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

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A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
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Two cylinders cutting

Find equation of non-circular cylinder $ f (y,z)=0 $ which cuts a circular cylinder $ x^2+y^2 = a^2 $ to produce an intersection curve of constant geodesic curvature on the circular cylinder. What is ...
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Constant K surfaces producing lines of constant geodesic curvature

When two spheres intersect or when a sphere intersects a plane the line of intersection has constant tangential/geodesic curvature $ k_g$ (small circle). How is this generalized? Or, Under what ...
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surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
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1answer
67 views

Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? ...
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Describe the graph of the function: Level Sets, Intersections

Describe the graph of the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow |y|$$ computing some level sets and some intersections. I have done the following: The level curves ...
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48 views

Level set for the function

Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$ I have done the following: The level ...
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34 views

Compute surface area about the $x$-axis

Compute the surface area swept out if the graph $y = e^{-x}$, for $0 \leq x \leq 1$, is revolved about the $x$-axis. The formula for the surface area is $$S = 2\pi\int_{a}^{b} f(x) \sqrt{1 + ...
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What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow ...
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35 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
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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 ...
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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 ...
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2answers
37 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 : ...
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2answers
45 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 ...
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22 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 ...
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1answer
18 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 ...
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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 ...
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53 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 ...
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24 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{ ...
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47 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 = ...
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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 ...
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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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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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68 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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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
149 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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120 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
74 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 ...
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30 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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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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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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36 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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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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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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Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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74 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
26 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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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 ...
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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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3answers
50 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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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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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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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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78 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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188 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 ...