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

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What is the object on the front of Larson and Edwards' calculus and pre-calculus textbooks called?

There is this incredible glass figure on the front of my Calculus textbook, I searched online for what this figure is called and the formula for creating it, but I can't find it. I think it is a ...
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1answer
63 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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27 views

Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
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32 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
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2answers
65 views

Polygonal presentations: why no two-letter words?

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...
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1answer
32 views

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
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1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
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19 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
4
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1answer
75 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
0
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1answer
25 views

Parameterizing part of sphere

the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$ first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...
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63 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
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1answer
84 views

About timelike surfaces with non-diagonalizable shape operator.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$ Take a differentiable surface $M \subset \Bbb L^3$, ...
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2answers
22 views

How to simplify this equation with change of variables,

I have the equation, after completing the square: $$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$ How can I further simplify this equation? I need to find the volume inside of this surface. ...
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2answers
35 views

Find the volume enclosed by the surface $S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$

Find the volume enclosed by the surface $$S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}.$$ My attempt was this: I moved the tricky $xy$-term over to the r.h.s. I now have $$x^2+y^2+z^2 = 1-xy,$$ ...
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1answer
51 views

Computing the volume inside a surface S, using a seemingly unrelated result,

Consider the surface $$S = \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$$. What is the volume inside S? This is actually part (b) of the question. I'm not sure which approach to take. But part (a) of the ...
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0answers
17 views

Rotation Invariant Descriptors for Bivariate Polynomial Surfaces

I start with a simple example. Consider: $$ z = x^2 + y $$ and $$ z = y^2 + x $$ Visually speaking, both of these are essentially the same surfaces rotated by 90 degrees about the z-axis. I am ...
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41 views

Complements of homeomorphic subsets of surfaces homeomorphic?

Let us consider a surface $S$ and a subset $T\subset S$, where $T\cong D^2$ are homeomorphic and $D^2$ is an open disc. Let $T\cong Q\subset S$ be homeomorphic. Are $S\setminus T $ and $S\setminus Q$ ...
3
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2answers
44 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
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1answer
17 views

How to graph the intersection of two surfaces

Another way of asking this is: How would one graph a 1 dimensional line (the intersection) in a 3 dimensional space. Some context for the question: Let's say I have two spheres ...
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1answer
28 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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12 views

Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
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1answer
28 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
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139 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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1answer
24 views

How to work out the angle of a line passing through a plane

I have a triangular plane composed of three points. From this it it easy to deduce that the plane is in fact composed of two vectors which must touch at some point. because all of this is relative, ...
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82 views

Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
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1answer
20 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
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25 views

Find the volume of the region by integration

I have to find the volume of the region bounded by $x + z = 1$;$ y + 2z = 2$;$ x = 0$;$ y = 0 $;$ z = 0$; I tried to sketch the graph separately in the $y-z$ plane and then in $x-z$ plane. But I am ...
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48 views

What substitution can be used to evaluate the integral giving the area of the surface of revolution of the curve $x = \sqrt[3]{y}$?

The question is: Find the area of the surface generated by revolving the given curve about the $x$-axis: $$x = \sqrt[3]{y}, \qquad 1 \leq y \leq 8.$$ Now, all is well, simple enough question, ...
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26 views

Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
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1answer
60 views

When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
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26 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
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2answers
17 views

Formula to find lenght of material I need to make a sprint

could somebody write me if you know how to calculate the lenght of a material i need to make a certain spring. I need any true formula you have.
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1answer
17 views

Parameterize the following surface: $x +y +z=1$, $x,y,z>0$.

I need to parameterize the following surface: $x +y +z = 1$, $x,y,z>0$. I tried to put: $\sigma(u,v)=(u+v,u-v,-2u+1)$, but does it solve the case of $x,y,z>0$?
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1answer
22 views

Ruled surface out of lines of curvatures

I'm trying to proof the following statement: Let $c$ be a curve inside a surface element $f:U\rightarrow\mathbb{R}^3$ (i.e $c=f\circ\gamma$ where $\gamma:I\rightarrow U$). Then $c$ is a line of ...
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1answer
60 views

Need help understanding a relation between the fundamental forms

The book I am reading briefly mentions this relation between the fundamental forms but gives no explanation of how they got it. Take the following as the Weingarten Map/Shape Operator where $\nu$ is ...
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28 views

Gradient of second fundamental form

In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented: The extramum values of $II(X,X)$ ...
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3answers
179 views

What surface is represented by the following equation

$$\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}+\sqrt[3]{z^{2}}=1$$ Taking cubes of both sides only leads to a more complicated formula. How should one interpret this one. And, also if you could point me to a tool ...
2
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2answers
70 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
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0answers
68 views

whether any shape can be placed on a tiled surface?

After read "Prove that any shape 1 unit area can be placed on a tiled surface",I think on a surface of equal square tiles where each tile side is 1 unit long,the shape ,less than some constant C>1 ...
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1answer
67 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
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1answer
30 views

“Uniqueness” of the Multi- Dehn-twist

I'm trying to writing down a proof for the following claim about Dehn-Twists: Let $\{a_1,...,a_m\}$ be a collection of distinct nontrivial isotopy classes of simple closed curves in a surface $S$ ...
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1answer
70 views

Surface area generated by revolving $r = \sqrt {\cos 2\theta}$

I've been giving a good time trying to solve this problem, I do not find a clear way to solve appreciate your help. \begin{array}{rcl} r& =& \sqrt{\cos 2\theta } \end{array} This Around to ...
2
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1answer
80 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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0answers
52 views

Any quartic in $\mathbb P^3$ contains only finitely many lines.

I want to prove thath any quartic $X$ in $\mathbb P^3$ contains finitely many lines, but I don't know any method for computing lines on a surface. What is the idea of the proof?
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3answers
28 views

Cornered sphere homemorphic to unit sphere

A buddy of mine asked me this question, to which I found it intuitively obvious, but was unable to come up with a proper proof. Consider the so-called cornered sphere, defined by $x^4+y^4+z^4=1$ in ...
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Can the same surface have minimal genus in both a 3-manifold and a 4-manifold?

By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$. ...
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2answers
46 views

How to find the tangent plane to a given point on a surface?

How can you find the tangent plane to a given point on a surface? (Verbal descriptions preferred) I'm thinking you can find the "vector versions" of two directional derivatives (maybe the partial ...
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1answer
29 views

Find the “surface vertices” of a collection of points.

I am currently doing some experiments in order to simulate liquids. I have a collection of 3D points that interact with each other to form a body of water. I would like to form a mesh from these ...
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41 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
0
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1answer
29 views

Finding all points on a surface where the tangent plane is parallel to the plane 5x+3y-z=0

Consider z=f(x,y)=x^3 + 2xy + y Find all points on the surface where the tangnent plane is parallel to the plane 5x+3y-z=0 So I took the gradient of f (x,y,z) and got (2x^2 i , 2x+1 j , -1 k) And ...