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

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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 ...
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34 views

How to find the side for the biggest area?

Let $x$ be one side of a rectangle and $a$ its perimeter. We know that it's area is given by: $$ S = x\cdot\left(\frac{a}{2}-x\right). $$ $$ S=-x^2+ax/2$$ where a=-1, b= a/2 and c= 0 $$D=a^2/4$$ ...
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Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
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22 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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73 views

Rational map on $\mathbb P^1$ and its fibers

Consider a non-singular complex projective surface $S$ and a rational map $\psi:S\longrightarrow \mathbb P^1$; moreover suppose that $\psi$ is not defined on $\Delta=\{x_1,\ldots,x_m\}\subset S$. Now ...
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44 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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60 views

Length of geodesic representative on hyperbolic surfaces

Let $S$ be a closed oriented hyperbolic surface. Let $x,y \in S$ and let $\alpha,\beta$ be two geodesic arcs with endpoints $x$ and $y$. Let $\alpha \beta$ be the closed piecewise geodesic curve ...
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21 views

Multivariable Calculus: Intersection of surface and planes

I have this math problem: Let $S$ be the surface that consists of all points $(x,y,z)$ that satisfy the equation $x^2+y^2=z^2$. 1) What are the intersections of $S$ with horizontal planes ...
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49 views

Principal directions bissect the asymptotic directions

How can one prove that at a hyperbolic point, the principal directions bissect the asymptotic directions?
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What is $H^0(\mathcal{O}_S(D))$

I know that there's a one-to-one correspondence between $\operatorname{Pic}(S)$, where $S$ is a smooth variety, and the ismorphism class of invertible sheaves. Let $\mathcal{O}_S(D)$ be the invertible ...
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Blow-up and base change

Consider a complex smooth (projective) surface $X$ and a blow-up $\epsilon:S\longrightarrow X$ at a point $x\in X$. Let $\sigma\in\text{Aut}(\mathbb C)$ be a field automorphism and moreover let ...
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Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
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63 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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Cubic surface as P$^2$ blowing up 6 points

This question is from the chapter 1 of Reid's note: Chapters on algebraic surfaces Suppose that L:(x=y=0), M:(z=t=0), and L$_5$:(y=t=0) lie on a nonsingular cubic surface X in P$^3$, define a ...
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An algorithm to find isometry between surfaces in $\mathbb{R}^3$?

Given two surfaces in $R^3$, i would like to find isometry between these two. Usually, in class, we did some examples, like bending the plane into a cylinder, or cone, and they were not hard, quite ...
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Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
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37 views

Calculating surface area

I have the following surface in $$R^3:{(x,y,z),(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2) \ ,\ x,y >=0}.$$ I want to find it's surface area. I've tried using spherical coordinates but calculating the ...
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71 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
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138 views

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
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1answer
49 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 ...
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106 views

Properties of fibers of a morphism of varieties

In this question, all varieties are supposed to be over an algebraically closed field $k$. Hypothesis: X is a smooth projective surface and $f:X\longrightarrow \mathbb P^1$ is a morphism with we ...
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Bertini's theorem for surfaces: informations about singular fibers.

Let $S$ be a complex non-singular projective surface embedded in some $\mathbb P^n$. Thanks to the Bertini's theorem (Hartshorne theorem II.8.18) there exists a hyperplane $H\subseteq\mathbb P^n$ ...
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78 views

Constant curvature metrics on the sphere

Are there Riemannian metrics other than the standard metric induced from the euclidean space on $S^2$ such that the sectional curvature is equal to 1 everywhere? Or is this the unique Riemannian ...
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132 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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Are there non-parametrizable surfaces?

Are there any surfaces that cannot be parameterized? (I'm in multivariable calc and we were talking about parametrizing surfaces for Stokes' Theorem so I was wondering if there are any surfaces that ...
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How do I convert a parametric surface $S(u,v) = (X(u,v),Y(u,v),Z(u,v))$ to a Monge representation

How do I convert a parametric surface $S(u,v) = (X(u,v),Y(u,v),Z(u,v)) $ to a Monge representation, $ S(x,y) = (x,y,Z(x,y))$?
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Finding the critical points of a quadratic form restricted to projective plane

I have a quadratic form $f(x) = x^t A x$ where A is 3x3 real symmetric and $f$ satisfies $f(x) = f(-x)$ and now restricted to $||x|| = 1$ this is a well defined map on the projective plane (when ...
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Does every differentiable ruled surfaces possess a global ruled parametrization?

According to my notes, a differentiable ruled surface of $\mathbb R ^3$ is a 2-dimensional $C^k$ submanifold of $\mathbb R ^3$ that can be described as a union of straight lines. I'm working on some ...
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89 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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177 views

Parametrised vs Regular Surfaces

Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry: Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an ...
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Derivative first fundamental form

Let $X,Y: I \rightarrow T_{\gamma}\Omega$ be vector fields along a curve $\gamma: I \rightarrow \Omega \subset\mathbb{R}^2.$ Now, in our lecture it was claimed that the derivative $\frac{d}{dt} ...
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Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
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Metric properties

Let $f: \Omega \rightarrow \mathbb{R}^3$ be a submanifold in $\mathbb{R}^3$ and also $f' : \Omega' \rightarrow \mathbb{R}^3$ another one. Now if $f(\Omega) \cap f' ( \Omega')$ is a regular curve $c: I ...
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Finding where a map between surfaces is a local diffeomorphism [duplicate]

Let $M=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{4}=1\}$, $\mathbb{S}^{2}=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}=1\}$ and $F:M\rightarrow\mathbb{S}^{2}$, $(x,y,z)\mapsto (x,y,z^{2})$. I have to ...
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Why is first fundamental form considered intrinsic

I am reading Kuhnel's differential geometry book, and in chapter 4, it says that "intrinsic geometry of a surface" can be considered to be things that can be determined solely from the first ...
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Parametrizing Surfaces

Can someone check my work? The question was: find a parametric representation of the portion of the surface $x+3y-z=5$ with $x\geq0, y\geq0$, and $x^2+y^2\leq 1$. I answered: $x=\cos\theta$, ...
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Evaluating stokes theorem $\int \vec{F} \cdot d\vec{r}$ on the surface $z=4-y^2$

Evaluate $\int \vec{F} \cdot d\vec{r}$ o the surface $z=4-y^2$ cut off by $x=0$, $z=0$, and $y=x$. I particularly need help with evaluating the integral on $C_3$. Please see picture I am ...
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How does the circumference of the top + bottom sides of a cylinder effect our calculations when working out the surface area?

I was watching a video tutorial on khan academy, (I've included the link at the bottom), and the question states that there is a 8cm cylinder, with a radius of 4. Part of the video shows a worked ...
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Evaluating the divergence theorem for region above $z=0$, below $z=x$ and inside $x^2+y^2=1$ where $\hat F=(xz,yz,z^2)$

Can someone please confirm my working below: The answer am getting look kinda crazy -Thanks. $$\color{green}{\hat F=(xz,yz,z^2)}$$ $1.$For the surface where $\color{green}{z=0}$ i.e. (flat ...
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Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
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Surface integration/Stokes/Divergence Theorem application

Let $\Sigma$ be a suitably well-behaved orientable surface in $\mathbb{R}^3$ whose boundary is a curve $\partial \Sigma$. Show that $$\int_{\Sigma}(dS\times \Delta)\times F=\int_{\partial \Sigma} ...
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how to determine the outward pointing normal (gauss divergence theorem)

I have a cone defined by $x^2+y^2=(1-z)^2$ i was trying to work out the normal vector on surface $s_1$ indicated on the plot On $s_1$: r=$\left<x,y,0\right>$ since $z=0$ on $x-y$ plane ...
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Finding the surface area $\iint_{s} f \, dS$ of $z=x^2-y^2$ cut off by $z=4-2y^2$

Finding the surface area $\iint_{s} f \, dS$ of $z=x^2-y^2$ cut off by $z=4-2y^2$ I have no idea which parametrization to use for this, however i did figure out the following: I think the ...
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Finding the surface area $\int \int_{s} f \, dS$ 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: ...
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Boundary of Mobius strip is $S^1$

I feel like this should be simple, and it is intuitively obvious by looking at the polygon with side identifictations version of the Mobius band, but how do we explicitly show, i.e find the ...
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Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
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Parametrization of folded surfaces with symmetry

How is parametrization done for folded surfaces which are smooth within restricted interval of fold, e.g., has is it been possible to define parametrization for plane faces of Platonic solid faces? ...
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89 views

Christoffel symbols of a surface of revolution

I am looking for a way to write down the Christoffel symbols for a surface of revolution. They are given by ...
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155 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 ...