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

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2
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Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
7
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4answers
332 views

Why is $\pi r^2$ the surface of a circle

Why is $\pi r^2$ the surface of a circle? I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why ...
40
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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 ...
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3answers
3k views

Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
8
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1answer
116 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 ...
7
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4answers
2k views

Drawing a thickened Möbius strip in Mathematica

I would like to have Mathematica plot a "thickened Möbius strip", i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be ...
8
votes
2answers
130 views

Prove that any shape 1 unit area can be placed on a tiled surface

Given a surface of equal square tiles where each tile side is 1 unit long. Prove that a single area A, of any shape, but just less than 1 unit square in area can be placed on the surface without ...
4
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1answer
119 views

How to find cubic non-snarks where the $\min(f_k)>6$ on surfaces with $\chi<0$?

Henning once told me that, [i]t follows from the Euler characteristic of the plane that the average face degree of a 3-regular planar graph with $F$ faces is $6-12/F$, which means that every ...
1
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2answers
589 views

Identify and sketch the quadric surface?

I'm stuck trying to figure out which type of quadric surface this equation is: $$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$ I have narrowed it down to a hyperboloid, but cannot ...
0
votes
1answer
351 views

How to integrate over an arbitrarily positioned spherical cap in spherical coordinates

If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for ...
17
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2answers
424 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. ...
20
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3answers
289 views

Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
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0answers
114 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
7
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2answers
89 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
0
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2answers
72 views

The equation of a 3D surface bounded by 3 known elliptical curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
0
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0answers
27 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
5
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1answer
212 views

Limit $\lim_{x\rightarrow x_0, x\in M} \int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{-1}{||y-x||} dS_y$

Ok I had a question I think I can almost answer it but I miss one step: Let $\partial M$ be a closed surface in $\mathbb{R}^3$, $x_0 \in \partial M$ than show this limit: ...
5
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1answer
1k views

The Gaussian and Mean Curvatures of a Parallel Surface

This is a homework problem from do Carmo. Given a regular parametrized surface $X(u,v)$ we define the parallel surface $Y(u,v)$ by $$Y(u,v)=X(u,v) + aN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ ...
5
votes
2answers
384 views

What function has a graph that looks like this?

I delete my file which I used to produce this graph. Does anybody have some idea how to produce it again? Thanks for a while.
4
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2answers
147 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
4
votes
1answer
166 views

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose ...
4
votes
1answer
396 views

Covariant derivative on hypersurface in $\mathbb{R}^n$

I saw in a talk that a surface gradient of $f:M \to \mathbb{R}$ where $M$ is a hypersurface in $\mathbb{R}^n$ defined as $$\nabla_M f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal ...
4
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2answers
429 views

Surface area of a flexible tube

Consider the hollow tube formed by sweeping a circle of radius $r(t)$ along a curve $\gamma(t)$ in $\mathbb{R}^3$; in other words, the set of points $$S=\{\gamma(t) + r(t) \hat{n}\quad \vert\quad ...
3
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1answer
229 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
2
votes
4answers
276 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
2
votes
2answers
662 views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
2
votes
1answer
245 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 ...
1
vote
1answer
46 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
1
vote
1answer
378 views

Need function for 2D sigmoid-shaped monotonic Surface

I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ ...
0
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1answer
24 views

Giving parametrization of Hyperboloid

I'd like to give a parametrization of an two sheet hyperboloid, such that this parametrization covers both sheets (Without using $±$ symbols). Does that parametrization exist?
7
votes
2answers
416 views

isolated non-normal surface singularity

I am looking for an isolated non-normal singularity on an algebraic surface. One obvious example occurs to me: the union of two $2$-dimensional affine subspaces of $\mathbb{A}^4$ which meet in a ...
5
votes
1answer
156 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
5
votes
1answer
102 views

Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.

Let $\partial M$ be $C^2$ closed surface in $\mathbb{R}^3$, $M$ is open. Show that $$ f(x) = \frac{\int_{\partial M} \left| \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} \right| ...
4
votes
0answers
236 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
3
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0answers
168 views

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
3
votes
2answers
133 views

What's the K-group of a surface?

What's the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.
3
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2answers
151 views

Equivalence of two definitions of differentiablitity on Regular Surfaces

When dealing with differentiable surfaces one defines a function $f:S\rightarrow \mathbb{R}$ as being differentiable if its expression in local coordinates is differentiable. But one could also define ...
3
votes
4answers
465 views

parametrization of surface element in surface integrals

I don't understand this How $ dS = \sqrt{ \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2 + 1 } \; dA \; \; $ ?? Is $ dA = dx\times dy$??
2
votes
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 ...
2
votes
2answers
68 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
votes
1answer
118 views

Topological surface thought experiment

Imagine a two-dimensional version of you lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you ...
2
votes
1answer
139 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
145 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?
2
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1answer
154 views

About fibers of an elliptic fibration.

Consider the following pencil of cubics: $\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of ...
2
votes
1answer
123 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
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vote
1answer
55 views

Need function for tunable sigmoid 2D surface

This question is similar to one that I asked 6 months ago, but I added some additional requirements and I'll try to ask it more concisely. Requirements: I need a $2D$ surface, $z = f(x, y)$ where ...
1
vote
2answers
90 views

Describe a twisted parabolic trough

I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that? Imagine this is the trough and the $z$ direction ...
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1answer
77 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
1
vote
1answer
133 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
1
vote
1answer
163 views

Orthogonal Parametrization of a Regular Surface

I was just wondering whether or not it is always possible to parametrize a regular surface $S$ via a function $X$ of local coordinates $u$, $v$ such that $X$ is an orthogonal parametrization- that is ...