/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Let \(S_{2}\) be an orientable r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 2

Let \(S_{2}\) be an orientable regular surface and \(\varphi: S_{1} \rightarrow S_{2}\) be a differentiable map which is a local diffeomorphism at every \(p \in S_{1}\). Prove that \(S_{1}\) is orientable.

Short Answer

Expert verified
By transferring the consistent orientation from \( S_{2} \) to \( S_{1} \) via the local diffeomorphism \( \varphi \), it is proven that \( S_{1} \) is orientable.

Step by step solution

01

- Understand Given Information

Identify the provided information: 1) There is an orientable regular surface denoted as \( S_{2} \), and 2) There is a differentiable map \( \varphi: S_{1} \rightarrow S_{2} \) which is a local diffeomorphism at every point \( p \in S_{1} \).
02

- Define Orientability

Recall that a surface is called orientable if it has a consistent choice of orientation at each point. This means that given any local region, an orientation (e.g., a choice of normal vector) on the surface can be extended to the entire surface without encountering any contradictions.
03

- Use Local Diffeomorphism

Understand that \( \varphi \) is a local diffeomorphism. This implies that for any point \( p \in S_{1} \), there exists a neighborhood \( U \subset S_{1} \) of \( p \) such that \( \varphi(U) \) is open in \( S_{2} \) and \( \varphi|_{U}: U \rightarrow \varphi(U) \) is a diffeomorphism. Therefore, \( \varphi \) locally preserves the structure of the surface.
04

- Transfer Orientation

Since \( S_{2} \) is orientable, there exists a consistent orientation on \( S_{2} \). For each point \( p \) in \( S_{1} \), the map \( \varphi \) can be used to transfer the orientation from \( \varphi(U) \subset S_{2} \) back to \( U \subset S_{1} \). This is possible because the diffeomorphism preserves the local geometric properties, including orientation.
05

- Prove Consistency

Show that the transferred local orientations from \( S_{2} \) to different neighborhoods in \( S_{1} \) are consistent with each other. Given that \( \varphi \) is a local diffeomorphism, overlapping regions in \( S_{1} \) will have their orientations matched by the corresponding regions in \( S_{2} \), ensuring no contradictions.
06

- Conclude Orientability

Since the orientation can be consistently transferred across all neighborhoods in \( S_{1} \) to form a global orientation without contradictions, this establishes that \( S_{1} \) is orientable.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diffeomorphism
Imagine a diffeomorphism as a smooth, flexible map that can be stretched or bent without tearing or gluing. This concept is pivotal in the problem since it ensures that our surface mappings preserve structure. Formally, a map \(\boldsymbol{\tau: M \rightarrow N}\) between two manifolds is a diffeomorphism if it is smooth, has a smooth inverse, and is bijective. This ensures that every local feature is preserved under mapping.
For our problem:
  • Let's say surface \(\boldsymbol{S_{2}}\) is oriented.
  • We use a map \(\boldsymbol{\boldsymbol{\tau}: S_{1} \rightarrow S_{2}}\) that is a local diffeomorphism at every point.
Since \(\boldsymbol{S_{2}}\) is oriented, it has a consistent way to define
Surface Orientation
Orientation of a surface means a consistent direction can be chosen at every point. Visualize a sphere with a choice of 'up' everywhere and maintaining that 'up' direction smoothly across the whole surface. If you move smoothly around the sphere, you will never feel like the direction flips.
To determine if \(\boldsymbol{S_{1}}\) is orientable:
  • Since \(\boldsymbol{S_{2}}\) is orientable, we know it has a consistently defined normal direction everywhere on it.
  • Through the diffeomorphism, the orientation on \(\boldsymbol{S_{2}}\) can be imposed back onto \(\boldsymbol{S_{1}}\) without contradictions.
We check by ensuring every piece of \(\boldsymbol{S_{1}}\) aligns without breaking this directional consistency, as implied by the map's smoothness and local structure preservation.
Regular Surfaces
Regular surfaces are smooth, continuous, and without any breaks, edges, or singularities. Think of them as perfectly even terrains without any bumps or holes that you might find in a mountain. Mathematically, a surface is regular if it is locally similar to a plane everywhere.
Concerning our exercise:
  • \(\boldsymbol{S_{1}}\) and \(\boldsymbol{S_{2}}\) are both regular surfaces.
  • Local diffeomorphisms mean every small patch on \(\boldsymbol{S_{1}}\) resembles a corresponding patch on \(\boldsymbol{S_{2}}\) without losing regularity.
Since no singularities or irregularities disrupt the orientation consistency in \(\boldsymbol{S_{2}}\), and the map maintains these properties, \(\boldsymbol{S_{1}}\) must also be orientable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that if \(p\) is a point of a regular surface \(S\), it is possible, by a convenient choice of the \((x, y, z)\) coordinates, to represent a neighborhood of \(p\) in \(S\) in the form \(z=f(x, y)\) so that \(f(0,0)=0, f_{x}(0,0)=0\), \(f_{y}(0,0)=0\). (This is equivalent to taking the tangent plane to \(S\) at \(p\) as the \(x y\) plane.)

(Tubular Surfaces.) Let \(\alpha: I \rightarrow R^{3}\) be a regular parametrized curve with nonzero curvature everywhere and arc length as parameter. Let $$ \mathbf{x}(s, v)=\alpha(s)+r(n(s) \cos v+b(s) \sin v), \quad r=\text { const. } \neq 0, s \in I, $$ be a parametrized surface (the tube of radius \(r\) around \(\alpha\) ), where \(n\) is the normal vector and \(b\) is the binormal vector of \(\alpha\). Show that, when \(\mathbf{x}\) is regular, its unit normal vector is $$ N(s, v)=-(n(s) \cos v+b(s) \sin v) $$

Show that the equation of the tangent plane at \(\left(x_{0}, y_{0}, z_{0}\right)\) of a regular surface given by \(f(x, y, z)=0\), where 0 is a regular value of \(f\), is $$ \begin{aligned} &f_{x}\left(x_{0}, y_{0}, z_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0}, y_{0}, z_{0}\right)\left(y-y_{0}\right)+f_{z}\left(x_{0}, y_{0}, z_{0}\right)\left(z-z_{0}\right) \\ &\quad=0 . \end{aligned} $$

Show that \(\mathbf{x}: U \subset R^{2} \rightarrow R^{3}\) given by $$ \mathbf{x}(u, v)=(a \sin u \cos v, b \sin u \sin v, c \cos u), \quad a, b, c \neq 0, $$ where \(0

(Theory of Contact.) Two regular surfaces, \(S\) and \(\bar{S}\), in \(R^{3}\), which have a point \(p\) in common, are said to have contact of order \(\geq 1\) at \(p\) if there exist parametrizations with the same domain \(\mathbf{x}(u, v), \overline{\mathbf{x}}(u, v)\) at \(p\) of \(S\) and \(\bar{S}\), respectively, such that \(\mathbf{x}_{u}=\overline{\mathbf{x}}_{u}\) and \(\mathbf{x}_{v}=\overline{\mathbf{x}}_{v}\) at \(p .\) If, moreover, some of the second partial derivatives are different at \(p\), the contact is said to be of order exactly equal to 1 . Prove that a. The tangent plane \(T_{p}(S)\) of a regular surface \(S\) at the point \(p\) has contact of order \(\geq 1\) with the surface at \(p\). b. If a plane has contact of order \(\geq 1\) with a surface \(S\) at \(p\), then this plane coincides with the tangent plane to \(S\) at \(p\). c. Two regular surfaces have contact of order \(\geq 1\) if and only if they have a common tangent plane at \(p\), i.e., they are tangent at \(p\). d. If two regular surfaces \(S\) and \(\bar{S}\) of \(R^{3}\) have contact of order \(\geq 1\) at \(p\) and if \(F: R^{3} \rightarrow R^{3}\) is a diffeomorphism of \(R^{3}\), then the images \(F(S)\) and \(F(\bar{S})\) are regular surfaces which have contact of order \(\geq 1\) at \(f(p)\) (that is, the notion of contact of order \(\geq 1\) is invariant under diffeomorphisms). e. If two surfaces have contact of order \(\geq 1\) at \(p\), then \(\lim _{r \rightarrow 0}(d / r)=0\), where \(d\) is the length of the segment which is determined by the intersections with the surfaces of some parallel to the common normal, at a distance \(r\) from this normal.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.