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

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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 ...
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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 ...
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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
14 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 ...
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1answer
22 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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351 views

Circular Helicoid

A helicoid has the following parametric equation: $$ r(u,v)=\left \langle v\cos u,v\sin u,cu \right \rangle,\qquad u,v,c\in\mathbb{R}. $$ In ruled form, $$r(u,v)=\alpha(u)+v\Lambda(u),$$ it has ...
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1answer
386 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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17 views

Riemann Sphere Topology [on hold]

To preface this, I study algorithms as a hobby. I was looking at an algorithm in a proof of a "unique analytic function," which in contemporary terms is a "holomorphic" function. I used Mathematica ...
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1answer
183 views

Is conformal equivalence the same as topological equivalence?

Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?
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13 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 ...
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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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Exponential map on the ellipsoid.

Consider the ellipsoid $M \subseteq \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{x^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian metric ...
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1answer
38 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
64 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 ...
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1answer
30 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 as illustrated here. When the surface is viewed from the side as shown here, such that the Y axis is ...
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1answer
14 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 ...
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15 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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22 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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15 views

how to verify a surface patch

Hi Im struggling to understand how to answer this question. Let $S=X(\mathbb{R}^2) $ where $X(u,v)=(u-\frac{u^3}{3}+uv^2, v- \frac{v^3}{3}+ vu^2,u^2-v^2), u,v \in \mathbb{R}^2 $ Show that $X(U) ...
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1answer
31 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
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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 $z$?
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42 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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linear approximation of surface

Suppose a surface determined by 4 points in the 3-D space like the following: Plane described by four points I would like to make a linear approximation in order to determine the z-dimension of point ...
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1answer
23 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 ...
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3answers
246 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 ...
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1answer
465 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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20 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 ...
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1answer
57 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 ...
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2answers
30 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 ...
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1answer
430 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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34 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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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 ...
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2answers
25 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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1answer
13 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 ...
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2answers
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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
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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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46 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 $K_P\leq0$?
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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 ...
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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 ...
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1answer
55 views

Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?

The question I am working on asks me to construct a homeomorphism $\phi : T^2/A \to X/B$ where $T^2$, $A$, $X$ and $B$ are given as follows: $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by ...
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0answers
21 views

Conformal mapping in terms of first fundamental form

I'm trying to understand conformal mappings in terms of the first fundamental form. I believe that two surfaces with fundamental forms $I_1$, $I_2$ are conformal if $\exists \lambda \not= 0$ such that ...
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2answers
23 views

Fundamental form of explicit surface

I'm trying to derive the formulae for the fundamental forms of an explicitly given surface $f(x,y)=z$ however I don't see how to set up my initial parametrisation. My intuition is that perhaps ...
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Vector analysis help - Regulated

If a surface is defined as regulated what does this mean and can I have an explained example please. Thanks
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269 views

How many parametrisations are needed to cover a sphere?

I have seen that a sphere can be covered with 6 parametrisations, but is it possible to totally cover a sphere with less parametrisations/charts?
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23 views

Solve for $y$ in terms of $x$ at $z=0$

Feeling like a complete idiot here (seems this should be easy)...the equation of a surface I fit is: $$ sf(x,y) = p_{00} + p_{10}x + p_{01}y + p_{20}x^2 + p_{11}xy + p_{02}y^2 $$ What is the ...
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407 views

What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ...
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1answer
29 views

How to find the matrix for $dN_p$, the differential of the Gauss map?

Suppose that $x:U\rightarrow \mathbb{R}^3$ is a chart for a regular surface $S$. Using the notation (from Shifrin P.39, 46) that $N_p$ is the Gauss map at point $p$, whereas the matrix with ...
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59 views

What is the cap body produced by the unit sphere?

For ecach number a > 1, let C(a) be the cap body produced by the unit sphere in E^(3) and the points (+-a,0,0). Calculate the volume, surface area and mean width of C(a). For this question, I don't ...
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Why this two surfaces have one end?

I want to prove that the infinite-holed torus and the infinite-jail cell window have one end but the doubly infinite-holed torus doesn't, my definition of one end is the following: A locally ...