Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [surfaces]

For questions about two-dimensional manifolds.

Formally, a surface is a two-dimensional topological manifold. Some examples of surfaces are the plane, the cylinder, the sphere, and the graph of a real-valued function of two variables.

More generally, the term "hypersurface" can be used to denote an $(n-1)$-dimensional submanifold of an $n$-dimensional manifold.

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determining equation of a surface

I was wondering if there is a way to determine the equation of a surface if three R2 linear equations are known. I work in a research lab that produces a lot of correlation equations (mx+b), and we have three mx+b equations. Is there a way to…
Jake
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maximum number of faces with n lines

I was wondering a formula F(n) to guess the maximum number of faces made with n lines, for example: with 1 line, we cant create a face; F(n) = 0; with 2 lines, we also cant create a face F(n) = 0; with 3 lines, the best we can do is a…
Daniel
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Parameterize the following surface: $x +y +z=1$, $x,y,z>0$.

I need to parameterize the following surface: $x +y +z = 1$, $x,y,z>0$. I tried to put: $\sigma(u,v)=(u+v,u-v,-2u+1)$, but does it solve the case of $x,y,z>0$?
user 242964
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Is that a regular surface?

Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2$ be a injective application, given by: $$\mathbf{r}(u,v)=A(u)+v\cdot B(u), \forall\ (u,v)\in (a,b)\times (0,1)$$ where $A,B:(a,b)\to\mathbb{R}^2$ are two functions of class $C^1((a,b))$. Is it true…
Bogdan
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Equation for multiple peak surface to test particle swarm optimising algorithm

i have developed a particle swarm optimisation algorithm that i am running some tests on. It is able to solve simple equations like this: $x^2 + y^2 + 300y - 254x + 3$ with only one optimum but when i apply it to my real-world problem, i think it is…
Gus Kenny
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How many feet of rope to wrap a column

A heating pipe in my bathroom measures 105" in height. It is 8" in circumference (so about 2.55" diameter). I want to wrap it with a 1/4" thick rope. How many feet should I buy? (All measurements in inches).
user166306
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Local diffeoorphism and orientability of surfaces

I need some help to prove this: Let $S_2$ be an orientable regular surface and $f : S1 \rightarrow S2$ be a local diffeomorphism. Then $S_1$ is an orientable surface. Thanks.
Rachel
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Normal lines of a regular surface

I need to prove this: If all the normal lines to a regular surface pass through a fixed point, then the surface is a portion of the sphere. I haven't really tried much since I don't know what to do. Thanks
Rachel
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Surface parameterization of a cylinder?

Suppose I wanted to parameterize the cylinder $x^2 + y^2 = R^2$ (for the purpose of computing a surface integral). Say $z$ is in range $-z_0 \le z \le z_0$. The standard parameterization I see everywhere is $G(\theta,z) =
Grid
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Is there a definition of cylinder that these equations satisfy

Our teacher is claiming that (in $\mathbb{R}^3$) the following surfaces are "cylinders": $3x+y+\frac{7}{2}=0$ $y=x^2$ $z^2 = y$ $\frac{x^2}{4} + \frac{y^2}{4} = 1$ Is there any definition of cylinder that can justify this statement, and if so,…
Dre
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genus of connected sum G(Q1#Q2)

could you tell me a a mathematical publication where i can find the proof of why the genus of connected sum G(Q1#Q2) is G(Q1) + G(Q2) for orientable surface and 2G(Q1) + G(Q2) otherwise. I need it to make a bibliographic set of a thesis. Thanks!
Pietro Girotti
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Revolve a line about $z$-axis

I have to find the parametrization of the surface which comes from the revolution of the line $$(0,4+3t,1-3t)$$ about the $z$-axis. I tried to do $$\sigma(t,x)=(0\cdot\cos(x),(4+3t)\sin(x),1-3t)$$ with $x\in[0,2\pi]$ but when I try to plot, I don't…
mvfs314
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Are surfaces (equivalent to) continuous maps from rectangles?

Curves are continuous maps from an interval to a topological space. So a natural question is: is there any surface whose image is not the image of a continuous surjection from a rectangle to a topological space? Wikipedia's page about surfaces says…
Carla is my name
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Q: The surface with equation z = x^3 + y^3 - 6xy has two stationary points...

one at the origin and the second at point A. Determine the Coordinates of A I was going to try to solve it but after searching it up it looks like I have to use partial differentiation which I haven't studied. Is this true? Also, I still want to…
draw electric
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Cartesian equation of a surface

Given the following surface $$\sigma(u,v)=(\cos u,2\sin u,v)$$ I gotta find a cartesian equation of it. What I could conclude was $$4x^2+y^2=4$$ Is there a way to associate $x$ and $y$ with $z$?
mvfs314
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