# Questions tagged [surfaces]

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### Awkward triple integral

I have a rotationally-symmetric surface in $(r,z)$ (i.e. cylindrical coordinates) defined as follows: $$\frac{3r^2 - a}{(r^2 + z^2)^{3/2}} + \frac{3az^2}{(r^2+z^2)^{5/2}} - b = 0$$ where $a,b$ are ...
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### Classification of weak del Pezzo surfaces

In this question I work over the complex numbers. Up to isomorphisms, the del Pezzo surfaces are the blow-ups of the projective plane $\mathbb{P}^2$ at $0 \leq n \leq 8$ points in general position (...
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### Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?

I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding: The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ with boundary ...
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### Differential of Gauss map

Let $S$ we a regular surface, differential of Gauss map is $\mathrm{d} \mathrm{N}_{\mathrm{p}}: \mathrm{T}_{\mathrm{p}}(\mathrm{S}) \rightarrow \mathrm{T}_{\mathrm{p}}(\mathrm{S})$. To evaluate the ...
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### Find 3-manifold based on parameter space of 1D curves

I'm trying to find the equation for the 3-manifold based on the phase space of some one dimensional curves... The space of 1D curves are found from the intersections between $\log(x)\log(y)\log(z)=-a$ ...
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### What does it mean to integrate a partial derivative in the “other” direction?

Intro: A continuous, differentiable function $F(x,y)$ can be pictured as a surface over the $(x,y)$ plane. At each point in space, we could plot another function $\frac{\partial F}{\partial y}(x,y)$, ...
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### Voss Weyl formula and divergence theorem in curvilinear coordinates

I am unable to reconcile the divergence theorem in curvilinear coordinates, and what I get by an application of the Voss Weyl formula and the divergence theorem in $\mathbb{R}^2$. Could someone help ...
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### Is this induced homomorphism surjective?

Some preliminaries: Given a surface $S$ and a covering space $q: \tilde{S} \rightarrow S$, we say that $q$ is abelian if its deck transformation group is abelian. Naturally, the universal abelian ...
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### Is there a useful $\mathbb{H}^3$ analogue for a “pair of pants”?

2-dimensional surfaces with negative Euler characteristic have a pants decomposition which cuts the surface into pairs of pants. I'm curious if there is an analogue a dimension up. A pair of pants has ...
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### Reference request: systole of hyperbolic surface increases in covering space

I'm wondering if anyone knows of a quick proof or reference of the following fact: Let $S$ be a compact hyperbolic surface and $l > 0$. Then there exists a finite covering space $S' \rightarrow S$ ...
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### Intersection points of the 27 lines on smooth cubic surface

Let us work over $\mathbb{C}$. It is a classic result that if $S$ is a smooth cubic surface, then there are 27 lines contained in $S$. My question is: can we compute the number of points which are ...
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### Characterization of non-separating simple closed curves

Let $\{\alpha, \beta\}$ be a pair of non-separating simple closed curves on a compact, oriented surface. Without directly cutting, is there a way to check if the union of the curves in the pair is non-...
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### Find a parametric representation for the lower half of the ellipsoid $4x^2 +4y^2 +z^2 = 4$. Then find the area of this surface. [closed]

Find a parametric representation for the lower half of the ellipsoid $4x^2 +4y^2 +z^2 = 4$. Then find the area of this surface. (I couldn't do my school homework, can you help me in detail, thank you ...
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### What's the number of planes whose intersection with $Q$ is a reducible conic in $\mathbb{P^3}_{\mathbb{C}}$?

Let $\mathbb{P^3}_{\mathbb{C}}$ be the complex projective space and let $l$ and $Q$ be a line and a non-singular quadric surface. Now we consider the pencil of planes that have $l$ as common line. ...
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### Orientability between diffeomorphic surfaces

From Do Carmo's Differential Geometry of Curves and Surfaces, 2nd edition, Now, when I tried to prove this, I reached this point: Up until this box, I couldn't understand the philosophy behind ...
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### Find the equation of the diametral plane that is perpendicular to the given plane.

Find the diametral plane of the surface $$x^2+2y^2-z^2+2xy-2yz-2xz-4x-1=0$$ that is perpendicular to the plane $$x+y+z-3=0$$ I know that the diametral plane passes through the center of the surface (...
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### Find the equation of the tangent-plane to the surface

Find the equation of the tangent-plane to the surface given with the equation $$2x^2+5y^2+2z^2-2xy+6yz-4x-y-2z=0$$ that passes through the line $$4x-5y=0, \ \ z-1=0.$$ The equation of the slope-...
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### How are Gerstner wave normals derived?

I'm looking at this GPU gems article on water rendering. It gives the following tangent space vectors. I understand the normals N are calculated as the cross product B x T, but I can only seem to ...
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### Meaning of conormal vector

In my situation let $S$ be a surface in $\mathbb{R}^3$ with $\partial S\not=\emptyset$ (enough regular). What is the meaning of $\eta$: conormal vector to $\partial M$ in M? I know it should be normal ...
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### Explicit Equation for Quadratic Funnel Surface

I've been trying to figure out an explicit form for a quadratic funnel surface, but I'm not succeeding. Essentially, I'm wanting an explicit form for the surface that would be created by revolving the ...
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### Find rectangular Cartesian coordinate system

Find rectangular Cartesian coordinate system in $\mathbb{R}^3$ to bring quadratic surface $2y^2-3z^2+4xz-12y+15=0$ to standard form. After splittig off square the equation can be rewritten as ...
Let be a function $f:(u^{1},u^{2})\rightarrow f(u^{1},u^{2})\in\mathbb{R}$ of class $r\geq 2$ defined in a open $U\subset\mathbb{R}^{2}$. Let be a Monge's parametrization: \$X:u\in U\rightarrow (u,f(u))...