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

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Show curve is tangent to a surface using gradient

The question is this: 'Show that the sphere $h(x,y,z) = x^2+y^2+z^2-8x-8y-6z+24=0$ is tangent to $f(x,y,z)=x^2+3y^2+2z^2=9$ at the point (2,1,1).' My approach was that grad(f) at P should give a ...
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42 views

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 ...
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1answer
31 views

Fundamental group of complement of image of diagonal embedding from $S_g$ to $S_g \times S_g$?

Let $S_g$ be a surface of genus $g \ge 0$. Let $\Delta \subset S_g \times S_g$ be the image of the diagonal embedding $x \mapsto (x, x)$. Let $X$ be the complement of $\Delta$ in $S_g \times S_g$. My ...
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Definition of Gauss curvature

In GR class a definition of Gauss curvature was introduced which I cannot understand. It says: In a curved surface in $R^3(x,y,z)$. For any small area (circle?) $\Delta A_1$ of the surface. ...
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39 views

I would like to find the inverse of $X$

Let $X(u,v)=(v−u,u^2−v^2,u+v)$, $(u,v)\in U=\mathbb R^2$ and $S=X(U)$. what is the inverse of $X$ where $X$ is the function which maps a $2$D object into a $3$D object $X^{-1}(x,y,z)$
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46 views

Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of $F(\...
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1answer
49 views

Surface of a polynomial

How can I find the surface represented by the polynomial $$x^2-y^2-2xz=0$$ any clue please?? I have tried to plot it using Maple
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31 views

Parametrisation of a surface and a cylinder

I have been asked to find C which is the curve I need to integrate over and C is the intersection of the cylinder $x^2+y^2=2y$ and the plane $y=z$. I assume you have to find a parametrization that ...
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1answer
63 views

Coverings of connected sum of four copies of $\mathbb{R}P^2$

G.Baumslag in one of his papers asserts that a group $G = \langle a,b,c,d | a^2b^2c^2d^2 = 1 \rangle$ contains all fundamental groups of closed compact orientable surfaces of genus $g\geq 2$? I think ...
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322 views

Exponential map on the ellipsoid.

Consider the ellipsoid $M \subseteq \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{x^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian metric ...
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30 views

Find the equation of the following curve

Suppose normal lines are drawn at all the points on the surface $z = ax^2 + by^2$, where $a$ and $b$ are some positive constants, that are at a given height $h$ above the $xy$-plane. Find an equation ...
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1answer
29 views

Finding velocity vector field over 3D surfaces

I have a surface, $S$, defined in three-dimensional space. For the sake of this question, let's assume it is a sphere with unit radius although the surface in my problem can be any shape. I have a ...
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5answers
47 views

Describing the motion of a particle (sphere)

If I have the following position at time t : $\hat{r}(t) = 3\cos(t)\hat{i} + 4\cos(t)\hat{j} + 5\sin(t)\hat{k}$ , then how can I tell if the particle's path lies on a sphere or not? If e.g. the second ...
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1answer
42 views

Why don't you add the top and bottom circle when finding the surfaces of revolution

I am studying Calculus II and I had a question about finding the surfaces of revolution. Why don't you add the top and bottom circle when you find the surface? If one is trying to find the surface ...
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2answers
30 views

How to prove a differentiable function from a surface to a surface

$S$ is the unit sphere without its north and south poles. $H$ is the hyperboloid $x^2+y^2-z^2=1$. For $p\in S$, $l_p$ is the ray perpendicular to the $z$ axis that starts on the $z$ axis and passes ...
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1answer
27 views

Finding the Surface that a Vector Function Lies On

I've been studying vector functions and one thing that I don't quite understand is finding the surface that some vector function r(t) lies on. For instance, in my textbook (Calculus Early ...
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26 views

Calculating surface integral (with gauss's theorem (?))

I would like to solve the following problem: Let $B_1$ be the unit ball in $R^3$ and $A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\})$. Let $F(x,y,z) := (-y+e^{x+z}, 0, e^{x+z})$. ...
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1answer
42 views

Surfaces in dimensions higher than $3$, how to compute the normal vector?

In my book surfaces are defined as maps $\sigma\ \colon A \subseteq \mathbb R^2 \to \mathbb R^n$ ($n \geq 3$). Then the book goes on to define regular surfaces and the normal vector. The latter is ...
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45 views

Degree of a ruled surface

If $X = C\times C' \subset \mathbb{P}^3$ for $C$ of genus $g$ and $C'$ of genus $g'$ (both smooth), then we know from Hartshorne exercise V.1.5 that $$ 8(g-1)(g'-1) = d(d-4)^2 $$ where $d = \text{deg}(...
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1answer
54 views

Does there exists any $x$ such that $x\geq AM\geq GM?$ If not,how do I prove it is unbounded?

We know $AM\geq GM$.Or,in words,$AM$ has a minimum value when it is equal to $GM$?But,by any chance is there any way to find $x$ such that it is the upper limit of $AM$? My inspiration for this ...
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1answer
50 views

Where does the following equation for the area of an ellipsoid come from?

I have been reading an old draft that I made a few years ago where I used the following expression for the area of an ellipsoid: $$A=2 \pi r^2 + \frac{2 \pi \tanh^{-1}(\sqrt{1-r^{-6}})}{r^4 \sqrt{1- ...
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24 views

How can I prove that a point on a surface can be parametrized by a certain parametrization? [closed]

So the specific question is as follows: how can I show that the surface $y(x-a)+zx=0$ can be parametrized by $\alpha(u,t)=(au,ut,t-ut)$? Or equivalently, the set of points defined by the first ...
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1answer
56 views

Klein Bottle as the connected sum of two projective planes

Can someone explain intuitively why the Klein bottle is diffeomorphic to the connected sum of two projective planes? I can do this using origamis/fundamental graphs )w/e they are called. is it ...
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1answer
33 views

Normal and geodesic curvatures of intersection curve of two given surfaces

I was recently presented this in differential geometry class which I cannot seem to solve involving curve of intersection of two surfaces in $ R^3 $ which reads as Let us define two surfaces in $ ...
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1answer
36 views

A regular curve on regular surface in $ \mathbb R^3 $ being of constant tangent length

I was given this in diff. geometry on which I am stuck: Let $ S \subset \mathbb R^3 $ be a regular orientable surface in $ \mathbb R^3 $ such that on it we define the regular curve $ \gamma : I \...
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1answer
47 views

Can there be a closed geodesic on surface embedded in R3 with zero Gaussian curvature

I was asked this in differential geometry class and it.is bugging me as i do not know the answer I know a surface of constant Gaussian curvature zero is locally isometric to a plane on which no ...
2
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1answer
92 views

Determining if there can be smooth closed geodesic given its curvature

In my class on differential geometry I have been given the following question on which I am stuck: Let S be a regular orientable surface in $ R^3 $ with Gaussian curvature $K$ (not necessarily ...
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22 views

Proving That $\alpha^{\prime}(t)$ is orthogonal to $N(\gamma(t))$

Let $\Phi$ be a surface in $\mathbb{R}^3$ with parameter domain $K\subset\mathbb{R}^2$ and let $\gamma:[a,b]\to K$ be a $\mathcal{C}^1$-curve. Also let $\alpha=\Phi\circ\gamma$. Prove that $\alpha^{\...
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How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
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39 views

Intersection of a surface and a line.

I have a surface ($H$) that passes every corner points(one coordinate gets its maximum $1$ while others $0$), such as $(1,0,...,0), (0,1,0,...,0),....,(0,...,1).$ H is characterized by $A\sum_{...
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100 views

Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
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1answer
18 views

Possibility of regular surface with specific first and second fundamental form matrices

I have met this in diff. geometry class which states: We are to determine if there exists a regular surface in $ R^3 $, $ S = f(u,v) $ with fundamental forms as follows: $ I = \begin{bmatrix} ...
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1answer
22 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
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38 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
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1answer
40 views

Computing the approximate or exact area of an isosurface

The isosurfaces I'm reading about are defined by a constant value v in a scalar field. The scalar field is defined by placing n vectors in k-space such that $iso(\vec{x})=\sum_{i=1}^{n}\sum_k(x_k-p_{...
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1answer
53 views

Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
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23 views

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 ...
2
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0answers
32 views

Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
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0answers
30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that $X$...
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2answers
25 views

Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
2
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1answer
37 views

Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
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1answer
38 views

Proving subset of regular surface - hyperboloid - is a regular surface

I have stumbled upon this in differential geometry dealing with regular surfaces: We define the following surface (a hyperboloid) as $ K = \{ (x,y,z) \in R^3 | x^2+y^2-z^2 = 1 \} $ and ...
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48 views

Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
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1answer
64 views

Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
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2answers
158 views

How to find the surface area of a spherical cap by integration?

I don't really understand how they derived the formula in the following picture. The aim is basically to find the formula for the surface area of a spherical cap. Why do you differentiate the $x=\...
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55 views

Proof of Theorem 3.2 - Elementary Differential Geometry, O'Neil.

I am going through Elementary Differential Geometry by O'Neil, and I am at Theorem 3.2 on page 151. O'Neil comments that a rigorous proof of this theorem requires the methods of advanced calculus, and ...
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1answer
80 views

Gradient in terms of first fundamental form

In Do Carmo's Differential Geometry of Curves and Surfaces, I'm having a quite hard time trying to solve Excersise 14 on pages 101-102. He defines the gradient of a differentiable function $f:S\to \...
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25 views

Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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20 views

Area of ​​the surface of revolution of the ellipsoid

I need to find the surface area of an ellipsoid using the equation of an ellipse. I believe my calculations are correct but the formulas I meet on the Internet are complex and have arcsin or arctan in ...
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78 views

Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ (x,0)...