For questions about surfaces.

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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
62 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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1answer
20 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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1answer
55 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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1answer
297 views

Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is \begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation} and it is easy to estimate the curvature numerically for given ...
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1answer
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Verify Divergence Theorem (using Spherical Coordinates)

I'm very used to calculating the flux of a vector field in cartesian coordinates, but I'm still getting tripped up when it comes to spherical or cylindrical coordinates. I was given the vector field: ...
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1answer
38 views

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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1answer
71 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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125 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 ...
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1answer
32 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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2answers
41 views

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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2answers
68 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
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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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1answer
121 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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2answers
42 views

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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1answer
362 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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0answers
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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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ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
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2answers
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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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1answer
27 views

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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1answer
66 views

If Möbius band embeds then $\mathbb RP^2$ is a (connected) summand!

This is exercise 6-4 on page 181 in John Lee's Topological Manifolds book which asks me to prove the above, that is, if $M$ is a boundaryless surface which contains a subset $B$ which is homeomorphic ...
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1answer
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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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2answers
90 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
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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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1answer
279 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
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2answers
123 views

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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2answers
58 views

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 ...
3
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1answer
37 views

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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1answer
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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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2answers
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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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0answers
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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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1answer
146 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 ...
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0answers
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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 ...
5
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2answers
63 views

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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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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1answer
55 views

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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1answer
68 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 \ ...
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1answer
38 views

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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3answers
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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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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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1answer
48 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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0answers
19 views

Calculating the equation of a multivariable surface of revolution

I'm stucked with a surface equation problem so I would be very thankful if someone could help me with it. What the excercise says: Find the equation of the revolution surface that is spanned when ...
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1answer
163 views

How to solve this questions about regular surfaces?

I'm trying to solve the following: $i)$ Show that if all normals to a connected surface pass trough a fixed point, the surface is contained in a sphere. $ii)$ Prove that if a regular surface $S$ ...
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2answers
61 views

Function for a sphere

I believe that there is something fundamentally wrong with my understanding of functions but I can't pin point what it is, so I would greatly appreciate any guidance. Consider a unit sphere, ...
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1answer
35 views

What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
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2answers
37 views

Moebius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Moebius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
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0answers
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Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
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1answer
41 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 ...
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3answers
4k views

How to calculate surface area of a curved plane?

could anyone explain how to calculate the surface area of a curved plane? I am trying to calculate the surface area of a "vaulted" ceiling that is 24' long, 7' wide, and the height of the curve is 4' ...
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1answer
73 views

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 ...