Homework set 6 - Solutions
نویسنده
چکیده
I state without proof a few results concerning the local properties of smooth maps. Section 11, beginning at page 89, of our class notes contains more information, examples, and proofs. All functions and maps will be assumed here to be smooth (infinitely continuously differentiable) but the discussion in the notes is more general. By a neighborhood of a point I mean an open set containing the point. A map f : U → V is said to be a smooth diffeomorphism between open subsets U ,V of Rn if it is bijective and both f and f −1 are smooth maps; a map f : U → Rn where U is an open subset of Rn is said to be a local diffeomorphism if for each p ∈U there are open neighborhoods V ⊂U and W of f (p) such that the restriction f |V : V →W is a diffeomorphism. Theorem 1 (Inverse function theorem). Let f : U →Rn be a smooth map where U is an open and non-empty subset of Rn . Suppose that at some p ∈U the differential d fp is a linear isomorphism. Then f is a smooth diffeomorphism from a neighborhood V of p (an open subset containing p) onto a neighborhood W of f (p).
منابع مشابه
Effects of Annotations and Homework on Learning Achievement: An Empirical Study of Scratch Programming Pedagogy
In Taiwan elementary schools, Scratch programming has been taught for more than four years. Previous studies have shown that personal annotations is a useful learning method that improve learning performance. An annotation-based Scratch programming (ASP) system provides for the creation, share, and review of annotations and homework solutions in the interface of Scratch programming. In addition...
متن کاملCS 5314 Randomized Algorithms
Homework 1 (Suggested Solutions) 1. Ans. Use principle of deferred decision. Let X i denote the outcome of the i-th die so that Pr(X i = k) = 1/6, where 1 ≤ k ≤ 6.
متن کاملOutlines of Solutions to Selected Homework Problems
This handout contains solutions and hints to solutions for many of the STA 6505 homework exercises from Categorical Data Analysis, second edition, by Alan Agresti (John Wiley, & Sons, 2002). It should not be distributed elsewhere without permission of the author. Additional solutions for odd-numbered exercises are available at the website for the text, http://www.stat.ufl.edu/∼aa/cda/cda.html. ...
متن کاملIncom: a Web-based Homework Coaching System for Logic Programming
Programming is a complex process which usually results in a large space of solutions. However, existing software systems which support students in solving programming problems often restrict students to fill in pre-specified solution templates or to follow an ideal solution path. In this paper, we introduce a web-based homework coaching system for Logic Programming (INCOM) which allows students...
متن کامل