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

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Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $$r(t)=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$$ how can I find a equation for quadric surface where the curve ...
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9 views

Parametrization for intersection curve of catenoid and cylinder

I have calculated intersection of two surfaces, the catenoid of revolution and displaced/eccentric cylinder. $$ (x^2 + y^2) = c^2 \cosh ^{2} (z/c) ; \, (z-a)^2 + x^2 = a^2$$ The cylinder generator ...
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1answer
22 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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118 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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22 views

How to calculate the surface of overlapping ellipsoids

I want to calculate the surface of a body made of at least 3 overlapping ellipsoids. Below there is a picture of the cross section of the body. I already know how to calculate the surface of single ...
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1answer
50 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
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0answers
19 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
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1answer
47 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
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1answer
24 views

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Find all surfaces that can be obtained from an octagon by identifying edges in pairs. I think there are many many surfaces. Can anyone give some hints for the question?Thanks.
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1answer
13 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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56 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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Find unit normal vector to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$

I've been trying to solve this question: Find a unit vector with positive $z$ component which is normal to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$ on the surface. My working: Let ...
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2answers
20 views

How to simplify this equation with change of variables,

I have the equation, after completing the square: $$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$ How can I further simplify this equation? I need to find the volume inside of this surface. ...
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0answers
36 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
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2answers
52 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
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2answers
41 views

Geodesics on a generalized cylinder

I want to prove that given a generalized cylinder $C(s,t)=\alpha(s)+t\hat{z}$ , where $\alpha$ is a curve on the $xy$ plane and $\hat{z}$ is the $z$-axis vector, then a geodesic curve $\gamma$ has the ...
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0answers
64 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
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21 views

Surface mentioned by Sophie Germain

Sophie_Germain_Surface Can someone please help going through the original paper no.22 in French? Does the doubly curved surface mentioned by Sophie Germain (Wiki) $$ 4 \frac{z^p-1}{z-1} = y^2 \pm ...
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20 views

Closest points on two skew surfaces [closed]

I am trying to find the two closest points on two surfaces. The surfaces are not extended infinitely but are restricted by some boundaries. Th possible set of surfaces could be combination of ...
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29 views

decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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571 views

What is the object on the front of Larson and Edwards' calculus and pre-calculus textbooks called?

There is this incredible glass figure on the front of my Calculus textbook, I searched online for what this figure is called and the formula for creating it, but I can't find it. I think it is a ...
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27 views

Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
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58 views

Polygonal presentations: why no two-letter words?

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...
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31 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
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1answer
139 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
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1answer
24 views

The set of fixed points of a $C^1$ involution is a surface.

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^1$ involution i.e $f\circ f=id$. Show that the set $Fix(f)=\{x \in \mathbb{R}^n;f(x)=x\}$ of fixed points of $f$ is a surface of $\mathbb{R}^n$ ...
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1answer
41 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
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1answer
31 views

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
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19 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
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68 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
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1answer
45 views

Calculus 3 - Level surfaces

I know how to find the level surfaces for a $2$ variable functions, $z=(x,y)$, by finding the $3$ planes. How would you find the level surfaces for a $3$ variable function, $w=(x,y,z)$. Would you find ...
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1answer
23 views

Parameterizing part of sphere

the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$ first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...
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41 views

Complements of homeomorphic subsets of surfaces homeomorphic?

Let us consider a surface $S$ and a subset $T\subset S$, where $T\cong D^2$ are homeomorphic and $D^2$ is an open disc. Let $T\cong Q\subset S$ be homeomorphic. Are $S\setminus T $ and $S\setminus Q$ ...
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1answer
59 views

How to mathematically formulate the surface of a spring?

I would like to mathematically map the surface of a cylinder constructed like a coil pot (or compressed spring), where the surface area and height of the pot is a function of the length of the coil, ...
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1answer
49 views

Computing the volume inside a surface S, using a seemingly unrelated result,

Consider the surface $$S = \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$$. What is the volume inside S? This is actually part (b) of the question. I'm not sure which approach to take. But part (a) of the ...
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1answer
82 views

About timelike surfaces with non-diagonalizable shape operator.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$ Take a differentiable surface $M \subset \Bbb L^3$, ...
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33 views

Find the volume enclosed by the surface $S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$

Find the volume enclosed by the surface $$S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}.$$ My attempt was this: I moved the tricky $xy$-term over to the r.h.s. I now have $$x^2+y^2+z^2 = 1-xy,$$ ...
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180 views

Best book for learning multiple integrals, line integrals, Green's 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 ...
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17 views

Rotation Invariant Descriptors for Bivariate Polynomial Surfaces

I start with a simple example. Consider: $$ z = x^2 + y $$ and $$ z = y^2 + x $$ Visually speaking, both of these are essentially the same surfaces rotated by 90 degrees about the z-axis. I am ...
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1answer
303 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
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Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
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1answer
15 views

How to graph the intersection of two surfaces

Another way of asking this is: How would one graph a 1 dimensional line (the intersection) in a 3 dimensional space. Some context for the question: Let's say I have two spheres ...
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1answer
27 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
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1answer
27 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
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1answer
22 views

How to work out the angle of a line passing through a plane

I have a triangular plane composed of three points. From this it it easy to deduce that the plane is in fact composed of two vectors which must touch at some point. because all of this is relative, ...
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When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
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69 views

Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
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1answer
19 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
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23 views

Find the volume of the region by integration

I have to find the volume of the region bounded by $x + z = 1$;$ y + 2z = 2$;$ x = 0$;$ y = 0 $;$ z = 0$; I tried to sketch the graph separately in the $y-z$ plane and then in $x-z$ plane. But I am ...