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.

3209 questions
2
votes
1 answer

Why is the cross product of partial derivatives of a parametric surface give the normal vector?

Why do I get the normal vector to a point on a parametric surface when I use the cross product on the partial derivatives? For example, $r(u,v)$ is our surface and the normal vector on the point $(a,b)$ is given by partial derivatives with respect…
Hdhehd
  • 21
2
votes
2 answers

Circles and surfaces

We say a subset $\overline{D}$ of $\Bbb R^2$ is $x\textbf{-normal}$ is there exist $a,b\in \Bbb R$ with $a
À toa
  • 23
2
votes
4 answers

Orientation of Surfaces

I'm having a little trouble understanding how to orient a surface in $\mathbb{R}^3$ For example, how would I orient the ellipsoid given by: $$x^2+y^2+z^2+xy+xz+yz=\frac12$$ for $(x,y,z) \in \mathbb{R}^3$ ?
user61974
  • 43
2
votes
0 answers

Given an arbitrary surface, what is the smallest distance from two points "walking" along the surface?

Yesterday, I was thinking about one characterization of the line segment in euclidean geometry: The shortest distance between two points is a line segment. We can use this to describe a "line" in spherical geometry, for example. But how do we do to…
Red Banana
  • 23,956
  • 20
  • 91
  • 192
2
votes
1 answer

Finding surface area of cone inside a cylinder

So I am presented with the following problem: Find the surface area of the cone $z=\sqrt{ x^2 + y^2} $ that lines inside the cylinder $x^2 + y^2 = 2x$. Im pretty sure a double integral is involved, but I have no clue how to even go about starting…
Nick
  • 271
  • 1
  • 7
  • 13
2
votes
2 answers

Create a smooth surface based on 4 points?

To understand this question, please first understand the question and answer here: Create a formula that creates a curve between two points We are essentially transcending a 2d problem into a 3d problem. I have 4 points arranged as the corners of a…
Lorry Laurence mcLarry
  • 1,051
2
votes
1 answer

Give the equation of the surface

Given $$z = y^2 + 3,$$ give the equation of the surface if rotated around the $z$-axis. After I plot this out, I get a simple parabola in the $yz$-plane... so flipping it about the $z$-axis is just a parabola opening down instead of opening up.. and…
Nick
  • 271
  • 1
  • 7
  • 13
2
votes
1 answer

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) $$ x^4 + y^4 + z^4 - 2 x^2 y^2 - 2 x^2 z^2 - 2 y^2 z^2 + 4 x^2 y^2 z^2 = 0 $$ resembles the octahemioctahedron in the same…
Anton Sherwood
  • 628
2
votes
1 answer

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 find the surface area of the shape, can I just…
kay
  • 21
2
votes
2 answers

question about closed disc and closed surfaces.

Question: is a closed disc is a example of closed surface. I know that, the boundary of an open disk viewed as a manifold is empty, while its boundary in the sense of topological space is the circle surrounding the disk. as by definition of…
Akash Patalwanshi
  • 3,269
1
vote
1 answer

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 $4$ traces? $(wxy, wxz, wyz, xyz)$? Here is a…
Winndie
  • 13
1
vote
1 answer

Rational parameterization of surface

The surface $$ (x^{2} + y^{2} - 2 y + 1) \cdot (x^{2} + y^{2} + 2 y + 1) \cdot (x^{2} + y^{2} - 2 x + 1) \cdot (x^{2} + y^{2} + 2 x + 1) - z^2 = 0$$ has the points $\left(\dfrac{t}{s},\dfrac{t}{s},\dfrac{s^{4} + 4 t^{4}}{s^{4}}\right)$, $…
jorox
  • 113
1
vote
1 answer

Randomly distributing points over a curved surface

I have a shape that is a cylinder of radius $r$ and height $h$ with a hemisphere at one end. Total area of interest is $2πr^2+2πrh$ (bottom is not included) I would like to randomly distribute points over that surface (not including the 'bottom')…
Spehro Pefhany
  • 145
1
vote
3 answers

How to calculate surface area of a curved plane?

could anyone explain how to calculate the surface area of a curved plane? I am trying to calculate the surface area of a "vaulted" ceiling that is 24' long, 7' wide, and the height of the curve is 4' at the mid-point so that I can figure out how…
Neal L
  • 13
1
vote
1 answer

How to write the implicit equation of a surface when given the parametric equations?

How can I find the implicit equations of a surface if I have the parametric equations? For example, if the surface $(S)$ is given by: $$x = u+\sin v$$ $$y=u+\cos v$$ $$z = u+a$$ what are the implicit equations of this surface?
Andrei0408
  • 103
1
2
3 4 5 6