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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Proving that surface is not elementary

I need help with proving or disproving that the following surface $x^2+y^2+z^2=a^2,a>0$ is elementary. If I had another condition $z>0$ then the surface would be elementary because $rank g'=2$ where $g(x,y)=(x,y,\sqrt{a^2-x^2-y^2})$. But here I know…
Trevor
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Parameterizing the surface formed by lines through the points of a helix and orthogonal to its axis

Given the following curve, a helix, $$\alpha(t)=(\cos(t),\sin(t),t)$$ I consider, for each $t$, the line that pass through $\alpha(t)$ and it's orthogonal to $z$ axis. Then, I have to parametrize the surface that is the union of all this lines. I…
mvfs314
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How to find a surface if we know the normal vectors?

I have the positions of 3D particles at each z-section each color represents a particle. The lateral projection of the data looks approximately like this imageThe top end of these particles form a curved surface I want to generate a surface locally…
Elizabeth
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Math question related to three dimensional surfaces?

So I have to determine and draw the surfaces $$z-2x^2-4y^2 ≥0,\qquad \mbox{and}\qquad 4y^2-x^2+4z^2-1 ≥0$$ so the first one in my opinion should be transformed like this $$z ≥2x^2+4y^2$$ then we multiply by two both sides and we have $$2z…
mathlover484
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Surface area of an ellipsoid above a given plane

The problem: $ \text{Set-up an integral for the surface area of the portion of the ellipsoid } 4x^2 + 9y^2 + z^2 = 64 \text{ that lies above the plane } z=-1 \text{. Do not simplify or evaluate the integral.}$ Professor's Answer: $$ SA =…
financial_physician
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Three (or more) surfaces with a common locus of intersection -- What to call it?

Consider three different functions, $F_1(x,y,z)$, $F_2(x,y,z)$ and $F_3(x,y,z)$, in a 3D Euclidean space with Cartesian coordinates x, y and z. Setting $F_1$ to some value, say $v_1$, defines an isosurface of $F_1$ in this space. Similarly, let…
Nominally Brain-Dead
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Why is it obvious that the plane $z=0$ is tangent to the surface $z=x^{2}+y^{2}$

Why is it obvious that the plane $z=0$ is tangent to the surface $z=x^{2}+y^{2}$ I don't quite understand, is this obvious? I have a problem with the background knowledge, I don't even know how to deal with the surface $z=x^{2}+y^{2}$.
2018 xyz
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Shortest distance between a point and a surface

What I actually have to do is find the point on the surface which is closest to the point $P$, both of which are given below. Function $z(x,y)=x^2+2y^2$ Point $P = (1,-1,1)$ This is what I have tried: Consider a point $Q = (a,b,c)$ on this…
Yak Attack
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Surface area and volumes, please solve

The length of the diagonal of a cuboid is $5\sqrt{5}$ cm and the sum of its length, width and height is $19$ cm. Find its surface area.
Rohit Verma
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Minimal measured foliation

Let $M$ be a closed connected surface and $\mathcal{F}$ a minimal (every leaf is dense) measured foliation (as, for example, in Thurston's work on surfaces) on $M$. Let $\tilde{M}$ be the universal cover of $M$ and $\tilde{\mathcal{F}}$ the pullback…
ah--
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Intersection of two surfaces.

So I was doing some problems looking for curves of an intersection between two surfaces. I was wondering how do I know when i "got" it? What is the algorithm of finding these curves? For example, Take the surfaces $z=4x^2+y^2$ and $y=x^2$. So the…
Sorfosh
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Question about homeomorphic surfaces

I am wondering if there exist an example of two surfaces such that they are orientable (or not orientable) and have same Euler characteristic but they are NOT homeomorphic. Thanks for your support.
LH8
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Surfaces where all four feets chairs stand

It's well known that a three feet chair can stand correctly on any (regular enough) surface (let alone the intersection between the chair and the surface!and chairs orientation). How about four feets chairs. I obviously found plane and sphere. I…
Regis Portalez
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In there a common name for the surface $z = x^y$?

Is there a standard name for the surface $z=x^y$, the way we have 'plane' for $z=x+y$, and 'hyperbolic paraboloid' or 'saddle' for $z=x y$ (which is congruent to the surface $z=\frac{x^2 - y^2}{2}$), as examples?
bosilveira
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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$ and $ c=0.125 $ on Mathematica.
Narasimham
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