Questions tagged [surfaces]

For questions about two-dimensional manifolds.

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61 views

Awkward triple integral

I have a rotationally-symmetric surface in $(r,z)$ (i.e. cylindrical coordinates) defined as follows: $$\frac{3r^2 - a}{(r^2 + z^2)^{3/2}} + \frac{3az^2}{(r^2+z^2)^{5/2}} - b = 0$$ where $a,b$ are ...
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1answer
29 views

Finding $z(x,y)$ from $z(x,\text{constant})$ and $z(\text{constant},y)$

I asked this question earlier, but I got no answer, maybe due to my bad English and bad explanation. Now, trying to ask in a different way, hope you can help me: In a surface, the form of $z(x,y=\text{...
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Is Gauss map a diffeomorphism onto its image in this case?

Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$. This means that $D$ is a smooth disk contained in this ball, $D \cap \partial \mathbb{B}^3 = ...
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31 views

Classification of weak del Pezzo surfaces

In this question I work over the complex numbers. Up to isomorphisms, the del Pezzo surfaces are the blow-ups of the projective plane $\mathbb{P}^2$ at $0 \leq n \leq 8$ points in general position (...
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1answer
49 views

Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?

I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding: The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ with boundary ...
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1answer
91 views

Classification of diffeomorphisms by association of differentials with Lie groups

Suppose we are given an oriented Riemannian manifold $S \subset \mathbb{R}^3$ (which I'll refer to as a surface) and a diffeomorphism on $S$, $\Psi: S \rightarrow S$ where $d\Psi\vert_{\bf q}:T_{{\bf ...
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1answer
20 views

Local diffeomorphism between a disk and a sphere

This may be a silly question, but I’ll make it anyway. Let $f: D^2 \to S^2$ be a local diffeomorphism between the closed unit disk and the unit sphere. Is it necessarily injective?
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differential of normal vector

For regular surface $S\subset \mathbb{R}^3$,we have differential of normal vector $dN_p:T_p(S) \to T_p(S)$. What's the size of linear map $dN_p$?I was a bit confuse here. Since we know dimension of $...
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1answer
50 views

Differential of Gauss map

Let $S$ we a regular surface, differential of Gauss map is $\mathrm{d} \mathrm{N}_{\mathrm{p}}: \mathrm{T}_{\mathrm{p}}(\mathrm{S}) \rightarrow \mathrm{T}_{\mathrm{p}}(\mathrm{S})$. To evaluate the ...
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Find 3-manifold based on parameter space of 1D curves

I'm trying to find the equation for the 3-manifold based on the phase space of some one dimensional curves... The space of 1D curves are found from the intersections between $\log(x)\log(y)\log(z)=-a$ ...
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1answer
35 views

Decomposition of a locally free sheaf as tensor product of sheaves

The setting is as follows: Let $X$ be an algebraic surface, $\mathcal{F}$ a locally free sheaf of rank 2 on $X$ contained in $\Omega^1_X$, and $D$ a divisor on $X$ such that $\mathcal{F}\otimes\...
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Two integrals in d-dimensional spherical coordinates depending only on the relative angle

I have the following problem: I have two d-dimensional integrals, where the integrand has the following dependence: $$\int\mathrm{d}^{d}x\mathrm{d}^{d}y\,f(\vert\vec{x}\vert,\vert\vec{y}\vert,\vert\...
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1answer
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Implicit curve/surface definition of a polynomial function that's rotated and translated

Supposing I have an $n^{th}$-order polynomial curve $$y = \sum_{i=0}^n c_ix^i$$ and an $n^{th}$-order polynomial surface $$z = \sum_{i,j\in\mathbb{Z}^+\!,\ i+j=n} c_{ij}x^iy^j.$$ Now suppose that in ...
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1answer
34 views

Addition of differentiable function on regular surface is still differentiable

Given two differentiable functions $f,g:V\to \mathbb{R}^n$ where $V\subset S$ is open subset of regular surface $S$. Prove that $f+g$ with $(f+g)(x) = f(x)+ g(x)$ is still differentiable. My attempt:...
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1answer
31 views

Efficient way to set up surface integral for a section of a sphere

Let $P$ be the polygon with vertices $(0,0), (1,0), (\cos \alpha, \sin \alpha)$ where $\alpha \le \pi/2.$ I wish to find the surface area of the portion of the unit sphere that lies above $P$ or its ...
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1answer
41 views

Surface of infinite genus deformation retracts onto a graph

This is an exercise in Hatcher's Algebraic Topology. There is a solution in this link: Fundamental Group of surface with infinite genus is free on infinite generators, which uses the van Kampen ...
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1answer
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Hyperbolic structure on surface gives a complex structure

My question is from A primer on mapping class group, p.295: I can see $X=\Delta/\Gamma$ has an induced hyperbolic structure, but why conversely any such hyperbolic structure gives a complex structure ...
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25 views

Computing surface area of Hyperboloid with $2 \pi \int r(z)dz$ vs using normal vector

I want to compute the surface area of a hyperboloid: $$ x^2 + y^2 -(z-11)^2 = a$$ For $0 < z < 14$, where $a$ is some positive number such that the hyperboloid is of 1 sheet. What I did was: $$ ...
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1answer
30 views

Curves on a surface intersect under constant angle

I had this exam question a few days ago and I still cannot figure out how to solve it: Do note that I was taught differential geometry in a different language. If something doesn't make sense, point ...
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2answers
49 views

Determining the genus of the compact orientable surface $S =\{[x_0, x_1, x_2, x_3]\in \Bbb RP^3 : x^2_0+ x^2_1- x^2_2-x^2_3=0\}$

Consider the subset $S =\{[x_0, x_1, x_2, x_3]\in \Bbb RP^3 : x^2_0+ x^2_1-x^2_2-x^2_3=0\}$ of $\Bbb RP^3$. Clearly $S$ is an embedded submanifold of $\Bbb RP^3$ of codimension $1$, so it is a compact ...
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1answer
41 views

Area of Projection to a Plane

I need to find all planes of the form $ax+by+cz=0$ such that the projection of: $$ S=\left\{x^2+y^2+z^2=4\middle|x^2+y^2\leq 2x\right\} $$ Onto that plane is one-to-one, and then use it somehow to ...
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Surface Integral of Maximum Area

Find the maximal area of a surface obtained by intersecting a sphere of radius 2 with a ball of radius 1. My Progress: I am quite sure that this problem will involve spherical coordinates, but I do ...
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1answer
31 views

Second fundamental form of asymptotic line [closed]

Why the second fundamental form at a point of a surface should vanish in the asymptotic direction?
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2answers
37 views

Computing curvature along planar intersection curves on a general surface $z = S(x,y)$

I am implementing a code that computes the curvature along curves of intersection between vertical planes and parametrised surfaces in 3D. In order to test this code, I have used the surface $S(x,y) = ...
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38 views

What does it mean to integrate a partial derivative in the “other” direction?

Intro: A continuous, differentiable function $F(x,y)$ can be pictured as a surface over the $(x,y)$ plane. At each point in space, we could plot another function $\frac{\partial F}{\partial y}(x,y)$, ...
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1answer
114 views

Voss Weyl formula and divergence theorem in curvilinear coordinates

I am unable to reconcile the divergence theorem in curvilinear coordinates, and what I get by an application of the Voss Weyl formula and the divergence theorem in $\mathbb{R}^2$. Could someone help ...
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1answer
26 views

Computing the signed curvature of a surface in an arbitrary direction

If I have a surface defined as the graph of the function $z = f(x,y)$, is there a closed-form expression for the signed curvature of this surface in an arbitrary direction? That is, if $x(t) = x_0 + t\...
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Constraint on two functions being equal for every point in a volume

Consider the following volume \begin{equation} T_\epsilon=\{(x,y)\in\mathbb{R}^2|-\epsilon\le x^2-y^2\le \epsilon\}, \end{equation} and the following statement: For every $\epsilon>0$, there ...
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Signature of a link vs signature of a manifold

In his book, Quantum Invariants of Knots and 3-manifolds, chapter 2 - Invariants of closed 3-manifolds pages 78-79, Turaev defines the signature of a framed link $L$ as being the signature of the $D^4$...
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29 views

Methods for computing boundaries of 3D sets

When I compute the boundary of a 3D domain, I always do it in a sort of intuitive way, if I can visualize it. But how does one proceed in general to compute the boundary of an abstract set? Are there ...
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10 views

Deriving the bivariate polynomial basis functions

Is there a means of computing the bivariate polynomial basis functions for an $n^{th}$-order polynomial from the associated $n^{th}$-order univariate polynomial basis functions? E.g. How is $\{1, x, y,...
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56 views

Is this induced homomorphism surjective?

Some preliminaries: Given a surface $S$ and a covering space $q: \tilde{S} \rightarrow S$, we say that $q$ is abelian if its deck transformation group is abelian. Naturally, the universal abelian ...
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Is there a useful $\mathbb{H}^3$ analogue for a “pair of pants”?

2-dimensional surfaces with negative Euler characteristic have a pants decomposition which cuts the surface into pairs of pants. I'm curious if there is an analogue a dimension up. A pair of pants has ...
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1answer
25 views

Reference request: systole of hyperbolic surface increases in covering space

I'm wondering if anyone knows of a quick proof or reference of the following fact: Let $S$ be a compact hyperbolic surface and $l > 0$. Then there exists a finite covering space $S' \rightarrow S$ ...
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1answer
49 views

Intersection points of the 27 lines on smooth cubic surface

Let us work over $\mathbb{C}$. It is a classic result that if $S$ is a smooth cubic surface, then there are 27 lines contained in $S$. My question is: can we compute the number of points which are ...
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26 views

Characterization of non-separating simple closed curves

Let $\{\alpha, \beta\}$ be a pair of non-separating simple closed curves on a compact, oriented surface. Without directly cutting, is there a way to check if the union of the curves in the pair is non-...
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1answer
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Find a parametric representation for the lower half of the ellipsoid $4x^2 +4y^2 +z^2 = 4$. Then find the area of this surface. [closed]

Find a parametric representation for the lower half of the ellipsoid $4x^2 +4y^2 +z^2 = 4$. Then find the area of this surface. (I couldn't do my school homework, can you help me in detail, thank you ...
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What's the number of planes whose intersection with $Q$ is a reducible conic in $\mathbb{P^3}_{\mathbb{C}}$?

Let $\mathbb{P^3}_{\mathbb{C}}$ be the complex projective space and let $l$ and $Q$ be a line and a non-singular quadric surface. Now we consider the pencil of planes that have $l$ as common line. ...
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1answer
37 views

Orientability between diffeomorphic surfaces

From Do Carmo's Differential Geometry of Curves and Surfaces, 2nd edition, Now, when I tried to prove this, I reached this point: Up until this box, I couldn't understand the philosophy behind ...
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Find the equation of the diametral plane that is perpendicular to the given plane.

Find the diametral plane of the surface $$x^2+2y^2-z^2+2xy-2yz-2xz-4x-1=0$$ that is perpendicular to the plane $$x+y+z-3=0$$ I know that the diametral plane passes through the center of the surface (...
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1answer
35 views

Find the equation of the tangent-plane to the surface

Find the equation of the tangent-plane to the surface given with the equation $$2x^2+5y^2+2z^2-2xy+6yz-4x-y-2z=0$$ that passes through the line $$4x-5y=0, \ \ z-1=0.$$ The equation of the slope-...
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1answer
22 views

How are Gerstner wave normals derived?

I'm looking at this GPU gems article on water rendering. It gives the following tangent space vectors. I understand the normals N are calculated as the cross product B x T, but I can only seem to ...
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21 views

Meaning of conormal vector

In my situation let $S$ be a surface in $\mathbb{R}^3$ with $\partial S\not=\emptyset$ (enough regular). What is the meaning of $\eta$: conormal vector to $\partial M$ in M? I know it should be normal ...
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21 views

Guidance for a surface integral problem

I've been dealing with a problem of surface integrals, I hope you could give me an advice. I have this parameterized function $G = (r\cos\theta, r\cos\theta, r)$ with $0\le r \le 1$ and $0\le\theta\...
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46 views

Why is the Narrow Mordell-Weil group a subgroup of finite index?

I am trying to understand the proof of Theorem 4.1 in this article: https://www.researchgate.net/publication/247043151_On_the_Mordell-Weil_Lattices I could understand every argument except for some ...
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1answer
31 views

Surface integral $\int_S{((x-1)~dy\wedge dz+y~dz\wedge dx+z~dx\wedge dy)}/{((x-1)^2+y^2+z^2)^{3/2}}$

Let $S = \{(x, y, z) \in \Bbb R^3:\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1\}$ (oriented by outward normal). I want to compute the surface integral $\displaystyle\int_S \dfrac{(x-1)~dy\wedge ...
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14 views

Explicit Equation for Quadratic Funnel Surface

I've been trying to figure out an explicit form for a quadratic funnel surface, but I'm not succeeding. Essentially, I'm wanting an explicit form for the surface that would be created by revolving the ...
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3answers
50 views

Find rectangular Cartesian coordinate system

Find rectangular Cartesian coordinate system in $\mathbb{R}^3$ to bring quadratic surface $2y^2-3z^2+4xz-12y+15=0 $ to standard form. After splittig off square the equation can be rewritten as ...
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1answer
74 views

Need a plain language style to understand this example from a book about Archimedes surface area example,

after trying so much time and effort I still can't understand what this is page says. I understand by very simple language, Can someone please explain it in plan English what this book is saying, what ...
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0answers
31 views

First and second fundamental form of a surface given a Monge's parametrization

Let be a function $f:(u^{1},u^{2})\rightarrow f(u^{1},u^{2})\in\mathbb{R}$ of class $r\geq 2$ defined in a open $U\subset\mathbb{R}^{2}$. Let be a Monge's parametrization: $X:u\in U\rightarrow (u,f(u))...

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