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

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Second fundamental form

Is it correct to say that .. the second fundamental form of surface theory determines the Euler characteristic and the genus of the surface ? If not how is it determined?
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123 views

Surface Area of the top half of an Astroid

How would I go about beginning this question? I have applied the standard surface area integral formula but it becomes complicated quickly.
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1answer
91 views

Asymptotic lines

I have a surface $f : \Omega \rightarrow \mathbb{R}^3$ that is represented by $$f(t, \phi) = (ae^t \cos(\phi),ae^t \sin(\phi), \int_0^t \sqrt{1-a^2 e^{2x}} dx)$$ I also calculated the matrix ...
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45 views

Elongation of edges of Moebius strip

Find length of edges of Moebius strip formed by cutting along length and re-joining from a circular cylinder segment: $( a \cos \theta, a \sin \theta, z), ( \theta, 0, 2 \pi ),( z,0,b) $ after half ...
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2answers
78 views

Second partial derivatives from first and mixed derivative of bicubic Bezier Surface Patch?

Given the definition of the bicubic Bezier Surface Patch function: $$f(u,v) = \begin{bmatrix} u^3 & u^2 & u & 1 \end{bmatrix} \begin{bmatrix} -1 & 3 & -3 & 1 \\ 3 & -6 ...
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43 views

Identify the Following Parametric Surfaces

1. $r(u,v) = ui+(u\cos v)j+(u\sin v)k$ 2. $r(u,v) = u\ cos(v)i+u\ sin(v)j+u^2k$ 3. $r(u,v) = ui+vj+(2u-3v)k$ 4. $r(u,v) = vi+\cos vj+\sin vk$ My Guess: Plane Circular Cylinder Cone ...
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39 views

what is the definition of “two parallel copies of a surface S”

As indicated in the title, suppose $S$ is a surface with genus $g$, then what is the definition of "two parallel copies of S"?
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229 views

Confusing Analysis proof

I have a question about a proof of the Beltrami-Enneper theorem: In the following $\nu$ is the surface-normal and $e_1,e_2,e_3$ the Frenet 3-frame. It states: Every asymptotic curve $c: I \rightarrow ...
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1answer
71 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
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1answer
51 views

Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area. Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$. I decided to approach ...
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61 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
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126 views

A “trivial” implication I don't understand.

I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial): ...
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64 views

Surfaces on which not every pair of points is connected by a geodesic

Let $S$ be a surface in $\mathbb{R}^3$. I believe that, if $S$ is smooth, bounded, and closed, then, for every pair of points $x,y \in S$, there is at least one geodesic $\gamma$ connecting $x$ to ...
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1answer
159 views

Proof of the existence of Lefschetz Pencils.

Let $S$ be a smooth complex projective surface. A Lefschetz pencil over $S$ is a rational map (which is not a morphism) $f:S--\rightarrow\mathbb P^1_{\mathbb C}$ with the following property: All but ...
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75 views

Inverse function theorem and parametric surfaces as graphs

Let $\psi$ be a regular surface at the point $(u_{0}, v_{0})$ ($\psi \in C^{1}, T_{u} \times T_{v} \neq 0$ at $u_{0}, v_{0}$). Use the implicit function theorem to show the image of $\psi$ near ...
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3answers
100 views

surface area of the solid (column side)

I made a problem But I'm stuck in solving .. :-( the problem is following. Find the surface area of the solid that lies under the paraboloid $z =x^2 + y^2$, above the $xy$-plane, ...
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58 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
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125 views

minimum number of points on the surface of a 3D ellipsoid to define it uniquely

An ellipsoid in 3 D is described by 9 independent parameters: 3 for the coordinates of its centre + 6 independent components of a symmetric 3 x 3 matrix. What is the minimum number of points on the ...
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75 views

Parameterizing a surface

The question I was asked goes like this: The part of the hyperboloid $5x^2 − 5y^2 − z^2 = 5$ that lies in front of the yz-plane. Let x, y, and z be in terms of u and/or v. Find a parametric ...
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Integrate Gaussian over elliptic area

I have got a two-dimensional gaussian distribution, where $\sigma_x = \sigma_y$ and $\mu_x = \mu_y = 0$. $ f(x,y) = \frac{1}{2\pi \sigma^2} e ^{-\left( \frac{x^2 + y^2}{2\sigma^2} \right)}$ I would ...
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1answer
34 views

Determine the best cup of coffee to have faster cooling possible

Assume that a cup of coffee is a cylinder. The coffee machine at my workplace always produces the same amount of coffee, so the volume is constant. The coffee is always really hot, so I'm looking (out ...
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1answer
135 views

Surface integrals and surface areas of arbitrary parameter domains

I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've ...
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134 views

Intersection and Curvature of Surfaces

a) Describe the intersection $(C)$ of sphere $x^2 + y^2 + z^2 = 1$ and the elliptic cylinder $x^2 + 2z^2 = 1$, and find out the total arc-length of this intersection. b) Determine the points on the ...
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Embedding Klein Bottle in $\mathbb{R}^4$ using a figure 8 loop.

I'm trying to show that we can embed the Klein bottle in $\mathbb{R}^4$. I've previously shown that a figure 8 curve can be embedded in $\mathbb{R}^3$ by a bump function that pushes away the ...
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625 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
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120 views

Maps between Riemann surfaces are open and continuous

I'm having some trouble with a couple of concepts in Riemman surfaces that I would really appreciate some help clarifying! Firstly, is it true that a holomorphic map between two Riemann surfaces $f:R ...
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2answers
730 views

Function for a sphere

I believe that there is something fundamentally wrong with my understanding of functions but I can't pin point what it is, so I would greatly appreciate any guidance. Consider a unit sphere, ...
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1answer
269 views

Surface has Euler characteristic 2 iff equal to sphere

Let $\Sigma$ be a connected (not necessarily compact) surface with or without boundary. Is it true that $\Sigma$ is homeomorphic to the sphere if it has euler characteristic $\chi(\Sigma)\geq 2$? I ...
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1answer
107 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
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141 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
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47 views

Understanding why Euler's Formula applies to planar graphs

I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$. I can prove this by induction directly on the edges except ...
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63 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
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3answers
3k views

Find the intersection of two surfaces

I have been looking into this question : we have two surfaces : $$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$ we need to draw or describe the ...
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299 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...
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58 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
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1answer
151 views

Surface of an onion-shaped church tower

I am wondering how to calculate surface of the church tower in the picture, for painting purposes. Especially, I am interested in the two 'onion-shaped' parts. I am thinking, that it is not really ...
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36 views

Volumes and surfaces of revolution?

Please can someone explain to me why we use $dx$ in a volume of revolution i.e. $$\pi \int{f(x)^2 dx}$$ but $ds$ (an elementary bit of arc) in a surface of revolution i.e. $$2\pi \int{f(x)ds}$$ does ...
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207 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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1answer
119 views

Number of fibrations over a curve.

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism): There is a fibration ...
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518 views

Surface area of a sphere limits

If I am finding the surface area of a sphere in spherical coordinates my intergral would be like this: $$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\phi d\theta =4\pi R^2$$ But if I do the ...
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51 views

Analytically isomorphic fibers.

Suppose that $S$ is a non-singular complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Suppose also that: There are only finitely many points $y_1,\ldots,y_n\in\mathbb ...
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790 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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51 views

a problem with Stokes' theorem(curl)

If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to? but i dont understand how the ...
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Given the equation for a surface, how to find enclosed volume?

Suppose we give an equation of the form $f(x_1,x_2,..., x_n)=C$, with $f$ a smooth function, and assume this is such that defines a closed surface in $\mathbb{R}^{n+1}$. Assume also that the equation ...
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69 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
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1answer
85 views

Equation of ellipsoid surface obtained by revolving an ellipse

I'm working through the following example from the Princeton Review book: If the ellipse $x^{2} + x^{2/9}=1$ in the $xz-$plane is revolved around the $z-$axis, what's the equation of the resulting ...
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1answer
90 views

Asymptotic Directions of a Cylinder

Say I am looking at a cylinder. I have found the shape operator and I have found the eigenvalues to be k1 = -1/a and k2=0. I have also found the principal directions {1,0} and {0,1}. I know that if ...
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185 views

Differential Geometry: Is a closed disk a surface?

An open disk is clearly a surface, in the sense that it is locally homeomorphic to a part of $\mathbb{R}^2$. But what about a closed disk, even though it still looks like a surface, I am starting to ...
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69 views

Find a normal to a surface defined by $F(u,v) =0$, $u = xy$, $v = \sqrt{x^2+z^2}$

This is a problem from Apostol, Calculus, Volume II (p. $302$, Chapter $9.8$). The three equations $F(u,v) = 0$, $u = xy$, and $v = \sqrt{x^2 + z^2}$ define a surface in $xyz$-space. Find a ...