Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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3
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1answer
25 views

Is an open subset of a compact surface with connected boundary completely determined by its fundamental group?

Is an open, connected subset of a compact surface with connected boundary determined (up to homeomorphism) by its fundamental group? If we weaken the hypotheses, I can see how this can fail: A ...
0
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1answer
16 views

find the equation of a tangent surface which is parallel to a plane surface

How to get the equation of the tangent surface of $F=x^2 + 2y^2 +3 z^2=21$, which is also parallel to the plane surface $ x+ 4y + 6z=0 $? Here is what I've tried: $n = \{1, 4, 6\}$ from $ x+ 4y + ...
3
votes
7answers
888 views

Software to display 3D surfaces

What are some examples of software or online services that can display surfaces that are defined implicitly (for example, the sphere $x^2 + y^2 + z^2 = 1$)? Please add an example of usage (if not ...
-2
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1answer
23 views

Characterizing a surface

can somebody help me get started with this problem? I don't even know how to start the proof. Say $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable. Prove that $z=xf(y/x)$ belongs to a surface ...
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2answers
31 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
0
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1answer
23 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
0
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2answers
65 views

Calculating integral using Stokes theorem and directly

Here is my task: Calculate directly and using Stokes theorem $\int_C y^2 dx+x \, dy+z \, dz$, if $C$ is intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, orientated in positive ...
1
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1answer
44 views

Line integrals - Surface area

Here is my task: Calculate surface area of $2(x^{2}+y^{2})^{2}=xy$ between surface $x^{2}+y^{2}=z$ and $z=0$. Here is my attempt to solve this problem. Firstly, I transformed line ...
0
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1answer
37 views

Is the inside of a sphere a hyperbolical surface?

So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? ...
0
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0answers
13 views

Sorting Bigger Boxes to Smaller Boxes

I am currently working on a Bin Packing program and need to know what would be the most efficient way of getting boxes (arbitrary width, length, and height) to be sorted in the manner below? ...
1
vote
1answer
42 views

Tangent plane and tangent lines to curves through a point

Let $S$ be the surface that is the graph of a continuous function $f: U \rightarrow \mathbb{R}$ on an open $U \subset \mathbb{R}^2$. Let $p = (x, y, f(x, y)) \in S$. One usually defines the tangent ...
0
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0answers
28 views

Calculus | Parametrization of boundary in $\mathbb{R^3}$

The Problem Given the volume $$ K = \left\{ (x,y,z)\in \mathbb{R^3} \big| \frac{x^2}{9} +y^2 \le z^2 +1, -\frac{1}{3}\sqrt{\frac{x^2}{9} +y^2} \le z \le 3 \right\} $$ What are $a$, $b$, and $K(z)$ ...
0
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1answer
310 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
2
votes
1answer
26 views

Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
0
votes
2answers
59 views

How to derive the 3D equation of a torus?

I'm doing a presentation on 3D surfaces for college and one of the equations I am using is a Torus. I know that the equation is $$z^2 = 25 - \left(10 - \sqrt{x^2 + y^2}\right)^2$$ For a torus with ...
2
votes
1answer
72 views

Finding the surface area $\iint_{s} f \, dS$ of $z=\sqrt{x^2+y^2}$ lying inside $x^2+y^2=x$

$z=\sqrt{x^2+y^2}$ is the surface we working on. I am a bit stuck on choosing the limits for this problem, I have done the following: ...
0
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0answers
10 views

What are the intersection curves of two quadric surfaces of revolution?

Is it possible to determine the type of curve formed by the intersection of two hyperboloids of revolution (of 2 sheets) or the intersection of two ellipsoids? Also, any books or papers on quadric ...
5
votes
1answer
71 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
0
votes
1answer
24 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
3
votes
2answers
45 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
1
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1answer
23 views

Arcs and surfaces. Why are there finitely many arcs on the surface up to the action of MCG?

Given a bordered surface $S$ (I imagine this is true for non-orientable surfaces too, but you may restrict to the case of orientable surfaces) with finitely many marked points on each boundary ...
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2answers
26 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
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0answers
136 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
1
vote
1answer
109 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
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0answers
21 views

Parametrization for intersection curve of catenoid and cylinder

Required to obtain equation of intersection line of two surfaces.. the catenoid of revolution and displaced or eccentric cylinder..in a parameterized form. $$ (x^2 + y^2) = c^2 \cosh ^{2} (z/c) ; \, ...
2
votes
1answer
34 views

How to calculate the surface of overlapping ellipsoids

I want to calculate the surface of a body made of at least 3 overlapping ellipsoids. Below there is a picture of the cross section of the body. I already know how to calculate the surface of single ...
0
votes
1answer
51 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
2
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0answers
23 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
2
votes
1answer
53 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
2
votes
1answer
37 views

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Find all surfaces that can be obtained from an octagon by identifying edges in pairs. I think there are many many surfaces. Can anyone give some hints for the question?Thanks.
1
vote
1answer
22 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
1
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1answer
60 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
2
votes
3answers
3k views

Find unit normal vector to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$

I've been trying to solve this question: Find a unit vector with positive $z$ component which is normal to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$ on the surface. My working: Let ...
0
votes
2answers
21 views

How to simplify this equation with change of variables,

I have the equation, after completing the square: $$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$ How can I further simplify this equation? I need to find the volume inside of this surface. ...
1
vote
0answers
39 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
2
votes
2answers
56 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
2
votes
2answers
43 views

Geodesics on a generalized cylinder

I want to prove that given a generalized cylinder $C(s,t)=\alpha(s)+t\hat{z}$ , where $\alpha$ is a curve on the $xy$ plane and $\hat{z}$ is the $z$-axis vector, then a geodesic curve $\gamma$ has the ...
2
votes
0answers
66 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
0
votes
0answers
31 views

decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
4
votes
2answers
583 views

What is the object on the front of Larson and Edwards' calculus and pre-calculus textbooks called?

There is this incredible glass figure on the front of my Calculus textbook, I searched online for what this figure is called and the formula for creating it, but I can't find it. I think it is a ...
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0answers
27 views

Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
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2answers
61 views

Polygonal presentations: why no two-letter words?

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...
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0answers
32 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
2
votes
1answer
141 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
2
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1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
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1answer
31 views

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
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0answers
19 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
4
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1answer
72 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
0
votes
1answer
48 views

Calculus 3 - Level surfaces

I know how to find the level surfaces for a $2$ variable functions, $z=(x,y)$, by finding the $3$ planes. How would you find the level surfaces for a $3$ variable function, $w=(x,y,z)$. Would you find ...
0
votes
1answer
23 views

Parameterizing part of sphere

the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$ first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...