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

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Effective divisor vs curve on surface

Hartshorne in his book, with the term "a curve $C$ on a surface $S$" (over an algebraically closed field $k$) means that $C$ is an effective divisor on $S$. So, can I conclude that a "a curve $C$ on ...
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1answer
23 views

Surface Area of cylinder

A roller $150$cm long has a diameter of $70$ cm.To level a playground it takes $750$ complete revolutions. Determine the cost of leveling the playground at the rate of $75$ paise per sq. metre.
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1answer
63 views

Linking surface integral of a gradient field to a contour integral [duplicate]

I have a vector field $F$ deriving from a scalar potential $f$, i.e. $F=\text{grad}(f)$. I want to compute the integral of $F$ over a surface (To evaluate the flux of $F$). I think there exists a ...
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1answer
23 views

From a family of projective curves to a surface

Suppose that $$\mathcal F:\;(X^3+Y^3+Z^3)\lambda+Z^2X\mu=0$$ is a family of projective plane curves parameterized by $(\lambda:\mu)\in\mathbb P^1(\mathbb C)$. This family of curves forms a surface ...
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1answer
136 views

Parametrization where coordinates lines are lines of curvature

I am asked to prove that given a surface $S$ and a point $p\in S$ non-umbilical, then there exists $U$ open in $\mathbb{R}^2$, there exists $Y:U\subset \mathbb{R}^2\longrightarrow \mathbb{R}^3$ a ...
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1answer
104 views

Find the surface area generated when the curve is revolved around the x-axis

Find the surface area generated when the curve is revolved around the x-axis $y=\frac{x^3}{10}$ on $[0,\sqrt{10}]$ This is what I have so far: $$f'(x)=\frac{3x^2}{10}$$ $$f'(x)^2=\frac{9x^4}{100}$$ ...
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1answer
66 views

Get 4 points lying on the plane by given normal

I would like to create plane using 4 points (which I need to find out), when I know the intersection point of the 2 diagonals in the plane. Next thing I know, that the Y coord of 2 bottom points will ...
2
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29 views

Curve minus a point on a surface

Let $S$ be a smooth complex projective surface and let $C\subseteq S$ a curve (maybe not integral). Suppose for example that $C$ is a fiber of a certain fibration of $S$ over $\mathbb P^1$. Now ...
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1answer
37 views

Evaluate a double integral over a domain

I'm asked to evaluate $$\iint(x^3+y)$$ over the ellipse on the xy plane such that $2x^2+y^2<2y$ I figured that the ellipse can be parametrized by $$\vec r(t)=\left(\frac{\cos t}{\sqrt2};1-\sin ...
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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 ...
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1answer
35 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 $$D=a^2/4$$ ...
2
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2answers
108 views

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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1answer
23 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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1answer
75 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 ...
3
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1answer
45 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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1answer
68 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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1answer
23 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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1answer
50 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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0answers
45 views

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 ...
5
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1answer
149 views

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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3answers
46 views

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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1answer
100 views

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 ...
2
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1answer
69 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 ...
2
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0answers
60 views

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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56 views

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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45 views

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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1answer
46 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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1answer
76 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
174 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
60 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
137 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
46 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
96 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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1answer
157 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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2answers
54 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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1answer
24 views

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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2answers
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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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2answers
91 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 ...
4
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1answer
191 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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1answer
54 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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2answers
73 views

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
41 views

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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17 views

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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2answers
212 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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1answer
38 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
60 views

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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46 views

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 ...