For questions about surfaces.

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3
votes
2answers
59 views

Are there odd-sheeted coverings of non-orientable surfaces by orientable surfaces?

For any non-orientable surface (compact,connected) $X$ with genus $h$, we have a $2n$-sheeted cover of $X$ by an orientable surface $Y$ first by covering $X$ by $\Sigma_{h-1}$ (a double cover) and ...
1
vote
1answer
85 views

bounds for surface integral of a plane

I need to calculate the next surface integral, but i'm having troubles with the bounds; $$\iint -2 \, dS,$$ where $S$ is the part of the plane $x+2y+z=2$ that is cut off in the first octant. My ...
1
vote
0answers
107 views

Question about integral on hypersurface

Let $S$ be a (n-1) dimensional hypersurface in $\mathbb{R}^n$. I see something like this written: $$\int_S f(s)d\sigma(s)$$ where $d\sigma$ means the surface measure. Now if $\Phi\colon T \to ...
2
votes
2answers
75 views

Height at 2D coordinate on a 3D rectangular surface

The Problem: How can I obtain every 3D coordinate on a rectangular surface given x and z? For those who are visual, picture looking down on the surface, and finding the height at where the x and z ...
0
votes
0answers
21 views

Relationship between the set of all canonical knots and the set of all knot genuses

What is the relation between the set of all canonical knots and the set of all knot genuses? I understand that there is at least not bijection between them.
2
votes
1answer
80 views

Mathematically explaining a trapped surface?

I'm currently writing my thesis, and the general area is that of minimal surfaces. I have a deep interest in cosmology so have directed it towards space-time, ie applying minimal surfaces in space, ...
2
votes
0answers
24 views

Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$ S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}. $$ If we denote by ...
1
vote
1answer
113 views

Surface area of a sphere bounded by a pyramid

I'm trying to calculate the surface area of a sphere that is bounded by the walls of a triangular pyramid. $o$ is the origin and centre of a sphere of radius $R$, $a$ is a point at the tip of the ...
1
vote
2answers
106 views

Is there a common name for the surface z = xy?

I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?
1
vote
2answers
29 views

Does $g(x,y,z)$ (the equation of the surface) need positive $z$ or negative $z$ when doing a surface integral?

$\quad$If a smooth surface $S$ is defined by $g(x,y,z)=0$, then recall that a unit normal is $$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$ where $\nabla g=\dfrac{\partial g}{\partial ...
2
votes
0answers
64 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
1
vote
2answers
56 views

Surface Integral of a Vector Field Over a Torus

Let $S$ be the surface obtained after rotating $(x-2)^2+z^2=1$ around the $z$-axis. What is the value of $$\int_{S}\mathbf{F\cdot n } dA$$ where $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y))$$
2
votes
0answers
95 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 ...
0
votes
1answer
36 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
1
vote
1answer
46 views

Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis

So here is my problem: Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis Just thinking about the problem I feel ...
2
votes
1answer
67 views

Calculate the surface of revolution (area) for $x^2/4+y^2/2=1$

$$x^2/4+y^2/2=1$$ The curve is rotating around the x-axis and I'm suppose to calculate the area. My attempt: $$y=\sqrt{2-x^2/2}$$ The integral goes from -2 to 2. I tried to simplify the integral ...
0
votes
1answer
45 views

Paremetric surface revolved around y-axis

if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation $2\pi x\sqrt{(x')^2+(y')^2}$ and I'm given $$x=(2/3)t^{3/2}$$ $$y=2\sqrt{2}$$ and I got ...
1
vote
2answers
49 views

Is there a Covering Map $\Sigma_3^1\to \Sigma_2^1$

Let $S_{g,n}^b$ be a genus $g$ surface with $b$ boundary components and $n$ punctures. I'm having some trouble with these past qualifying exam questions: Is there a covering map $p\colon ...
8
votes
1answer
179 views

Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous ...
1
vote
0answers
111 views

Surface Integral

The glass dome of a futuristic greenhouse is shaped like the surface $z = 8 - 2x^{2} - 2y^{2}$. The greenhouse has a flat dirt floor at $z = 0$ Suppose that the temperature T, at points in and around ...
17
votes
2answers
358 views

How to determine that a surface is symmetric

Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane. The special case where $f$ is a polynomial is of some interest. ...
-1
votes
2answers
128 views

Identifiying the next point on the surface of a cube ( or 3D object )

I have a cube of unit length. Each face of the cube is divided into 10 x 10 equal segments. Consider an object of size equal to that of a segment moving through the surface of the cube ( or any 3D ...
38
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
0
votes
1answer
115 views

Recognize the equation of a surface of revolution

Yesterday, I asked a question about the critic points of the surface $$z = (x^2 + y^2)e^{-(x^2 + y^2)}$$ and my question was if I had a easier way to classify the critic points of this surfaces ...
3
votes
2answers
165 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
0
votes
1answer
67 views

Volume of hyperboloid limited by two planes (multivariable calculus).

As usual the teachers solution sheet takes leaps and bounds over steps in the solution that I need to understand it. Q: Determine the volume of the body limited by $x^2+y^2-3z^2=1$, $z=1$ and ...
0
votes
1answer
45 views

Prove that $\textbf{II}_p\equiv0$ on $M:=f(U)$ if and only if $M$ is contained in a plane

Let $U\subset\mathbb R^2$ be open and connected, and let $\ f:U\rightarrow\mathbb R^3$ be the parametrization of a regular surface. Prove that $\textbf{II}_p\equiv0$ on $M:=f(U)$ if and only if $M$ ...
1
vote
1answer
58 views

Proof that the infinite cylinder is a regular surface.

I have to proof that the circular cylinder $M=\{(x,y,z)\in\mathbb{R}^3\mid x^2 + y^2 = r^2\}$ is a regular surface, where $r$ is a constant, $r>0$. Then I have to see also that $\mathrm x\colon ...
1
vote
2answers
124 views

Maximum surface area of cylinder (1-variable)

In a given sphere of radius $R$, it is required to find the cylinder with maximum surface area that we can inscribe in this sphere. Using that the radius of the cylinder is $r$, with Pythagoras ...
0
votes
1answer
21 views

Show that the tangent plan pass through the origin

Show that all the tangent plans to the conic surface $z = xf(\frac{y}{x})$ at the point $M(x_o,y_o,z_o)$, where $x_o \neq 0$, pass through the origin of the cordinates First, I've found the tangent ...
0
votes
2answers
456 views

Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in ...
1
vote
1answer
70 views

Why $xyz = e^x$ can be seen as the level surface $f(x,y,z) = xyz - e^x$?

That does not make sense to me. I recognize a level surface from the form $f(x,y,z) = k$. Where is the $k$ there? It looks just like a $3$ variables function to me.
1
vote
0answers
52 views

Do there exist double points on a surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In ...
0
votes
0answers
55 views

Did I solve this surface area of revolution problem correctly $y=\frac{1}{3}x^3,$ when $0 \le x \le 2$

My Professor hasn't posted the solutions to our practice exam just yet, but I'd like to know if I solved the following surface area of revolution problem correctly? $y=\frac{1}{3}x^3,$ when $0 \le x ...
2
votes
1answer
145 views

Show that 2 surfaces are tangent in a given point

Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + ...
1
vote
1answer
47 views

Enlarge a quadratic surface equally in all directions

I have a quadratic surface defined by $$ Ax^2 + By^2 + Cy^2 + Dxy + Eyz + Fzx + Gz + Hy + Iz + J = 0 $$ I know the values of the constants $A,B,C,D,E,F,G,H,I,J$. I need to make another surface $1$ ...
0
votes
1answer
27 views

Non-zero terms in a B-Spline surface

Questions For question 4 I know that in the u direction there is at most 5, non-zero basis functions N_(i,4) to N_(i-4,4) and in the v direction there is at most 4 non-zero basis functions N_(j,3) to ...
0
votes
1answer
48 views

Splitting of a surface group over the subgroup associated to a closed geodesic

Let $G=\pi_1 (S)$ for a closed surface $S$, consider a closed geodesic $c$ on $S$ and let $H$ be the subgroup of $G$ induced by $c$. Is it true that $G$ splits over $H$, i.e. $G$ can be written as a ...
1
vote
1answer
20 views

Formula for extracting bounding points from a given set of points

I have a set of geographic locations, i.e. points defined by latitude and longitude. Given this set of points, I need to select only those of them that belong to the surface bounds. The simplest ...
1
vote
0answers
101 views

Average arc length between two random points on a unit sphere?

I'm trying to find the average arc length between two random points on a unit sphere. The solution I've come up with is rather ugly. Consider a parametric surface: $$X(u,v)=\sin u\cos v\\Y(u,v)=\cos ...
0
votes
1answer
227 views

compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...
3
votes
0answers
47 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
0
votes
1answer
335 views

Outer Unit Normal: Cylinder

I have a cylinder occupying the region $x_{1}^{2}+x_{2}^{2} = R^2$ and $-G< x_3 < 0$ All I want to do is define the outer unit normal on the curved face. I thought about just calling it $e_1$ ...
0
votes
1answer
41 views

Is there a definition of cylinder that these equations satisfy

Our teacher is claiming that (in $\mathbb{R}^3$) the following surfaces are "cylinders": $3x+y+\frac{7}{2}=0$ $y=x^2$ $z^2 = y$ $\frac{x^2}{4} + \frac{y^2}{4} = 1$ Is there any definition of ...
4
votes
1answer
209 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
2
votes
1answer
136 views

Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
0
votes
1answer
102 views

Parametric surfaces - Parameterization of torus

A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This ...
0
votes
2answers
126 views

surface vs differentiable manifold

Every surface is a smooth manifold, but the reciprocal is verified? some concrete example of a differentiable manifold is not surface? Thanks in advance for the suggestions.
2
votes
1answer
112 views

Is conformal equivalence the same as topological equivalence?

Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?
1
vote
0answers
48 views

Gauss and Stocks teory

Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=${$(x,y):y=\phi(x),0\le x\le 1$} $S=${$(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1$} a.I need to prove that $A(S)=2\pi\int_\gamma ...