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

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closed surface with a part of the cone $z^2=x^2+y^2, 1\leq z\leq 2$ [on hold]

Let $S$ be the closed surface consisting of the part of the cone $z^2=x^2+y^2, 1\leq z\leq 2$ together with the top and bottom disks in the planes $z=1,z=2.$ Show that the vector fields ...
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Finding the points where a tangent plane is parallel to another plane?

Find all points on the surface ${\bf{r}}(\mu,\nu)=(\mu^2\nu,\mu\nu^2,1)$ where the tangent is parallel to the plane $z=x-y.$ Two planes are parallel if their normal vectors are parallel. That is, ...
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Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
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3answers
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Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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Surface of the intersection of $n$ balls

Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, ...
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2answers
871 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
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17 views

Method of characteristics - integral surface formation

On page 2 of this PDF from Standord, which describes the Method of Characteristics for first-order PDEs, it is written at the end of the page: "In doing so, we see that $z(x,t)$ is constant along ...
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26 views

Confusion of classification of closed surfaces

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct? But what confuses me is that the Klein ...
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21 views

Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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26 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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104 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
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37 views

Surface integral of a scalar over a unit cube.

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$ where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$ Honestly, I don't know what to do. All I know is that you have to ...
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14 views

Induced connection on an oriented regular surface $S\subset\mathbb{R}^3$

Let $S\subset\mathbb{R}^3$ be an oriented regular surface - then we have an embedding $\iota:S\rightarrow\mathbb{R}^3$ where $\iota$ is the inclusion of $S$ into $\mathbb{R}^3$. We also have a smooth, ...
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12 views

Parametrise union of two curves

If we are given two curves, is there a way to parameterise the union between them? For example how to can we parameterise the region constructed by joining $y=x^2$ and $x=y^2$.
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67 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 ...
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19 views

Slicing a 3d surface using a 2d line equation

So what I'm trying to do is to find the equation of a 2d function on a 3d surface using a 2d line equation. With : $z = f(x, y)$ the equation of the surface and $ax + by + c = 0$ the line ...
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33 views

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 ...
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116 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 ...
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1answer
450 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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40 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
38 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 ...
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32 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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50 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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41 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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1answer
18 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 ...
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482 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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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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27 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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1answer
38 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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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\ ...
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1answer
32 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 ...
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31 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 ...
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2answers
51 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 ...
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1answer
28 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, ...
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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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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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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 ...
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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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29 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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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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21 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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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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366 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
433 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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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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15 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 ...