, 2018 5:24 pm
Forum: Multivariate Calculus
Topic: Surface area of an Elliptic Paraboloid
Replies: 1
Views: 687

Re: Surface area of an Elliptic Paraboloid

... $$A_P=ab\int_0^1\int_0^{2\pi} \sqrt{1+\frac{4r^2\cos^2\theta}{a}+\frac{4r^2\sin^2\theta}{b}}\,r\,d\theta dr$$ ... 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 ...
by Grigorios Kostakos
Fri Aug 31, 2018 4:23 pm
Forum: Multivariate Calculus
Topic: Volume, area & line integrals
Replies: 0
Views: 659

Volume, area & line integrals

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)\,, \en...
by Grigorios Kostakos
Tue Aug 14, 2018 6:41 am
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 1243

Re: Show that a vector field is not conservative (example)

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.
by Grigorios Kostakos
Sun Aug 12, 2018 11:01 am
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 1243

Re: Show that a vector field is not conservative (example)

...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...
by Grigorios Kostakos
Sat Aug 11, 2018 10:29 am
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 1243

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{...