Skip to main content

Questions tagged [surfaces]

For questions about two-dimensional manifolds.

Filter by
Sorted by
Tagged with
0 votes
0 answers
4 views

A detail in the proof of Killing-Hopf theorem for Euclidean surface.

I am reading the book Geometry of Surfaces by Stillwell. In chapter $2$, he proves the following theorem: Theorem: (Killing-Hopf) Each complete, connected Euclidean surface is of the form $\mathbb{R}^...
Zoudelong's user avatar
  • 347
0 votes
0 answers
22 views

Are planes through the origin the only position vector self-tangent surfaces?

Are there any surfaces in $\mathbb{R}^3$ (other than planes through the origin) such that each position vector lies in the respective tangent plane at that point? If the surface is given by say $\phi:...
Derso's user avatar
  • 2,811
2 votes
0 answers
71 views

Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
Justin Solomon's user avatar
2 votes
1 answer
44 views

How can I modify a surface to satisfy two distance conditions?

I have two variables, $\phi$ and $\theta$, and I'm trying to create a smooth surface such that the following rules are met for the distance between on the surface, $D$ \begin{align*} 1)& \: D[(\...
David G.'s user avatar
  • 292
1 vote
0 answers
61 views

A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.

Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$ Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
Thomas Finley's user avatar
1 vote
1 answer
41 views

Evaluate $\iint_S\vec{F}$ where $S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above and below by two planes.

Use the divergence theorem to evaluate $$\iint_S\vec{F}$$ where $\vec F(x,y,z)=\langle xy^2,x^2y,y \rangle$ and $\vec S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above ...
Thomas Finley's user avatar
2 votes
0 answers
78 views

Why does the volume comes out to be $\pi\frac{a^3}{8}$ instead of $\frac{3\pi a^3}{8}$?

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$ I tried solving the problem as follows: The equation ...
Thomas Finley's user avatar
0 votes
0 answers
60 views

Explicit linear system corresponding to rational map to $\mathbb{P}^1$

Let $L$ be a line in $\mathbb{P}^3$. Then we can define a map $$ \pi\colon \mathbb{P}^3 \dashrightarrow \mathbb{P}^1, \qquad x \mapsto \langle x, L \rangle \cap L' $$ where $L'$ is a line such that $...
fish_monster's user avatar
1 vote
1 answer
67 views

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$ I tried solving this problem as follows: Equation of the cylinder $x^2+(y-a)^2=...
Thomas Finley's user avatar
3 votes
1 answer
59 views

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$ I tried solving this problem as follows: The equation ...
Thomas Finley's user avatar
4 votes
1 answer
204 views

Calculating the arc length of a curve... Which formula?

Let $S$ be a surface parameterized by variables $u,v$ and $\alpha(t)=(u(t),v(t))$ be a curve on the surface. I am of the understanding that we can find the arc length of $\alpha$ by integrating it's ...
PhysicsIsHard's user avatar
2 votes
0 answers
41 views

What action on $\mathbb{R}^2$ yields a closed genus 3 surface

The closed, connected, orientable surface of genus 3 has universal cover $\mathbb{R}^2 \rightarrow \Sigma_3$. As such, $\Sigma_3$ can be described as a quotient of $\mathbb{R}^2$ by some properly ...
JMM's user avatar
  • 1,145
0 votes
0 answers
22 views

Finding the optimal surface enclosing a given volume

I would like to find, over the set of continuous surfaces that enclose a volume $V$, the one(s) that lead to the maximal value of a certain cost function. I'm working on a physics problem which ...
amrit 's user avatar
  • 36
1 vote
0 answers
22 views

area surface (Schwarz paradox)

Are there any good articles or publishing about area surface with polyhedral? I have to do a project about it for my analysis course at university but for now ive only read Cordoba's work. I have ...
Paz's user avatar
  • 11
1 vote
0 answers
47 views

minimal cuts and pastes to make a t-shirt?

(I have not found an exercise about this in do Carmo or Struik, and I think it could go a long way towards building intuition about curvature and develop skills in computing special 2D areas, as well ...
fromscratch's user avatar
0 votes
0 answers
42 views

Gauss Rodrigues map image

I'm trying to solve an old exam on differential geometry to prepare for my exam and I have doubts about the following point: Having chosen an orientation, describe the image of the Gauss Rodrigues map ...
Sigma Algebra's user avatar
0 votes
0 answers
58 views

Finite covering of Hirzebruch surfaces

Question Let $F_n = \mathbb{P}(\mathscr{O}_{\mathbb{P}_1}\oplus\mathscr{O}_{\mathbb{P}_1}(n))$ be an Hirzebruch surface and consider its finite covering(or double covering) $f: X \to F_n$. Let $f$ be ...
Tommk's user avatar
  • 11
0 votes
0 answers
19 views

Chain rule in differential of Gauss map

This is probably more of a calculus question than a geometry question. Let $N: S \rightarrow \mathbb{S}^2$ be the Gauss map. And let $\varphi : \mathbb{R}^2 \supseteq U \rightarrow S$ a ...
F13's user avatar
  • 173
0 votes
0 answers
9 views

Torus Circle Meridian modification for Cassinian Intersections

Is it possible to define a torus whose sections parallel to torus axis are Ovals of Cassini? For contextual reference sections of a Circular Torus and code (Mathematica) are given below: We can see ...
Narasimham's user avatar
1 vote
0 answers
22 views

Is there something like a compact surface classificator?

I'm currently studying compact surfaces and there are some exercises as the following: find a simple scheme equivalent to $abc, da^{-1}b,cef,e^{-1}f^{-1}d$ and classify the surface. After computations ...
Valere's user avatar
  • 1,344
1 vote
1 answer
64 views

Gauss–Bonnet Theorem for rectangular surface with rectangular holes: is the right hand side always $0$?

I want to use the Gauss–Bonnet theorem for a non-Euclidean rectangular 2D surface with discrete curved boundaries that has rectangular holes with discrete curves. I know that: $$ \oint k_g \, ds + \...
archrook's user avatar
0 votes
0 answers
32 views

Parameter curves are geodesic then $E_v=G_u = 0$

I'm trying to solve question $\text{4/II/12G}$ of this pdf: A special case $F=0$ of this question is proved in another post For $(\text{i})$: The tangent vector to the unit-speed curve $u=c$ is $\...
hbghlyj's user avatar
  • 3,045
0 votes
0 answers
48 views

surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.

I try to solve Exercise $122$ on page 40 of this pdf Show that the surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics. ...
hbghlyj's user avatar
  • 3,045
0 votes
1 answer
23 views

Condition of orthogonality of two solutions to a 2nd order ODE with constant coefficients on a smooth chart

I try to solve Exercise $100$ on page 37 of this pdf Let $\mathbf{r}:(u, v) \mapsto \mathbf{r}(u, v)$ be a smooth chart. Show that the solutions to the differential equation $$ A \dot{u}^2+2 B \dot{u}...
hbghlyj's user avatar
  • 3,045
1 vote
1 answer
98 views

Exercise about contact between a curve and a surface in $\mathbb{R}^3$

This excercise is taken from do Carmo, Differential geometry of curves and surfaces, section 3.3. A Curve $C$ and a Surface $S$ have contact of order $\ge n$ in a common point $p$ if there exists a ...
Giovanni Busdon's user avatar
-4 votes
1 answer
44 views

A surface passing through two different surfaces [closed]

Suppose I have two surfaces $f_1=k_1$ and $f_2=k_2$ in 3D. Then, how do I find the equation of a surface passing through (intersecting) the two surfaces $f_1,f_2$? Like, does $f_1-f_2=k_3$ help? But, ...
vidyarthi's user avatar
  • 7,085
1 vote
0 answers
109 views

Questions about a differential geometry exercise

The exercise comes from an old exam from the 90s at my university Let $\varphi(u,v) = (v^2-u,u,u-v) \quad u,v \in \mathbb{R^2} \quad S=im(\varphi)$ a) Prove that $S$ is a regular surface and that a ...
jackes gamero's user avatar
1 vote
0 answers
27 views

Characteristic class detecting "upward-facing" surfaces

Let $\Sigma \subseteq \mathbb{R}^3$ be a smoothly embedded compact oriented surface with boundary. Let $\vec{n}: \Sigma \rightarrow \mathbb{R}^3$ be the field of unit normal vectors associated to the ...
JMM's user avatar
  • 1,145
0 votes
0 answers
35 views

Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
zeta space's user avatar
4 votes
1 answer
167 views

Differential Geometry of Curves and Surfaces from Riemannian Geometry

I'm a relativist. Hence, I have a working knowledge of Riemannian geometry, but I never really studied differential geometry of curves and surfaces. I know the traditional path is to start with curves ...
Níckolas Alves's user avatar
0 votes
0 answers
82 views

A reference for a proof regarding the connected sum of two surfaces

I need a reference for the proof that any two connected sums of two nonempty, compact, connected 2-manifolds are homeomorphic. I am looking for a proof with the same construction as exercise 10-8 of ...
Lone Sloane's user avatar
0 votes
1 answer
60 views

How can I see that this surface is indeed closed

I recently came across a proof of the weak maximum principle (if a function $u:\mathbb{R}^3\to\mathbb{R}$ is such that $\nabla^2u=0$ and $u$ has only isolated critical points, then they can't be nor a ...
Andrés Vásquez's user avatar
3 votes
2 answers
321 views

Is it true if a face of a graph is not homeomorphic to an open disk, then we may find a noncontractible curve contained in the face?

Suppose we have a graph embedded on a surface $Q$ and one face $F$ of the graph is not homeomorphic to an open disk. Does there exist a closed (smooth nonselfinteresecting) curve $g$ contained in $F$...
Hao S's user avatar
  • 468
0 votes
0 answers
43 views

If $\gamma_v$ is geodesic, then $\gamma_{sv}$ is geodesic

I'm trying to solve the following: Let $p\in S$ ($S$ is a surface) and $v\in T_{p}S$ and a $\delta > 0$ s.t. $\gamma_{v}:\left(-\delta,\delta\right)\to S$ is a geodesic and $\gamma_{v}\left(0\...
Ludwig's user avatar
  • 395
2 votes
1 answer
42 views

Genus of a graph consisting of two faces homeomorphic to open disks

Suppose the graph $G$ is embedded in a surface $Q$ such that there are two faces $F_1,F_2$ of the embedding, each homeomorphic to the open disk, such that each node of $G$ lies on $F_1$ or $F_2$. Is ...
Hao S's user avatar
  • 468
0 votes
0 answers
24 views

Does $\int_{x_0}^{x_1} 2 \pi ydx$ give us any geometric information about the corresponding surface of revolution? [duplicate]

Consider the function $y=y(x)$. We can rotate the graph of this function around the x-axis to get a surface of revolution. The area of this surface of revolution between $x_0$ and $x_1$ is given by: $$...
PhysicsIsHard's user avatar
0 votes
0 answers
31 views

Free-Surface kinematic BC - Azimuthal suspended flow - cylindrical coordinates - How to define the free surface and derive its impermeability?

I am working on a linear stability analysis of an azimuthal free surface flow. However, I am stuck in the derivation of the kinematic BC to apply for the impermeability of the azimuthal free surface (...
Virgil 11's user avatar
0 votes
1 answer
69 views

Hessian and the second fundamental form

I am trying to relate the shape operator with the Hessian. I wish to motivate my question with an example. Consider $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $f(x,y) = x^2 + y^2 - 1$. Then $\nabla ...
user57's user avatar
  • 796
1 vote
0 answers
31 views

How many plane intersections to uniquely determine a surface

A brazilian math contest that ended yesterday posed the following problem (adapted and translated by me) Let $S$ be some surface in $3$ dimensional space. Given that: The intersection of $S$ with ...
Almeida's user avatar
  • 595
2 votes
0 answers
119 views

Differential Invariants of Surfaces

I'm reading Peter Olver's paper about Differential Invariants of Surfaces. In page 3, he defines two invariant differential operators $\mathcal{D}_1$ and $\mathcal{D}_2$, and he adds - "which ...
gipouf's user avatar
  • 55
0 votes
0 answers
24 views

Surface area element in 4-dimensions

In Dirac's "General Theory of Relativity" (p. 40) he says "If we take two small contravariant vectors $\xi^\mu$ and $\zeta^\mu$, the element of surface area that they subtend is ...
Khun Chang's user avatar
1 vote
2 answers
86 views

How do I calculate the parametrization of a 3D surface given its support function?

The support function $h_S$ of a non-empty, closed, convex set $S$ in $ℝ^n$ describes the distances of supporting hyperplanes of $S$ from the origin. A supporting hyperplane is a hyperplane that has ...
Lawton's user avatar
  • 1,861
0 votes
1 answer
53 views

Volume of logarithmic horns

I have been working on this set of problems involving logarithmic horns (basically just logarithmic spirals in $3$ dimensions); however I have been struggling with the final question of the set of ...
MM117's user avatar
  • 1
0 votes
0 answers
29 views

Which 2D surface has the largest area in 3D?

Is my thinking correct that the space-filling surface like this one [1] will have the largest area among all surfaces enclosed into a given volume? Or, more exactly, the area of $n$-th approximation ...
Vladislav Gladkikh's user avatar
0 votes
1 answer
17 views

Change paraboloid vertex coordinate

The paraboloid equation $x^2 + y^2$ It has its vertex at the point $x=0,y=0,z=0$. How can I get the equation of a paraboiloid with its vertex at the point $x=2.4,y=2.4,z=0$?
user3204810's user avatar
1 vote
0 answers
50 views

Any smooth compact 2-dimensional submanifold $S\subset\mathbb{R}^3$ is orientable

A manifold $M$ is orientable if the bundle of antisymmetric $n$-linear forms $\Lambda^n(M)$ is trivial. Equivalently, $\Lambda^n(M)$ admits a nowhere vanishing section. I suppose we must assume that $...
danimalabares's user avatar
3 votes
2 answers
152 views

Open sets on a surface with locally connected boundary

Let $\Sigma$ be a surface and $\Omega$ be an open subset of $\Sigma$. Suppose that $\Omega$ is homeomorphic to the open unit disk $\mathbb{D}$ and is relatively compact in $\Sigma$. I'm interested in ...
Dilemian's user avatar
  • 1,097
0 votes
0 answers
21 views

Determining embedded Parameterization in 3D of a 2D manifold when only the metric is given, by exploiting symmetries

I have the second fundamental form $$E(h,\gamma) = \frac{1}{16} \frac{h \gamma^2}{ (h^2-1) (h^2+\gamma^2-1)^{\frac{3}{2}}}, $$ $$F(h,\gamma)= -\frac{1}{16}\frac{ \gamma}{ (h^2+\gamma^2-1)^{\frac{3}{2}...
prikarsartam's user avatar
1 vote
1 answer
36 views

To prove that $k_1 + \dots + k_m = mH/2$, where H - average curvature: $H = \lambda_1 + \lambda_2$

Faced with such a task: Let $k_1, \dots , k_m$ be the normal curvatures of the surface in the directions dividing the plane into angles $\frac{\pi}{m}$ To prove that $k_1 + \dots + k_m = mH/2$, where ...
Bogdan Witcher's user avatar
2 votes
2 answers
49 views

Determine a tangent plane on the surface $\psi(u,v)=(3u^2-2v^2,u-v,u+v)$ at the point $\psi(1,2)$.

So in this question basically I'm getting stuck after calculating the Jacobian of the function $\psi$, somehow I'm trying to find the values of a vector that is on this plane, but I'm not so sure of ...
Angelo's user avatar
  • 47

1
2 3 4 5
65