学年

質問の種類

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 ✅).

未解決 回答数: 0
英語 高校生

英語の長文です。 文法表現のあるところが知りたいです。 よろしくお願いします。

UNIT 1 5 Reading Passage 10 15 20 20 25 Listening There are more than 37,000 known species of spiders in the world in a wide variety of shape's and sizes! The largest spiders in the world live in the rain forests of South America and are known by the people who live there as the "bird-eating spiders." These spiders can grow up to 28 centimeters in length- about the size of a dinner plate, and, as their name suggests, have been known to eat small birds. In comparison, the smallest species of spider in the world is native to Western Samoa. These tiny spiders are less than half a millimeter long — about the size of a period on this page and live in plants that grow on mountain rocks. - Some people like to keep spiders as pets, particularly tarantulas, which are native to North America and can live for up to twenty-five years, Most people, on the other hand, do not like touching spiders, and a significant number of people are afraid of them, mainly because of their poison. However, despite their bad reputation, only thirty of the 37,000 known species of spiders are deadly to humans. Spiders actually provide benefits to humans, by catching and eating harmful insects such as flies and mosquitoes. - - The main thing that makes spiders different from other animals is that they spin web's to catch the small insects they feed on. The unique silk of a spider's web is produced by special organs found spider web is five times in the lower part of the spider's body. It is light, elastic, and strong stronger than steel. Additionally, it is completely biodegradable. This means that the web will making it perfect for uses completely decompose¹ and eventually return to nature over time such as making fishing nets. Some people have tried to raise spiders commercially in order to collect the silk these spiders produce, but no one has ever really managed to make a go of it. One reason why these businesses never stand a chance is because it takes 670,000 spiders to produce half a kilogram of silk, and all of these spiders need living insects for their food. In addition, spiders are usually solitary² animals, and need to be kept alone. Researchers at an American company working together with two U.S. universities may have found a solution to making artificial spider web. Using genetically modified silkworms,³ the company hopes that in the long run it will be able to make large quantities of very light, very strong fiber for medical as well as other uses. Additionally, because the manufacture of the artificial web is from living silkworms, the industry potentially would be non-polluting and less harmful to the environment

回答募集中 回答数: 0