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Questions tagged [surfaces]

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

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About the 3d Surface

I am creating the Enneper Surface in Geogebra with the following equations. Is the surface or equation correct, and does the Enneper surface look like this? I'm confused because Wikipedia has a ...
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Do Carmo Section 2.2 Proposition 2 Proof Clarification

I am having some difficulties in understanding the proof given by Do Carmo of the following proposition. I will run through the proof and comment on it as it progresses and place my questions ...
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Gaussian curvature with Laplacian

In a lot of papers and books, I have seen the following expression of Gauss Curvature in $2$-dimensional surfaces with a conformal metric $$\overline{g} = e^{2u}g$$ $$K - \overline{K} e^{2u} = \...
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Exterior derivative of a 1-form on a surface for non-regular mappings

I would appreciate some help for this problem. I have no idea how to start. Let $M\subset \mathbb{R}^3$ be a smooth surface. Let $\phi$ be a $1$-form on $M$. By definition, the exterior derivative of ...
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Surface area of part of sphere $x^2+y^2+z^2=a^2$ enclosed by cylinder $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Note the given cylinder $a>b>0$ is elliptical. What I did: I took one fourth of the ellipse in the $xy$-plane and called it $K$, with $$K= \left\{ (x,y):~ 0 \le x \le a,~ 0 \le y \le b\sqrt{1-\...
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What is a simple parametric surface?

What is the formal definition of simple parametric surface? We are dealing in $\mathbb R^3$. I came across the term in Apostol's Calculus (2nd volume), where he states that the parametric function $\...
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Gluing hyperbolic convex polygon

How can I prove the existence of a regular convex $4n$-gon, $C$, with the angles $\pi /2n$ and how would I be able to show that $C$, is a gluing diagram for a hyperbolic surface after is has been ...
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Unit circle and cone

I know that in the following question, I would consider finitely many cases depending on where the points are located, but I would appreciate help on this: a) How would I define a unit circle in a ...
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union of horizontal planes with non discrete heights is not a surface

I trying to prove that the union of horizontal planes with non discrete heights is not a surface. My intuition says that we have some problem with the topology. My best attempt is the following: ...
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28 views

Curve length on a surface

Consider a surface with the equation $z=x^3 − 3xy^2$. So I have parametrization $p(u,v) = \begin{bmatrix} u \\ v \\ u^3-3uv^2\end{bmatrix}~.$ I have found that the first fundamental form of this ...
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Is there something missing in the usual calculation by integral for hyper-surface volume of a $3$-D ball?

Where is my $4$-D intuition going wrong about hypersurface volume of a $3$-D ball? There are plenty of examples of how to calculate the hypersurface volume and hypervolume of a $3$-D ball (eg. ...
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Calculating the directional derivative without finding an explicit curve on the surface

If we have a surface, say $S$, and a smooth map $f: S \to R$ (or $\mathbb{R}^3$), and a parametrization of that surface $ x: U\subseteq \mathbb{R}^2 \to S$, how do we find the directional derivative ...
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do Carmo Chapter 3.3 Question 14 Surface of Revolution and its planar points

Consider the surface obtained by rotating the curve $y = x^3$ , $−1 < x < 1$ around the line $x = 1$. Show that the points obtained by rotation of the origin $(0, 0)$ of the curve are planar ...
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1answer
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Homology condition - bounding a disk in a handlebody?

Suppose that $\gamma_1,...,\gamma_n$ are a set of disjoint simple closed curves on a closed orientable surface $\Sigma$ that all bound disks in some handlebody $H$ with $\partial H = \Sigma$. Let $\...
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Show that all triangulations of a compact surface are equivalent

I'm having trouble with solving the following question: Let $T_1, T_2$ be two finite triangulations of a compact surface. Show that if $E_{T_1}\cap E_{T_2}$ is a finite set of points, where $E_{T_i}$ ...
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How can I prove a curve lies on an ellipsoid?

I am trying to prove a curve parameterized by $$\mathbf{r} (t) = \cos(t) \, \mathbf{i} + \sqrt{2} \sin(t) \, \mathbf{j}-\sin(t) \, \mathbf{k}$$ lies on an ellipsoid. How do I do this?
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Proving a weaker version of the Jordan Curve Theorem (by induction)

I'm trying to solve the following question: Suppose that the sphere $ \mathbb{S}^2 $ is given the structure of a closed combinatorial surface. Let $C$ be a subcomplex that is a simplicial ...
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Are there any mathematical methods which can analogize a discrete 2D matrix into a function (describing the continuous surface)?

It's a very naive and ignorant question, but let's say if we have a 2D matrix (with high dimensionality), are there any methods using which we could create a continuous surface (i.e. described by a ...
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Parametric equation of a hill shape surface

What is a simple parametric equation to plot a surface shaped like a hill? like the one made by the 3D gauss bell $e^{-(x^2 +y^2)}$ Any help is welcome.
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Components of $\mathbb{S}^2$ as a closed combinatorial surface.

I'm trying to solve the following problem, and I'm not having much luck: Suppose that the sphere $ \mathbb{S}^2 $ is given the structure of a closed combinatorial surface. Let $C$ be a ...
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Finding intersection of a sphere and a plane

Suppose I have the sphere $\ x^2 + y^2 + z^2 = 4 $ and the plane $\ x-y\sqrt{3} =0. $ How do I find the intersection curve and write it in polar terms? In polar coordinates the sphere is just $\ ...
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Isoperimetric problem on a generic surface

In the plane, fixed two points, the curves that minimize the length are straight lines, i.e curves with zero curvature. In the plane, fixed an area, the curves that minimizes length are circles, i.e. ...
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Does $x_0^4+x_1^4+x_2^4+x_3^4-ax_1x_2x_3x_4$ really define a surface?

I'm working through Shafarevich's Basic Algebraic Geometry book and in one of the exercises (number II.1.15 in my edition), he asks "for what values of $a$ does the surface $x_0^4+x_1^4+x_2^4+x_3^4-...
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Meridians and Parallels on a Unit Sphere

Let $S$ be the unit sphere in $\Bbb R^3$ with centre $(0, 0, 0)$ $\sigma(u, v) = (\cos v/\cosh u,\sin v/\cosh u,\tanh u)$ is a parametrization of $S$ minus the north and south poles. Show that ...
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Showing a Parametrization!

Show that $σ(u, v) = (\sin u,\sin v,\sin(u + v))$ is a parametrisation of $S$. $S= (x^2 − y^2 + z^2)^2 = 4x^2z^2(1 − y^2)$
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37 views

Isn't a smooth map having rank 2 injective by the corollary of inverse function theorem?

In the book of E. Bloch, at page 171, it is stated that Corollary: Let $U\subseteq \mathbb{R}^2$ be an open subset, and let $f: U \to \mathbb{R}^3$ be a smooth map. If for $p \in U$, $Df(p)$ ...
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How to construct a two-sheeted cover of a non-orientable surface?

Let $S$ be a non-orientable surface. Then there exists a two-sheeted covering map $p:S'\to S$ with $S'$ an orientable surface. I want to know how to construct $p$. I know that $\mathbb{R}P^2$ is 2-...
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On what condition the intersection of all tangent planes of the surface $z = y f(\frac{x}{y})$ is $\{ (0, 0, 0) \}$?

Problem Suppose $f \in C^1$, $A$ is the intersection of all tangent planes of the surface $z = y f(\frac{x}{y})$. Under what condition, we have $A = \{ (0, 0, 0) \}$ ? That is, to find a set $F$, s.t....
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Proving that level sets of a surface near local max/min are closed curves

Let $S$ be a surface in $\mathbb{R}^3$ defined by the graph $z=f(x,y)$ of a smooth function $f: \mathbb{R}^2 \to \mathbb{R}$. Suppose furthermore that $S$ satisfies the second-derivative test for a ...
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Systematic approach to triangulation closed combinatorial surfaces

I was wondering whether there is a systematic approach to the triangulation of closed combinatorial surfaces, which we know can be shown to be homeomorphic to polygons with complete set of side ...
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Coordinate-Free Computation of Covariant Derivative

Suppose $S$ is an oriented 2-dimensional manifold equipped with a Riemannian metric $\langle \cdot,\cdot \rangle$. Let $\theta$ be a 1-form on $S$. By non-degeneracy, there is an identification of the ...
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Compatible conditions on first fundamental form

The fundamental theorem of surface theory asserts that the existence of a simply connected surface in $\mathbb{R}^3$ whose first and second fundamental forms are $I$ and $II$, if the coefficients ($g_{...
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Finding angle of intersection between two curves

I am trying to solve a problem in Do Carmo, a book which I often find incomprehensible. Let $X(\varphi,\theta)=(\sin(\theta)\cos(\varphi),\,\sin(\theta)\sin(\varphi),\,\cos(\theta))$ be a ...
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Can a compact surface-with-boundary admit arbitrarily high genus subsurfaces-with-boundary?

More precisely, does there exist a compact surface-with-boundary $\Sigma$ with the following property? For every $g\geq 0$, there exists a subsurface-with-boundary $\Sigma'\subseteq\Sigma$, where the ...
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61 views

On the connected sum of a surface with a torus

I am studying the classification of Surfaces, and run into the notion of connected sum. We define it in terms of triangulations. I want to show the following. Let $S$ be a triangulated surface. I want ...
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Prove that $M = f(\mathbb{R}^{n})$ is a $C^{\infty}$ submanifold of $\mathbb{R}^{n}$ when $f \circ f = f$.

Let $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ be a $C^{\infty}$ function such that $f\circ f = f$. Prove that $M = f(\mathbb{R}^{n})$ is a $C^{\infty}$ submanifold of $\mathbb{R}^{n}$. Definition. A ...
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Shortest distance to a spheroid

I have a point $P'$ given by known Cartesian coordinates $(x',y',z')$ and an oblate spheroidal surface parameterized by \begin{align} x &= a \cosh{\mu} \cos{\nu} \cos{\phi} \\ y &= a \cosh{\mu}...
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44 views

Deriving the geodesic equations on a cone. Are these equations correct?

So I'd like to derive the geodesic equations of a cone which I call $\mathcal{C}$. I believe I've done this correctly but would like a second opinion. $\mathcal{C}$ can be described by taking the line ...
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1answer
43 views

Why can the gaussian curvature be computed this way?

I understood everything up to the last part ("It follows that the gaussian curvature $K = K(s, v)$ of the tube is given by..."). Why? What am I missing here? I know we can just compute it from the ...
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2answers
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Is $r^2-4r\cos(\theta)=14$ an equation of a circle or cylinder?

A question asks to identify the surface of the polar equation \begin{equation}\tag{1} r^2-4r\cos(\theta)=14. \end{equation} I converted $(1)$ into Cartesian coordinates: \begin{equation}\tag{2} (x-2)^...
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Showing that the unit sphere is a surface

I was going through Andrew Pressley's book and on the place where they have discussed surfaces, there is one example which deals with the unit sphere. I have understood to the point where they have ...
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Klein bottle as two Möbius strips with fundamental polygon

My question is the following. If we operate with the fundamental polygon of a Klein bottle in order to obtain two Möbius bands, really you don´t obtain two new Klein bottles? I think that the ...
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Sketch the surface of $y^2 +4z^2 = 4$

To clarify, the standard form of the above equation is in the form of an ellipsoid. However, as it is independent of its $x$ values, may I know if the sketch of the surface is still an ellipsoid or ...
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34 views

What's the name of the surface resulting from rotating the exponential curve?

In three dimensions, if we rotate the curve $(x,0,e^x)$ for $x\in\mathbb{R}$ around the axis through the points $(0,0,1)$ and $(1,0,0)$ by the angle $2\pi$, what's the resulting surface called? It ...
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32 views

What is the name of this surface? (resembles Boy's surface with holes)

Does anyone know the name of this surface? I feel like it has a particular name but can't seem to remember it for some reason.
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28 views

Understanding whether an equation gives 4D or 3D surface

Consider the following surfaces S1: xyz=10 and S2: z=x^2+y^2 I cannot understand whether S1 and S2 are three dimensional or 4 dimensional. S1 seems to be 4 dimensional, as I can consider x, y and z ...
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Parametric equation of a parabola rotating on its axis

Write the parametric equation of the surface generated by a parabola rotating around its axis. I guess it's simply getting from the parabola equation to the parametric equations of a generic ...
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how to calculate surface area bounded by $g(\theta) = (\cos(\theta) , \sin^3(\theta))$

let $g(\theta) = (\cos(\theta) , \sin^3(\theta))$ find the surface area bounded by this curve. $ 0 \leq \theta \leq 2\pi $ this was an exam question from today i had VERY hard time calculating the ...
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How does $\cos\alpha dA=dydz$ come?

In the red rectangle, author defined what is surface integral in terms of parametric form. I am confused with the expression in the yellow rectangle. Can you please explain? How does $$\cos\alpha\; dA=...
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31 views

Parametric equations for the surface of revolution generated by rotating the hyperbola $z^2−y^2=2$ about the $y$-axis

Write the parametric equations of the surface of revolution generated by the hyperbola $$\frac{z^{2}}{2} − \frac{y^{2}}{2} = 1$$ (in the $yz$ -plane) when it rotates around the $y$-axis. Any hint ...