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

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Parametrization and area of surface

I have not grasped the way to solve these kinds of problems yet. I need to parametrize the surface and find its area: $S:x^2+y^2+z^2=4$ with $z \ge\frac{\sqrt{x^2+y^2}}{3}$. I have already ...
3
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4answers
124 views

Parametrizing the surface $z=\log(x^2+y^2)$

Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$. I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times \textbf{T}...
3
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1answer
37 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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1answer
42 views

Can we find a regular ($C^k$) parametrization for this surface?

I have here a surface whose curvature properties I want to study, represented in cylindrical coordinates: $$f(r,\theta) = r^2\cos4\theta$$ The problem, however, is that the parametrization is not ...
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1answer
43 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
2
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1answer
19 views

Showing that, at an elliptic point, a surface lies on one side of the tangent plane.

Let $p\in S$ be an elliptic point of a surface $S$. I want to show that there exists a neighbourhhod $V$ of $p$ in $S$ such that all points in $V$ belong to the same side of the tangent plane $T_p(S)$....
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1answer
56 views

Proof of Euler's Theorem involving curvature.

Theorem: Let $φ$ be the angle, in the tangent plane, measured counterclockwise from the direction of minimum curvature $\kappa_1$ . Then the normal curvature $\kappa_n(φ)$ in direction $φ$ is given by ...
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0answers
18 views

Proof that the change of parameters between two regular surfaces is a diffeomorphism

I'm stuck on the proof that the change of parameters between two regular surfaces is a diffeomorphism. I'm using Do Carmo's book Differential Geometry of Curves and Surfaces, which can be found online ...
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1answer
29 views

Calculate the Euler-Poincaré characteristic of followin surfaces.

Calculate the Euler-Poincaré characteristic of: An ellipsoid. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\} $. Note: Not how to do this problem, I not ...
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0answers
10 views

Finding Surface area of a shape given its spherical coordinate equation

Is the surface area of the shape defined by $\rho = 4\cos(\theta)\sin(\theta) $ given by the following? $$\int_0^{2\pi}\int_0^\pi\sqrt{1 + 0 + 16\cos^2(2\theta)}\ \rho^2\sin(\phi)\ \ d\phi\ d\theta$$...
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1answer
35 views

Focal point and principal curvature of a surface

Suppose $S$ is a surface parametrized by $f$ and its Gauss map is denoted by $N$. Define a map $f_t(u,v)=f(u,v)+tN(f(u,v))$. Define a focal point $q$ of $S$ as follows: if there is $t\neq 0$ such that ...
2
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1answer
34 views

How to present $S_g$ as a $(4g+2)$–gon?

I know we can present $S_g$ (compact surface of genus $g$) as a $4g$–gon with opposite sides identified, but how to present $S_g$ as a $(4g+2)$–gon with opposite sides identified? There is an ...
3
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2answers
85 views

Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
1
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1answer
71 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point $x\...
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1answer
21 views

Prove existence of pair of points on compact surface such that distance is maximized?

Let $M_m$ be a compact $C^1$ surface in $\mathbb{R^n}$. Prove that there exists $x,y,\in M_m$ such that the distance between them is greatest among all pairs on the surface. Then show that the ...
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0answers
32 views

Finding a generating curve for a regular valued surface

Given a regular surface $X=\{(x,y,z):x^2+y^4+z^3=1\}$ and a point $p=(1,1,-1)\in X$ and a tangent vector $u = (2,-1,0)\in T_pX$ define a generating curve $\alpha (t):(-i,i)\rightarrow X$ for $u$ such ...
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1answer
29 views

Orientability of $ x(u,v)= \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, v\sin\frac{u}{2}\bigg) $

Consider the map: \begin{equation} x: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}: (u,v) \rightarrow \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, v\sin\frac{u}{2}\...
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1answer
26 views

Tangent plane of a surface at points with given gradient

So I'm stuck on the next problem: I need to find the tangent plane of the surface $$u=\ln\left( x+\frac{1}{y} \right)$$ at all the points where the gradient is equal to $$\nabla u=\hat i-\frac{16}{9}\...
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0answers
17 views

Proper name of a curve from “Vanishing Surfaces”

My question is maybe more about linguistics than maths... So, if you have a 3D surface that vanishes as a curve when projected on a 2D-plane (e.g. an axisymmetric surface projected to the r-Z plane). ...
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2answers
30 views

Is there a name for generalized ellipsoids?

In two dimensions, we have the following series of generalizations: circle $\rightarrow$ ellipse $\rightarrow$ smooth, convex, closed curve $\rightarrow$ smooth, simple, closed curve And in three ...
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1answer
51 views

Show that $(x\land y)z + (y\land z)x + (z\land x)y=0.$ where $x\land y=(x \times y)\cdot N$.

Let $P\subset \mathbb{R}^3$ be a plane through the origin and $N$ be a unit normal to $P$. For $x,y \in P$, set $x\land y=(x \times y)\cdot N$. Then for any three vectors $x,y,z \in P$, we have $$(...
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0answers
30 views

Given second fundamental form what is the geometric /topological invariant?

The Gauss Bonnet integrates the first and second forms into an elegant structure. But before that... If first fundamental form alone is given, a series of mutually bendables with isometric /intrinsic ...
3
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0answers
26 views

Given a first fundamental form, showing a particular second form cannot exist

If I have a first fundamental form $ \mathrm{d}u^2+\cos^2 u \mathrm{d}v^2$, I am trying to show that the second fundamental form cannot equal $f(u,v)\mathrm{d}v^2$ for a smooth function $f(u,v)$. I ...
2
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2answers
84 views

Circular cylinder $S=\{ (x,y,z) : x^2+y^2=1 \}$ can be covered with a single surface patch.

I somewhere found that we can take $U$ an annulus instead of a disc where $U=\{ (u,v): 0 < u^2+v^2 < π \}$. Can anyone please explain me that how a cylinder can be covered with a single surface ...
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1answer
25 views

Prove the boundary is a compact 1 manifold

A closed surface with boundary is a compact connected topological space $B$ with the property that each point $p \in B$ has an open neighborhood $U$ homeomorphic to either: $\{(x, y) \in \mathbb{R^2}|...
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1answer
28 views

Sketching the surface $z=\frac{x^2y}{3}$

I am trying to sketch the part of $x^2+y^2=9$ which lies in the first octant between the surfaces $z=0$ and $z=\frac{x^2y}{3}$. I understand that $x^2+y^2=9$ is a cylinder with radius three, ...
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1answer
20 views

Orthogonality of ruling and directrix of a ruled surface

Let $M$ be a ruled surface of $\mathbb{R}^3$ with a regular parametrization given by: $$x(u,v)= \alpha(u) + v\beta(u)$$ where $\alpha' \neq 0$ and $ ||\beta || = 1$. I want to show that $<\alpha'...
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0answers
16 views

Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
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1answer
47 views

Is the quotient space obtained by identifying the poles of a sphere homeomorphic to a closed surface?

I'm interested in the quotient space of $S^2$ obtained by identifying the poles, and in particular whether it is homeomorphic to a closed surface. I'm pretty sure its homotopic to one, just by ...
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2answers
75 views

Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
2
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1answer
39 views

How to derive 2D equation representing minimums of constrained 3d equation?

I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints. When the surface is viewed from the side (as below), such that the Y axis is not visible, there is ...
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1answer
23 views

Finding the surface area of $S={(r\cos\theta,r\sin\theta,3−r):0\leq r \leq 3, 0\leq \theta\leq2 \pi }$

So we've been given this set: $$S={(r\cos\theta,r\sin\theta,3−r):0\leq r \leq 3, 0\leq \theta\leq2 \pi }$$ and I can see that this is part of a cone but I'm not too sure how to find the surface area....
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1answer
19 views

Regular parametrization of a surface is conformal iff it preserves angles.

Can anyone give me some hints of how to start the proof, because I have no idea where to start. I know if a parametrization is conformal, then $E=G$ and $F=0$, where E,F,G are values in the first ...
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0answers
30 views

Difference between a Möbius Strip and a Simple Surface

I am trying to distinguish between a Möbius strip and a surface that has no separations, holes and a connected boundary (homeomorphic to a disk or a half-sphere). Since a Möbius strip also has all the ...
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1answer
40 views

Parallel surface

For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...
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2answers
16 views

What is the equation for the walls of a 3D cylinder?

If for example I have the circle $x^2 + y^2 = 4$ in the $x$-$y$ plane, and I want to extend it upwards into the $z$ dimension, how would I write the equation for the circular walls in terms of $z$?
4
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1answer
51 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
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1answer
27 views

Euler's formula about graphs embedded in $\mathbb{R^2}$

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$ I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the complement ...
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0answers
23 views

Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
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1answer
58 views

Finding the derivative of an equation

I am currently doing an investigation in which I am required to design the dimensions of a juice box (can be cube/cuboid) which has the least possible surface area that can hold 200 ml of juice. I ...
2
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2answers
43 views

Optimisation of a juice box: finding the least possible surface area that can hold the most volume

I have an investigation which requires me to design the dimensions of a juice box (cuboid) which has the least possible surface area that can hold the most volume. I am not sure as to how I should ...
0
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1answer
47 views

Definition of a regular surface

Here is the definition of a regular surface from Differential Geometry of Curves and Surfaces by Manfredo do Carmo: A subset $S ⊂ \mathbb R^3$ is a regular surface if, for each $p ∈ S$, there ...
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0answers
19 views

Proper application of Surface Area of Revolution formulae?

I'm a little confused about how to properly apply the integrals used to calculate the area of a surface of revolution. Find the exact area of the surface obtained by rotating the curve about the x-...
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2answers
56 views

Ray intersecting a quad mesh

I am trying to solve the math behind rendering a quad-mesh surface. MatLab for instance can take a regularly spaced (x,y) grid with arbitrary third-dimension (z) values, treat each four neighbouring ...
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2answers
32 views

How can we find a normal to this plane? [closed]

A vector is said to be normal to a surface (plane) if it is perpendicular to that surface. Consider a plane P, and let points K(2,1,1), L(3,-1,2) and M(1,1,2) be on this plane. How can we find a ...
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1answer
16 views

Need help understanding what the curve made by two or three intersecting surfaces looks like

I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $ <cos(t),sin(t),sin(t)$ > Clearly this would ...
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1answer
55 views

Suppose a surface contains a straight line. How can I prove that all the points on this line have non-positive Gaussian curvature?

Suppose a surface contains a straight line. How can I prove that all the points on this line have Gaussian curvature $K_P\leq0$?
5
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1answer
45 views

3D Analogue of a Catenary

When a cable is supported at its ends and droops due to its own weight, the resulting curve is called a catenary. However, is there a three-dimensional analogue of this shape? For example, let's say ...
0
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1answer
19 views

Representing results of CSG operations with spline-based surfaces

I've been playing with a few different CAD programs and have become interested in the math involed with CSG and spline-based surfaces. During my research, I found that the curve representing the ...
2
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0answers
40 views

Total Gaussian curvature

For a compact surface, $S$, in $\mathbb{R}^3$, how would I go about showing that the total Gaussian curvature $\int_S K da \leq 4 \pi$? I feel like Hopf's Umlaufsatz and the Gauss-Bonnet Theorem are ...