For questions about surfaces.

learn more… | top users | synonyms

1
vote
0answers
32 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
0
votes
1answer
43 views

Surface area of sphere in N dimensions; and a failed extension to ellipsoids

I'll present a calculation of the surface area of a sphere in $N$-dimensions. This calculation is performed in cartesian coordinates. I haven't seen the computation done this way before (though I ...
3
votes
1answer
34 views

Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
3
votes
1answer
95 views

Metric Tensors and its Taylor Expansion in Normal Coordinates

With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ...
1
vote
1answer
31 views

Name of the surface with two sides and three boundaries

Once i have seen a 3d visualization of a surface with the following characteristics: it had three circular borders. If you imagine the surface inscribed in the earth globe, one of the borders would ...
4
votes
1answer
35 views

What curvature conditions make a surface rigid?

Consider a compact surface $S$, possibly with boundary, embedded in $\mathbb{R}^3$, with the induced Riemannian metric. I believe that if $S$ has constant positive Gaussian curvature (that is, $S$ is ...
1
vote
0answers
27 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
0
votes
1answer
27 views

Local diffeoorphism and orientability of surfaces

I need some help to prove this: Let $S_2$ be an orientable regular surface and $f : S1 \rightarrow S2$ be a local diff eomorphism. Then $S_1$ is an orientable surface. Thanks.
1
vote
0answers
32 views

the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
3
votes
1answer
92 views

Homology subgroups generated by non-intersecting cycles

Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by ...
1
vote
1answer
36 views

Surface described by parametric equations

If I've got the surface in $\mathbb{R}^3$ described by: $x(s,t)=s^2-t^2$, $y(s,t)=s+t$, $z(s,t)=s^2+3t$ for $(s,t)\in\mathbb{R}^2$, and I'm told this surface is the graph of a function $f(x,y)$, how ...
0
votes
0answers
13 views

Scalar potential, vector field and line integral

I've been trying to get my head around this topic for a project where I need to reconstruct a 3D surface given the estimated normals of it (Photometric Stereo). I just want to be sure I'm ...
0
votes
1answer
23 views

Normal lines of a regular surface

I need to prove this: If all the normal lines to a regular surface pass through a fixed point, then the surface is a portion of the sphere. I haven't really tried much since I don't know what to do. ...
0
votes
0answers
12 views

Show that $\alpha_j=2iA_j$

I have a question, I don't understand that Why do we have $\alpha_j=2iA_j,\ P_j(z, \overline{z})=\text{Re}A_jz^j$ since proof of Lemma??? ======================================================= ...
1
vote
1answer
34 views

Questions about surface integrals and an example problem

It is- Double integral(x + y) dS Where S is the part of the cylinder y^2 + z^2 = 4 . With x being between 0 and 5 First question, if we want to get the integral of the surface of a cylinder, I ...
0
votes
1answer
38 views

oriented surface of genus g and m punctures

Let $S_{g,m}$ be an oriented surface of genus g and m punctures, what's the condition to ensure $S_{g,m}$ is hyperbolic? If $g\ge 2$, I know it is hyperbolic, how about g=0 and g=1? Thanks in advance. ...
1
vote
1answer
11 views

Double integrals using cylindrical coordinates

Suppose we had a vector field $F = (axy^2,ayx^2,x^2\cos(\pi z))$ and wanted to calculate the surface integral of $$\int\int_SF\cdot n \ dS$$ where $S = \{(x,y,z): x^2 +y^2 = 1, 0 \leq z \leq 1/2 \}$ ...
0
votes
0answers
23 views

Parametric equations of surfaces of revolution

I need to find a parametrization for the following two surfaces: (i) The surface obtained by revolving the curve $z = f(x),\ (a \lt x \lt b)$ in the $xz$ plane, around the $z$ axis, where $a \gt 0$. ...
1
vote
0answers
21 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
2
votes
1answer
37 views

Whats the general idea for finding the mapping class group of the twice punctured sphere?

Consider the twice punctured sphere. Now I know the mapping class group of the twice punctured sphere is $\mathbb{Z}_{2}$, the cyclic group of order 2. However, I know one applies the alexander trick ...
1
vote
2answers
129 views

How to parametrize a triangle?

How do I parametrize a triangle with vertices $A(1,1)$, $B(2,2)$ and $C(1,3)$? I have tried working with the equations of the lines that form it but am not completely sure how to link them together ...
2
votes
1answer
39 views

Metric Questions Using the First Fundamental Form

Consider the Poincare Disk with local coordinates $\rho=\log\frac{1+r}{1-r}$ and $\theta$, where $r$ is the radius and $\theta$ is the angle with $x$ axis. After finding the metric tensors as ...
1
vote
0answers
57 views

Showing that two surfaces are not isometric/locally isometric

I am trying to solve an exercise which asks to show that two surfaces are not isometric and additionally that they are not locally isometric. The two surfaces presented are graphs. I know that if two ...
0
votes
1answer
27 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
1
vote
0answers
101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
0
votes
0answers
56 views

compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
1
vote
1answer
52 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
2
votes
2answers
132 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
1
vote
1answer
59 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
0
votes
1answer
22 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
1
vote
0answers
36 views

Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
3
votes
1answer
140 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
0
votes
1answer
26 views

Surface parameterization of a cylinder?

Suppose I wanted to parameterize the cylinder $x^2 + y^2 = R^2$ (for the purpose of computing a surface integral). Say $z$ is in range $-z_0 \le z \le z_0$. The standard parameterization I see ...
0
votes
0answers
15 views

How to calculate the double tangential derivative of a surface?

Let z=u(x,y) is smooth surface in $\mathbb{R}^3$. $\vec{\tau}$ is a tangential vector which is orthogonal to the vector of gradient direction $\nabla u$. How to prove the equality $$ u_{\tau\tau} = ...
1
vote
1answer
41 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
2
votes
1answer
42 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
0
votes
0answers
43 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
2answers
68 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
0
votes
1answer
51 views

Find normal vector for the surface F(x,y)=0

I need to find the normal vector(in a point (a,b)) for a surface F(x,y)=0, that we can't write as y=f(x) and F(x,y) doesn't satisfies the conditions of the implicit function theorem. For example: the ...
2
votes
1answer
55 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
1
vote
1answer
102 views

Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
1
vote
1answer
43 views

Orthogonal Parametrization of a Regular Surface

I was just wondering whether or not it is always possible to parametrize a regular surface $S$ via a function $X$ of local coordinates $u$, $v$ such that $X$ is an orthogonal parametrization- that is ...
0
votes
1answer
32 views

GNU Octave draw figure of 2 planes

How can I draw two planes in same figure in GNU Octave? $$ x + y + z = 1\\ 2x - y + 3z = 4$$ Thanks!
0
votes
2answers
31 views

Show that the set is a surface

I would be appreciate some help with the following problem, which has been suggested (but not assigned), by my professor: Consider the mapping $f : \mathbb R^3 \rightarrow \mathbb R$ defined by ...
1
vote
1answer
43 views

Surfaces of Revolution with Constant Gaussian Curvature

Surface $S$ is parametrized by $$X(u,v) = (\varphi(v) \cos{(u)}, \varphi(v) \sin{(u)}, \psi{(v}))$$ with everywhere-constant Gaussian curvature $K$. Let $v$ be the arc length of the generating curve ...
1
vote
1answer
41 views

Proving the a Pseudosphere is a Regular Surface

I need to prove that a pseudosphere $S$ is a regular surface. I have found for $S$ the parametrization $X(t,\theta) = (\mathbb{e}^{t} cos{(\theta)}, \mathbb{e}^{t} sin{(\theta)}, \pm ...
1
vote
1answer
40 views

Proving that a surface is a Möbius strip

I have a given parametrization $X(u,v)$ of a surface $S$ in $\mathbb{R}$. I must prove that it is a Möbius strip. I cannot use graphical means and I am not to reparametrize the surface- essentially, I ...
1
vote
0answers
16 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
0
votes
0answers
29 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
0
votes
0answers
52 views

Surface Area cut from cone by cylinder

I genuinely have no idea how to do this. I've been struggling on this problem for about two hours now (more hours last night), and am not getting anywhere. The question is: Find the surface area ...