For questions about surfaces.

learn more… | top users | synonyms

1
vote
0answers
26 views

How to prove that dim$|L|=D^/2+1$ on a $K3$ surface?

Let $D$ be an irreducible curve on a $K3$ surface $S$, and let $L=\mathcal{O}_S(D)$. The Riemann Roch formula on the $K3$ surface is given by : $\qquad\qquad\qquad\qquad\qquad ...
1
vote
0answers
29 views

Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
2
votes
2answers
35 views

Why is $h^1(S,\mathcal{O}_S(-D))=h^0(D,\mathcal{O}_D)-1$ on a $K3$ surface?

Let $D$ be a divisor on a $K3$ surface $S$. We have an exact sequence : ...
2
votes
2answers
36 views

Why is this divisor on a K3-surface not effective?

Let $D$ be a divisor on a $K3$ surface and set $L:=\mathcal{O}_S(D)$. Riemann Roch theorem : $\chi(L)=\chi(\mathcal{O}_S)+\frac{1}{2}D.(D-K)$ reduces to ...
0
votes
0answers
20 views

Interval Range function for 3d implicit surface

Given the trivial implicit surface $$f = \sqrt{x^2+y^2+z^2},$$ what is an equation to return the interval range of the surface? Background: I'm a $3d$ artist moving into programming, so I don't have ...
0
votes
1answer
14 views

Pyramid Surface Area

Square based pyramid has a side length of $220$ (b) and a height of $105.$ Find the surface area. I tried by "doing" Pythagorean theorem $110^2+105^2=s$ then i did the equation for surface area ...
0
votes
0answers
9 views

About fibrations with isomorphic smooth fibers

Let $S$ be a smooth projective (complex) surface with a fibration $f:S\longrightarrow B$ over a base curve $B$. If all the fibers of $f$ are isomorphic to $F$, which is smooth, can I conclude that ...
2
votes
1answer
45 views

Connected Sum of Surfaces

I am trying to prove that the connected sum of surfaces is a surface. My definition of surface is: A topological space locally homeomorphic to $\mathbb{R}^2$, second countable, Hausdorff and ...
1
vote
1answer
22 views

Cylindrical coordinates - Surfaces

I found the following: Cylindrical coordinates $(\rho , \theta , z)$. This system consists of the following coordinate surfaces: Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ ...
0
votes
0answers
34 views

Cartesian Ovals formed by planar intersections of which surface?

Find a surface [ either parametric or $ f (x,y,z) = 0 $] in 3-space, symmetrical about (Z,Y) plane producing Cartesian Ovals in (X,Y ) plane projections on intersection with planes Y = 0 or Y/Z = ...
1
vote
0answers
41 views

Family of curves with one base point and a blow-ups.

Suppose that a smooth complex surface $X$ is covered by a family of curves $\{C_\alpha\}_{\alpha\in\mathbb P^1}$. Suppose moreover that $\bigcap C_\alpha=p\in X$, and that these intersection are ...
2
votes
0answers
49 views

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 ...
-2
votes
1answer
20 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.
1
vote
1answer
36 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 ...
1
vote
1answer
20 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 ...
1
vote
1answer
73 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 ...
1
vote
1answer
50 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: $$f'(x)=\frac{3x^2}{10}$$ $$f'(x)^2=\frac{9x^4}{100}$$ ...
0
votes
1answer
42 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
votes
0answers
27 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 ...
0
votes
1answer
28 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 ...
1
vote
1answer
43 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 ...
0
votes
1answer
33 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 $$D=a^2/4$$ ...
1
vote
2answers
72 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 ...
1
vote
1answer
20 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 $\Delta \subset PSL_2(\mathbb{R})$ ...
1
vote
1answer
70 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
votes
1answer
43 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
1
vote
1answer
52 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 ...
0
votes
1answer
17 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 ...
2
votes
1answer
48 views

Principal directions bissect the asymptotic directions

How can one prove that at a hyperbolic point, the principal directions bissect the asymptotic directions?
0
votes
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
votes
1answer
130 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 ...
1
vote
3answers
39 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 = ...
6
votes
1answer
70 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
votes
1answer
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 ...
2
votes
0answers
50 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 ...
2
votes
0answers
49 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 ...
2
votes
0answers
27 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 ...
0
votes
1answer
31 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 ...
0
votes
1answer
69 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 ...
0
votes
1answer
116 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 ...
1
vote
1answer
43 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 ...
2
votes
1answer
94 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 ...
3
votes
1answer
44 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 $H\subseteq\mathbb P^n$ ...
1
vote
1answer
65 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 ...
2
votes
1answer
107 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 ...
1
vote
2answers
51 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 ...
1
vote
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))$?
1
vote
2answers
53 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 ...
2
votes
0answers
46 views

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 ...
1
vote
2answers
83 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 ...