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

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When does there exist a isometric transform between the surfaces $S$ and $\widetilde{S}$?

Suppose there are two $E^3$ surfaces, $$S:\mathbf{r}(u,v)=(au,bv,\frac{au^2+bv^2}{2})$$ ...
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49 views

How can we calculate the unit normal $\textbf{N}$ of the sphere?

I want to show that the normal curvature of any curve on a sphere of radius $r$ is $\pm \frac{1}{r}$. $$$$ The normal curvature is $\kappa_n=\gamma '' \cdot \textbf{N}$, where $\gamma$ is a ...
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3answers
65 views

Compact surface

To check if the surface $x^2-y^2+z^4=1$ is compact, we have to check if the surface is closed and bounded. Could you give me some hints how exactly we check that? How can we check if it closed and ...
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2answers
92 views

Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
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0answers
72 views

Open subset of a plane [duplicate]

Suppose that the second fundamental form of a surface patch $\sigma$ is zero everywhere. How can we prove that $\sigma$ is an open subset of a plane? The second fundamental form of a surface patch ...
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133 views

Two twisted cubic curves in $\mathbb P^3$ intersect iff they lie in a common cubic surface

Let $C_1$ and $C_2$ be twisted cubic curves in $\mathbb P^3$. I want to prove that they intersect if and only if they lie in common cubic surface, perhaps singular. The second condition can be ...
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1answer
117 views

Show that it is a smooth surface

Show that the ellipsoid $$\frac{x^2}{p^2}+\frac{y^2}{q^2}+\frac{z^2}{r^2}=1$$ where $p$, $q$ and $r$ are non-zero constants, is a smooth surface. To do this do we have to take a parametrization of ...
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2answers
42 views

Why is this the equation of the tangent plane?

I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point ...
6
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41 views

Visualizing order 3 mapping class of genus 2 surface

Let $\Sigma_2$ be a closed genus $2$ surface. There exists an orientation-preserving diffeomorphism $f:\Sigma_2 \rightarrow \Sigma_2$ of order $3$. The diffeomorphism has $4$ fixed points (each, of ...
3
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58 views

Hypersurfaces containing given lines

Let $L_1, ..., L_n$ be non-intersecting (general, if necessary) lines in $\mathbb P^3$. I need to find the dimension of the space of polynomials of degree $d$, vanishing on these lines. ...
4
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1answer
75 views

Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?

I am told that for a surface with genus $g$, call it $F_g$, the abelianization of $\pi_1(F_g) = \langle a_1, b_1, \ldots , a_g, b_g \mid [a_1, b_1] \cdots [a_g,b_g] = e \rangle$ is $\pi_1(F_g)^{ab} = ...
2
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1answer
35 views

Geodesics on the Surface of Revolution from do Carmo's book

This is a question I encountered at DoCarmo's Differential Geometry of Curves and Surfaces p258. I do not know this sentence just below the second equation: "(Of course the geodesic may be tangent ...
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2answers
66 views

Determine the dimension of the set of surfaces of $\mathbb{P}^{3}$ that contain certain conic.

Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ I have to ...
4
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1answer
189 views

Curl of unit normal vector on a surface is zero?

I have a scalar field $\phi$. From this field, I define an iso-surface $\phi=\phi_{iso}$. The unit normal vector on this surface is ...
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19 views

Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
2
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1answer
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Applications of the generalized Gauss-Bonnet Theorem for surfaces

Quoting Do Carmo's 'Differential Geometry of Curves and Surfaces': "We have only to think of all possible shapes of a surface homeomorphic to a sphere to find it very surprising that in each case the ...
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16 views

surface has two geodesic with fixed angle must be a developing surface

It seems easy to show by Gauss-Bonnet theorem that a surface which has two families of geodesics with fixed angle $\theta$ must be locally flat, i.e. its Gauss curvature $K=0$. But to show it is in ...
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Gauss-Bonnet-Chern Theorem coincides with Gauss-Bonnet Theorem in the two dimensional case.

The generalized Gauss-Bonnet theorem says that: Let $M$ be a closed oriented Riemannian manifold with an even dimension $n$, then $$ \int_{M}\Omega=\chi(M). $$ In that formula, $\chi$ is the euler ...
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1answer
18 views

Surface areas of different submanifolds of $\Bbb R^3$

Can you circumscribe a continuous, smooth manifold in $\Bbb R^3$ with another manifold that completely encapsulates it but has a surface area which smaller than that of the one contained? Is there a ...
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83 views

Must $\vec{n}$ be a Unit Normal Vector (Stokes' Theorem)?

If $S$ is an oriented, smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation, then for some vector field $\vec{F}$: $$\oint_C \vec{F} \cdot d\vec{r} ...
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56 views

Parameterization of Ellipsoid

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed ...
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1answer
33 views

Parametrization of a cylinder that is parallel to x axis

The answer is no it does not matter. The surface is $y^2+z^2=4$, I parametrized it so: $\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$ But Pauls Outline works through the ...
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1answer
33 views

Evaluation of an oriented surface integral

I am having a bit of trouble understanding example 1 in Paul's Calculus Notes page on surface integral: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx I understand how to do ...
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1answer
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Decomposing a surface $S$ with a simple closed curve $\Gamma$

In class we learned about how the Euler characteristic changes when we take a connected sum of surfaces $M_1$ and $M_2$: $$\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - 2,$$ and it made me wonder how ...
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1answer
41 views

What exactly is a surface integral?

I'm learning surface integrals right now and I don't think I fully understand what they are. What exactly do surface integrals represent? Is it volume? The basis for surface integrals seems just like ...
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2answers
50 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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1answer
24 views

Extend simplicial homeomorphism in a PL surface

Let $S$ be a connected PL closed surface. How can I show that, given a 2-simplex $\Delta$ in $S$ and a simplicial homeomorphism $g:\Delta\to \Delta$ that preserves orientation, this can be extended to ...
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28 views

Prove a surface is a plane

Let $\Sigma$ be a $C''$ surface defined on an open connected set D in the UV plane. Suppose $d^2\Sigma=0$ in $D$,prove that $\Sigma$ is a plane. I know $\Sigma$ has the form ...
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1answer
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How to find contact points of a plane with an uneven surface

I try to find the contact points of a plane when it is placed on an uneven surface. For example a book that is placed on uneven terrain, where would it touch the ground? I already have some ideas how ...
3
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38 views

Images of linearly equivalent divisors are linear equivalent?

Let $f:X\longrightarrow Y$ be a finite morphism of degree $d$ projective surfaces over $\mathbb{C}$. Suppose that $X$ is smooth. Let $L'$ be a line bundle on $Y$, and let $L=f^*L'$. 1) Consider a ...
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How to convert density function to hermite data?

[Apologies for any mistakes in tagging, I'm new to this site and topic] I'm rendering a procedural surface in a game engine. A method returns density for a particular point in space. Density is a ...
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61 views

How to parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute a tangent plane

How do I parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute the tangent plane at $(1, \frac{1}{3}, 0)$ using the resulting parametrization? I know that the tangent plane should be ...
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Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
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1answer
182 views

Fundamental forms of constant mean curvature surfaces

For surfaces of constant mean curvature in $E^3$, prove that either they are all-umbilic-points surfaces or their fundamental forms can be represented as following: I $=\lambda(u,v)(dudu+dvdv)$ ...
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Isomorphism on Homology induced by Inclusion of CW-complexes

I want to prove the following and I do not know how. Let $X$ be a CW-complex of dimension $n-1$ and let $Y$ denote the space obtained from $X$ by attaching a finite number of $n$-cells. Then the ...
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63 views

How to interpret a 3D plot

I am trying to interpret 3D plots but its driving me nuts. Lets say I want to see the relationship between 3 variables and I have data as following: ...
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62 views

Compute flux of a vector field with divergence of zero (multivariable calculus)

I'm attempting to solve this problem: Compute the flux of the vector field $\vec F = (e^z − 2xy, y^2 − e^z, 2xy − y^2)$ through the surface S that is the part of the surface $ z = x^4 + e^{y^2} $ ...
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1answer
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Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
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1answer
47 views

Generally compute Gauss curvature

For a surface $F(x,y,z)=0$, we want to compute its Gauss curvature. I tried to suppose $z=f(x,y)$ locally and get a complicated expression. Is there any direct way to compute this? Thanks for your ...
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1answer
85 views

When are mapping tori isomorphic as bundles over the circle?

Suppose $\Sigma$ is an orientable genus-$g$ surface (possibly with boundary). The mapping torus corresponding to an orientation-preserving diffeomorphism $\phi: \Sigma \to \Sigma$ is the quotient ...
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Manifold is cut along hypersurfaces; how to define a connection on this?

May $M$ be a smooth manifold with boundary $\partial M$. Metrics and Connection can be defined everywhere. But now this manifold is cut by a smooth hypersurface $A$ and the cut goes along $M \cap A$ ...
3
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1answer
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Non-existence of embedded incompressible surfaces

I want to prove the following assumption: Let $g,h$ be natural numbers with $g > h$ and let $S_g$ be the closed, orientable surface of genus $g$. Then, there is no (smooth) map $f: S_g \to S_h ...
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Path Lengths on a Sphere example.

What does $\rho$ mean as to be measured along geodesics and more importantly how would i be able to parametrize this accordingly as being on the sphere's surface? I know that I have to use the First ...
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1answer
141 views

Find the Equation of the Envelope of a Family of Line (Plane) Segments

Consider the first quadrant in the $OXY$ plane in $\mathbb{R}^2$. Point $O$ is the origin and the points $P$ and $Q$ are chosen on the $y$-axis and the $x$-axis, respectively as it is showed in the ...
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1answer
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what is the minimum surface area shape required in order to contain a 1 meter line at all angles

been stuck on solving/proving the following puzzle: You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that ...
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1answer
34 views

For all unit vectors $\mathbf u$ and a positive definite $\mathbf C$, what surface do vectors $\mathbf u \mathbf u^\top \mathbf C \mathbf u$ form?

Let $\mathbf C$ be a positive-definite $k\times k$ matrix. For all vectors $\mathbf u\in \mathbb R^k$ of length $\|\mathbf u\|=1$, consider vectors $\mathbf {uu}^\top\mathbf{Cu}$; they form a surface ...
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2answers
66 views

A surface with Euler characteristic of $-1$ [closed]

Is it possible to a have a surface that has an Euler Characteristic of $-1$ and what would that surface be homeomorphic to? $\displaystyle \chi \left({M}\right) = -1$
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scalar curvature under conformal deformation of a two - dimensional Riemannian manifold

I am currently stuck with an identity that I'd love to derive myself. Suppose $(M,g)$ is a surface (a two - dimensional Riemannian manifold) without boundary. Let $\tilde g = e^{2u} g$ be a conformal ...
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The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
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1answer
55 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...