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

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3
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Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: (${\bf \sigma}_u=:\dfrac{\partial \sigma}{\partial u}$). The above text several times assuming that $\dfrac{\partial ...
0
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0answers
37 views

Is there affine surface of general type of the form $y^2=f(x) f(z)$ or $y^2=f(x) g(z)$?

Let $f,g$ be univariate polynomials with integer coefficients of degree $n$. Is there affine surface of general type of the form (1) $y^2=f(x) f(z)$ or (2) $y^2=f(x) g(z)$? I would expect for $n$ ...
3
votes
1answer
35 views

Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.
0
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0answers
23 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : ...
2
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1answer
28 views

Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. ...
0
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0answers
26 views

Computing Gauss curvature using Meusnier theorem

I have troubles with finding Gauss curvature and mean curvature in a certain point of an oblique cylindrical surface. I know the way using the fundamental forms, but I am supposed to use the Meusnier ...
0
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0answers
24 views

Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
2
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0answers
33 views

name this Romanesque surface

I happened to notice that the surface $$ x = \sin(u-v), y = \sin(v), z = \sin(-u) $$ or equivalently (if I haven't blundered) $$ x^4 + y^4 + z^4 - 2 x^2 y^2 - 2 x^2 z^2 - 2 y^2 z^2 + 4 x^2 y^2 z^2 = 0 ...
-1
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2answers
27 views

Vector parametrization of a surface intersection

How does one parametrize the following curve in 3-space to $\vec{g}(t): [a, b] \to \mathbb{R}^3$: the intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$ ? What I could come up with is as follows: ...
5
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33 views

Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
2
votes
1answer
44 views

What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions : 1) Can we find an ...
0
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0answers
16 views

Volume and surface of knock out drum

I have to calculate the volume and the surface of some KO drums (knock out drum). To avoid ambiguous understandings here's a picture of one: http://www.zamilsteel.com/ped/images/projects/11.jpg I ...
2
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0answers
56 views

What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$?

I am interested what is the type of the surfaces over the rationals $$ x^5 - y^5 + z^2 + x=0$$ and $$ x^5 - y^5 + z^2 + x+1=0$$ Magma's ...
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28 views

How to find coordinates of points on a 2D surface embedded in 3D space

kindly assist with this problem. Given an equilateral triangle in 2D plane (see figure 1) with origin (0,0) at point B, the coordinates of points A and C can be calculated as A(acos60,asin60) and ...
2
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0answers
63 views

What is the push forward of the canonical class?

Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, ...
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0answers
16 views

Surface area's part of the whole?

I just saw a commercial on tv where they said that something that they were trying to sell was 98 % meat and the only part that wasn't meat was the surface of the object which was made of something ...
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2answers
38 views

Showing that the image of a curve lies on a surface?

I am looking for an intuitive explanation to a problem in one of my practice tests. I'm given a parameterized curve from $\Bbb R$ to $\Bbb R^3$, called ${\bf r}(t) = (\sin t \cos t, \cos^2 t, \cos ...
0
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1answer
12 views

Offsetting a 2-D polynomial

I have a surface that is defined using a two dimensional polynomial: $$z = f(x) + g(y)$$ I want to offset the curve in the $XY$ plane from a point on the surface $\left(x_0, y_0, z_0\right)$ to a ...
2
votes
1answer
51 views

Times by $2\pi$ to find surface area using arc length

I am trying to find the surface area of a 'biconcave disc', which is the shape of a red blood cell. I know the formula/length for the curve, which I am integrating to find the volume of the shape. To ...
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0answers
20 views

theorem relating the genus of a surface to the Euler characteristic

One of the most basic facts of the theory of surfaces is $$ \chi = 2-2g $$ where $$ \chi \triangleq V - E + F $$ for orientables surfaces. Does this result have a name? I am also after a proof or a ...
4
votes
1answer
49 views

Is an open subset of a compact surface with connected boundary completely determined by its fundamental group?

Is an open, connected subset of a compact surface with connected boundary determined (up to homeomorphism) by its fundamental group? If we weaken the hypotheses, I can see how this can fail: A ...
0
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1answer
19 views

find the equation of a tangent surface which is parallel to a plane surface

How to get the equation of the tangent surface of $F=x^2 + 2y^2 +3 z^2=21$, which is also parallel to the plane surface $ x+ 4y + 6z=0 $? Here is what I've tried: $n = \{1, 4, 6\}$ from $ x+ 4y + ...
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1answer
25 views

Characterizing a surface

can somebody help me get started with this problem? I don't even know how to start the proof. Say $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable. Prove that $z=xf(y/x)$ belongs to a surface ...
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2answers
32 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
0
votes
1answer
24 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
0
votes
2answers
69 views

Calculating integral using Stokes theorem and directly

Here is my task: Calculate directly and using Stokes theorem $\int_C y^2 dx+x \, dy+z \, dz$, if $C$ is intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, orientated in positive ...
0
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1answer
38 views

Is the inside of a sphere a hyperbolical surface?

So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? ...
1
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1answer
45 views

Line integrals - Surface area

Here is my task: Calculate surface area of $2(x^{2}+y^{2})^{2}=xy$ between surface $x^{2}+y^{2}=z$ and $z=0$. Here is my attempt to solve this problem. Firstly, I transformed line ...
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15 views

Sorting Bigger Boxes to Smaller Boxes

I am currently working on a Bin Packing program and need to know what would be the most efficient way of getting boxes (arbitrary width, length, and height) to be sorted in the manner below? ...
0
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29 views

Calculus | Parametrization of boundary in $\mathbb{R^3}$

The Problem Given the volume $$ K = \left\{ (x,y,z)\in \mathbb{R^3} \big| \frac{x^2}{9} +y^2 \le z^2 +1, -\frac{1}{3}\sqrt{\frac{x^2}{9} +y^2} \le z \le 3 \right\} $$ What are $a$, $b$, and $K(z)$ ...
1
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1answer
43 views

Tangent plane and tangent lines to curves through a point

Let $S$ be the surface that is the graph of a continuous function $f: U \rightarrow \mathbb{R}$ on an open $U \subset \mathbb{R}^2$. Let $p = (x, y, f(x, y)) \in S$. One usually defines the tangent ...
0
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2answers
69 views

How to derive the 3D equation of a torus?

I'm doing a presentation on 3D surfaces for college and one of the equations I am using is a Torus. I know that the equation is $$z^2 = 25 - \left(10 - \sqrt{x^2 + y^2}\right)^2$$ For a torus with ...
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0answers
10 views

What are the intersection curves of two quadric surfaces of revolution?

Is it possible to determine the type of curve formed by the intersection of two hyperboloids of revolution (of 2 sheets) or the intersection of two ellipsoids? Also, any books or papers on quadric ...
2
votes
1answer
27 views

Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
0
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1answer
29 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
1
vote
1answer
24 views

Arcs and surfaces. Why are there finitely many arcs on the surface up to the action of MCG?

Given a bordered surface $S$ (I imagine this is true for non-orientable surfaces too, but you may restrict to the case of orientable surfaces) with finitely many marked points on each boundary ...
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2answers
27 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
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0answers
21 views

Parametrization for intersection curve of catenoid and cylinder

Required to obtain equation of intersection line of two surfaces.. the catenoid of revolution and displaced or eccentric cylinder..in a parameterized form. $$ (x^2 + y^2) = c^2 \cosh ^{2} (z/c) ; \, ...
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1answer
122 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
3
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2answers
45 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 ...
2
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1answer
42 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 ...
2
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0answers
26 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. ...
2
votes
1answer
58 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 ...
0
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1answer
51 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 ...
2
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1answer
40 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
34 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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42 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 ...
2
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2answers
43 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 ...
2
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0answers
68 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 ...
0
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0answers
33 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 ...