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 Forum: General Topology   Topic: Continuous functions

 Post subject: Continuous functions
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

 Post subject: Not closet set
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

 Post subject: Two examples
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 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 sub...

 Forum: Algebraic Structures   Topic: Symetry group of Tetrahedron

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

 Post subject: Area & surface integral
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{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: 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

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{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: 449


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: 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)

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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