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数学 大学生・専門学校生・社会人

多様体を構成するために、位相空間に完全アトラスを導入するところで質問です。 完全アトラスを導入するメリットとして、この文章の下線部を「異なる座標系を用いたのに同じ計算ができてしまうという問題が解消される」解釈したのですが、そこがよくわかりません。座標系を変えて計算する... 続きを読む

1 Two n-dimensional coordinate systems & and ŋ in S overlap smoothly provided the functions on¯¹ and ŋo §¯¹ are both smooth. Explicitly, if : U → R" and ŋ: R", then ŋ 1 is defined on the open set ε (ur) → ° (UV) V and carries it to n(u)—while its inverse function § 4-1 runs in the opposite direction (see Figure 1). These functions are then required to be smooth in the usual Euclidean sense defined above. This condition is con- sidered to hold trivially if u and do not meet. Č (UV) R" Ĕ(U) n(UV) R" S n(v) Figure 1. 1. Definition. An atlas A of dimension n on a space S is a collection of n-dimensional coordinate systems in S such that (A1) each point of S is contained in the domain of some coordinate system in, and (A2) any two coordinate systems in ✅ overlap smoothly. An atlas on S makes it possible to do calculus consistently on all of S. But different atlases may produce the same calculus, a technical difficulty eliminated as follows. Call an atlas Con S complete if C contains each co- ordinate system in S that overlaps smoothly with every coordinate system in C. 2. Lemma. Each atlas ✅ on S is contained in a unique complete atlas. Proof. If has dimension n, let A' be the set of all n-dimensional coordinate systems in S that overlap smoothly with every one contained in A. (a) A' is an atlas (of the same dimension as ✅).

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英語 中学生

(B)教えて欲しいです😭😭 答えは preserve culture になります。 解説では(7)の2文目と(3)の最後から3文目を参照とかいてありますがそれでも意味がわからないです。

3 次の文章を読んで, あとの各問に答えよ。 (*印の付いている単語·語句には, 本文のあとに [注] がある。) When we go to the library, we read books/*search for and/share information and have a *discussion with others. // Libraries are very convenient places. /The library has a long history of collecting and keeping books. /Books have been an important part of culture. Around 1445 Johann Gutenberg *invented the *printing machine./ Libraries began to collect the hooks *printed by the printing machine, and the number of libraries grew./ Now some libraries have begun to *digitalize a lot of books. Some people say most of the books will become digitalized *data/ When 声った all the books are digitalized, what will the future of the library be? / Some even say the library will disappear. Will that really happen? To answer this question, we first have to see how people have digitalized books. We can say the idea of digitalizing books began with Michael Hart in 1971/ He was able to use an expensive computer,/so he thought he could do something good for other people by using it. A computer can keep a lót of data/and it can search for the data in a very short time./When the computer has a lot of digitalized data from the books, these data become an important part of culture. / Michael Hart thought that people would use these data as they like, His idea became a *project. /He couldn't digitalize books which had *copyright, so he digitalized books which were *in the public domain and collected them in a computer./ People were able to read the distalized books without *paying any money. Hart named his project “Project Gutenberg," |He thought his project was as important as Gutenberg's printing machine, because the printing machine also spread knowledge 知識てめる all over the world. / Project Gutenberg continues even after Hart died in 2011. Now you can read - 4

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