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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Divergence on a parameterised surface

Given a smooth vector field, $\textbf{A}$, defined on a smooth parameterised surface, $\textbf{r}(s,t)$ say, how do we obtain an expression for its divergence, $\nabla\cdot\textbf{A}(\textbf{r}(s,t))$, in terms of its components, defined through…
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In "Curves and Surfaces" by Montiel and Ros, is $(dF_r)_p(e_i) = (1-rk_i(p))e_i$ correct?

Consider the following problem from Chapter 3 of Curves and Surfaces, 2nd edition, by Montiel and Ros: Now, for me it is easy to see that $$ (dF_r)_p(v) = v + r (dN)_p(v), \quad v \in T_pS. $$ If $v = e_i$ is a principal direction,…
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A question regarding orthogonal parametrizations

This passage is part of the solution of an exercise in Differential Geometry. Let $S$ be a surface, and let $X: U \longrightarrow S$ be an orthogonal parametrization. If $N^X = \frac{X_u \wedge X_v}{|X_u \wedge X_v|}$, then $$ \langle X_{uu}, N^X…
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Umbilical points of the ellipsoid.

The following is Exercise (11) of Chapter 3 of Curves and Surfaces, 2nd edition, by Montiel and Ros: Determine the umbilical points of the ellipsoid of equation $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ where $0 < a < b <…
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How to choose limits of integration when calculating the volume of a tridimensional object.

I am trying to compute the volume of the figure limited by these surfaces: $$z=xy$$ $$x+y+z=1$$ $$z=0$$ My attempt: Using the first two equations I get $y=\frac{1-x}{1+x}$, which is, I think, the projection in the $XY$ plane of the intersection…
ABC
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Reparametrization of an ordinary differential equation

The following is part of the solution of an Exercise in Curves and Surfaces, second edition, by Montiel and Ros, which asks the reader to prove that the only surfaces of revolution with zero mean curvature are catenoids and planes. After computing…
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Small path enclosing a point $p$

Can anyone help me see why this statement is true Let $p$ be a point of a Riemann surface $X$, and let $S$ be a subset of $X$ whose closure does not contain $p$, then theres is a closed path $\gamma$ on $X$ with the following properties, $\gamma $…
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Stokes theorem problem to find alpha and beta so that I is independent of the choice of S

I have a question that I got half through but can't finish it. If anyone could help I would appreciate it. Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle $x^2+y^2=1$, z=0 $y\le0$. Let S be the smooth surface…
user68203
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Find shape operator of a parallel surface

Let $S$ be a regular surface and $\mathbb{X}(u, v)$ a local parametrization of $S$. If $a$ is a small number, let's consider $$\tilde{\mathbb{X}}(u,v)=\mathbb{X}(u,v)+a\textbf{N}(\mathbb{X}(u,v))$$ where $\textbf{N}$ is the normal vector to the…
kubo
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What does the surface |z₁|² = |z₂|² look like?

In a quaternionic plane there are 2 axes and each point corresponds to $q \in \mathbb C^2$. Now, $|z_1|^2 = |z_2|^2$ should define a surface that divides a 3-sphere $|z_1|^2 + |z_2|^2 = 1$ into two pieces, where $z_1, z_2 \in \mathbb C$. What does…
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Are measurements using the first fundamental form preserved by re-parameterization?

I may be conflating a number of different concepts, but I am confused about measurements using the first fundamental form. Here is what I think I know: The first fundamental form can be used to compute distance on a parametric surface The…
marcman
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Why do we classify algebraic curves primarily by genus and not degree?

I read in a text the following, 'One such classification might be according to the degree of the curve. Although this is a reasonable idea for very low degree curves, it turns out to be unsatisfactory for higher degree curves. Instead of the degree…
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Zariski decomposition and vanishing cohomologies

On a projective surface $S$, a $\mathbb{Z}$-divisor $D$ admits a unique Zariski decomposition , $$ D = P + N \,, $$ where $P$ and $N$ are $\mathbb{Q}$-divisors called the 'positive' and 'negative' parts. The important aspect for my question is that…
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Local diffeomorphism preserves orientation if and only if the jacobian is positive everywhere

Consider the following problem from Curves and Surfaces, 2nd Edition, by Montiel and Ros: To me, the solution is pretty much straightforward, following directly from the definitions. My question is? Where is the connectedness hypothesis used?
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Characterization of star-shaped surfaces as diffeomorphisms of the sphere

Let $f: \mathbb{S}^2 \longrightarrow \mathbb{R}$ be a positive differantiable function. Define $$ S(f) = \{f(p)p \in \mathbb{R}^3 \ | \ p \in \mathbb{S}^2\}. $$ It can be shown that $S(f)$ is a compact surface diffeomorphic to the sphere…
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