Tagged Questions

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

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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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 ...
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 $M \cap A$ ...