# Tagged Questions

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

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### 1-dimensional surfaces classification

Hy friends! I need to classify all the 1-dimensional compact surfaces ( in fact, i need those with boundary) and I don't know how to do it. I now the classic books of Guillemin & Pollack or ...
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### Maximal offset distance for a surface

Let $\vec r = \vec r(u, v)$ be a regular (analytic) surface. Now we offsetting this surface to distance $d$ in normal direction; new surface is $\vec r' = \vec r + d\vec n$. New surface $\vec r'$ is ...
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### What are these quotient spaces homeomorphic to?

I would like to know what the following spaces $X$ and $Y$ look like. More precisely, I want to know if they are homeomorphic to some other known spaces. I define $X$ and $Y$ as a quotient of the ...
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### Orientation under local diffeomorphism

Given regular surfaces $S_1$ and $S_2$ such that $S_2$ is orientable and a local diffeomorphism $f: S_1 \rightarrow S_2$, then why is $S_1$ orientable? What I think that can be done is to choose an ...
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### Surface element area from constrains

Consider a surface in $\mathrm{R}^n$ defined by $m$ linear constrains: $$\sum_i c_{ki} x_i = 0$$ We assume that the $m\times n$ matrix $c_{ik}$ is full-rank. Then there exists a linear ...
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### Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
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### How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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### What is the difference between surface and algebraic curve in general?

The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode ...
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### Volume and surfaces

i need help. Be : $X =(x,y,z)$ and $$T=\{x\in R^3\mid X=\begin{bmatrix}(1+rsin(u))cos(v)\\(1+rsin(u))sin(v)\\rcos(v)\end{bmatrix} ,\\0.5\geq r \geq 0,\\ 2\pi\geq u \geq 0 \\ 2 \pi\geq v \geq0$$ ...
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### Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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### Is this spiral known?

Parametrized as $$\sec \theta \,( p \cos(\theta+ \alpha), \,p \sin(\theta+ \alpha) , c\alpha),$$ the spiral is plotted $(-\pi/4<\theta< \pi/4;\,\,0< \alpha < 3 \pi)$ for $p= 1$ ...
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### Normal to surface at point

I have this function: $F(x,y,z)=x^2−y^2−z^2+4$ where $z\ge 0,0\le x \le 2,0 \le y \le 2$. How can I find the normal at some point $P=(p_x,p_y,p_z)$? I have tried to calculate the derivatives of ...
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### surface of the saddle [closed]

i need Help. Determine the surface of the saddle $$S={(x,y,z)∈R^3; x^2+y^2<=2, z=x^2 -y^2}$$ and the flow of $v(x) = x$ , by S plane polar coordinates, dx dy = r dr dφ, are helpful. Thanx
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### Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel f(r)=r in 3D. I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the constructed ...
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### Existence of closed level sets on a surface for some field

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. ...
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### 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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### 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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### 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 ...
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 $b$?
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 ...