For questions about surfaces.

learn more… | top users | synonyms

2
votes
0answers
93 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
66 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
43 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
48 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
168 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
110 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
356 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
127 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 ...
37
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
114 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
158 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
63 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
57 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
123 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
411 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
69 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
141 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
41 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
47 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
99 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
222 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
45 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
328 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
40 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
195 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
129 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
98 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
123 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
110 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 ...
0
votes
1answer
87 views

Surface equation for a triangle when vertices are given

How to find equation for surface of a triangle when vertices are given? Such as when vertices are $(1,0,0),(0,1,0),(0,0,1)$. Surface given by $x+y+z=1$.
2
votes
1answer
63 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
2
votes
1answer
83 views

Whitney umbrella birational to $\mathbb{A}^2$ but not isomorphic

Define the Whitney umbrella as the affine surface $V(z^2 - yx^2) \subset \mathbb{A}^3$. I've come across an exercise that asks me to show that this surface is birational, but not isomorphic, to ...
1
vote
1answer
137 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
1
vote
0answers
103 views

Surface orientation

Let $S_{1}$ and $S_{2}$ be two oriented surfaces ($N_{1}$ and $N_{2}$ their normal fields, respectively). We say that a local diffeomorphism $f$ : $S_{1}$$\rightarrow$$S_{2}$ preserves orientation if ...
0
votes
1answer
41 views

Search for sharp maximums of 4D surface

I need to find sharp local maximums of numerically defined 4D surface. I have a surface with lots of maximums. I already know how to find them all. Some of them look like this: wide extremum, others ...
3
votes
1answer
51 views

What does it mean for a surface to evolve with divergence-free velocity?

Suppose we have an evolving hypersurface which evolves with a velocity field $V$, such that $\nabla_S \cdot V = 0$ where $\nabla_S$ is the surface or tangential gradient. What does this mean? What ...
7
votes
1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
4
votes
1answer
170 views

Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...
3
votes
0answers
82 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
1
vote
1answer
72 views

The equation of a surface created by the extrusion of a 2D closed curve along a path

How do I obtain the equation of a surface created by the extrusion of a circle (or ellipse) created on the XY plane along a parabola or a parametric curve which lies on the YZ plane. The goal is to ...
1
vote
2answers
38 views

Boy surface parameterization confusion

I'm looking at equations 10, 11 and 12 here. What do the letters I and R represent in these equations?