For questions about surfaces.

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4
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2answers
163 views

Why is first fundamental form considered intrinsic

I am reading Kuhnel's differential geometry book, and in chapter 4, it says that "intrinsic geometry of a surface" can be considered to be things that can be determined solely from the first ...
3
votes
1answer
38 views

Parametrizing Surfaces

Can someone check my work? The question was: find a parametric representation of the portion of the surface $x+3y-z=5$ with $x\geq0, y\geq0$, and $x^2+y^2\leq 1$. I answered: $x=\cos\theta$, ...
1
vote
1answer
52 views

Evaluating stokes theorem $\int \vec{F} \cdot d\vec{r}$ on the surface $z=4-y^2$

Evaluate $\int \vec{F} \cdot d\vec{r}$ o the surface $z=4-y^2$ cut off by $x=0$, $z=0$, and $y=x$. I particularly need help with evaluating the integral on $C_3$. Please see picture I am ...
2
votes
2answers
55 views

How does the circumference of the top + bottom sides of a cylinder effect our calculations when working out the surface area?

I was watching a video tutorial on khan academy, (I've included the link at the bottom), and the question states that there is a 8cm cylinder, with a radius of 4. Part of the video shows a worked ...
0
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0answers
43 views

Evaluating the divergence theorem for region above $z=0$, below $z=x$ and inside $x^2+y^2=1$ where $\hat F=(xz,yz,z^2)$

Can someone please confirm my working below: The answer am getting look kinda crazy -Thanks. $$\color{green}{\hat F=(xz,yz,z^2)}$$ $1.$For the surface where $\color{green}{z=0}$ i.e. (flat ...
5
votes
0answers
64 views

Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
0
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0answers
48 views

Surface integration/Stokes/Divergence Theorem application

Let $\Sigma$ be a suitably well-behaved orientable surface in $\mathbb{R}^3$ whose boundary is a curve $\partial \Sigma$. Show that $$\int_{\Sigma}(dS\times \Delta)\times F=\int_{\partial \Sigma} ...
0
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2answers
99 views

how to determine the outward pointing normal (gauss divergence theorem)

I have a cone defined by $x^2+y^2=(1-z)^2$ i was trying to work out the normal vector on surface $s_1$ indicated on the plot On $s_1$: r=$\left<x,y,0\right>$ since $z=0$ on $x-y$ plane ...
0
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1answer
59 views

Finding the surface area $\iint_{s} f \, dS$ of $z=x^2-y^2$ cut off by $z=4-2y^2$

Finding the surface area $\iint_{s} f \, dS$ of $z=x^2-y^2$ cut off by $z=4-2y^2$ I have no idea which parametrization to use for this, however i did figure out the following: I think the ...
2
votes
1answer
52 views

Finding the surface area $\int \int_{s} f \, dS$ of $z=\sqrt{x^2+y^2}$ lying inside $x^2+y^2=x$

$z=\sqrt{x^2+y^2}$ is the surface we working on. I am a bit stuck on choosing the limits for this problem, I have done the following: ...
1
vote
3answers
28 views

Boundary of Mobius strip is $S^1$

I feel like this should be simple, and it is intuitively obvious by looking at the polygon with side identifictations version of the Mobius band, but how do we explicitly show, i.e find the ...
5
votes
2answers
68 views

Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
0
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0answers
12 views

Parametrization of folded surfaces with symmetry

How is parametrization done for folded surfaces which are smooth within restricted interval of fold, e.g., has is it been possible to define parametrization for plane faces of Platonic solid faces? ...
0
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1answer
77 views

Christoffel symbols of a surface of revolution

I am looking for a way to write down the Christoffel symbols for a surface of revolution. They are given by ...
3
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0answers
146 views

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
3
votes
0answers
31 views

Calculating the equation of a multivariable surface of revolution

I'm stucked with a surface equation problem so I would be very thankful if someone could help me with it. What the excercise says: Find the equation of the revolution surface that is spanned when ...
1
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1answer
76 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
1
vote
1answer
68 views

If Möbius band embeds then $\mathbb RP^2$ is a (connected) summand!

This is exercise 6-4 on page 181 in John Lee's Topological Manifolds book which asks me to prove the above, that is, if $M$ is a boundaryless surface which contains a subset $B$ which is homeomorphic ...
5
votes
2answers
74 views

What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
0
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0answers
41 views

Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
0
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2answers
47 views

Möbius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Möbius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
0
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2answers
56 views

The equation of a 3D surface bounded by 3 known elliptical curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
2
votes
2answers
33 views

question about closed disc and closed surfaces.

Question: is a closed disc is a example of closed surface. I know that, the boundary of an open disk viewed as a manifold is empty, while its boundary in the sense of topological space is the circle ...
4
votes
2answers
57 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...
0
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0answers
39 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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0answers
38 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
0
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0answers
26 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
1
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1answer
197 views

How to solve this questions about regular surfaces?

I'm trying to solve the following: $i)$ Show that if all normals to a connected surface pass trough a fixed point, the surface is contained in a sphere. $ii)$ Prove that if a regular surface $S$ ...
2
votes
1answer
88 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
1
vote
1answer
62 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
1
vote
1answer
54 views

Surface fitting to a mesh grid of data points

I wonder if there is a technique for fitting a surface to a given mesh grid of data points? I've seen interpolating a polynomial to $2$D data, but not $3$D. E.g. say I was given the matrix $$ ...
1
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1answer
53 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
3
votes
2answers
103 views

Differential Geometry of Curves and Surfaces

I'm self-studying differential geometry using Lee's Intro to Smooth Manifold and Do Carmo's Riemannian Geometry. However, I've never studied the subject so-called "differential geometry of curves and ...
1
vote
1answer
24 views

Evaluate Surface Integral over this triangular surface

When I solving the practice excericse problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral,but cant. Can someone please ...
2
votes
1answer
36 views

Is this cylinder a regular surface?

Let $C$ be a figure $‘‘8"$ in the $xy$ plane and let $S$ be the cylinder surface over $C$; that is, $$S=\{(x,y,z)\in\mathbb{R^3}:(x,y) \in C \}$$ Is the set $S$ a regular surface? I know that the ...
0
votes
1answer
49 views

Integrate the gaussian curvature

Let $T$ be a torus. We have a parameterization by $((c+a \cdot cos(v))cos(u),(c+a\cdot cos(v),a\cdot sin(v))$ for $u,v \in [0,2\pi)$. The first fundamental form is given by $E=(c+a\cdot cos(v))^{2}, ...
1
vote
1answer
43 views

Computing the first fundamental form

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a ...
1
vote
1answer
41 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
0
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0answers
40 views

Equation for flat circular/lenticular surface

We know that $x^2+y^2= r^2$ is equation of circle curve but I want to draw a flat circular surface....not curve. i.e. to explain the problem-lets draw a flat circle not move it out of screen (positive ...
0
votes
1answer
48 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
2
votes
0answers
57 views

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
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1answer
29 views

Is an open disk complete?

See the definitionS of complete surface, first definition: without edges. second definition: Any line segment can be continued indefinitely. By the first, open disk seems to be a complete surface, but ...
2
votes
1answer
41 views

Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
0
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0answers
53 views

Basic question: degree of normal bundle is not self-intersection number

For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$. Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular? Why do ...
2
votes
1answer
169 views

How to get ellipse cross-section of an ellipsoid

I'm trying to get the major and minor radius of an ellipse which represents the cross-section of a given ellipsoid. This is particularly of interest in the field of RF propagation in terms of Fresnel ...
0
votes
1answer
35 views

Surface integrals where normal changes?

I am having problems getting my head around this problem: Evaluate the surface integral $$\int_S \vec F\bullet d\vec s$$ where $\vec F=x \vec i-y \vec j +z \vec k$ and where the surface S is ...
0
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0answers
15 views

Calculate area of subsection of a three-dimensional surface

For the following three-dimensional surface, z = -4.76 + 2.78x + 2.97y - 1.18xy, I would like to calculate the area (or percent of total area) for each of three subsections of this surface: (1) for ...
0
votes
1answer
36 views

Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
0
votes
1answer
84 views

Find the equation of the plane through a point which is perpendicular to a curve

Find the equation of the plane through the point $(1, -1, 2)$ which is perpendicular to the curve of intersection of the two surfaces $x^2 + y^2 - z = 0$ and $2x^2 + 3y^2 + z^2 - 9 = 0$. And would ...
0
votes
1answer
52 views

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating.

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating. I know that I need to make the genus covered by infinitely many triangle then ...