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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

(1)の答えがchoosing、(2)の答えがウ、(5)の答えがアで、それぞれなんでその答えになるのかと、 5⃣の本文を上から4行、4行、5行、4行、3行、2行で分けた時それぞれに題名を付けるとしたらどうなりますか? 教えて欲しいですm(_ _)m

5 次の英文を読んで, あとの問いに答えなさい。 <川越東改> Origami is the Japanese art of folding paper. To do origami, the artist starts with a square piece of paper. Some people like to use special origami paper that is two colors. The front of the paper is one color, and the back of the paper is another color. Other people like to use origami paper that has patterns on it. After ①(choose) your paper, you can find many instructions for folding the paper into different things. Many people enjoy (make) origami flowers, animals, or things ( 3 ). For all of these things, there are a few kinds of folds that you need to use. For example, sometimes you need to fold the paper in half. Sometimes, the paper must be folded from corner to corner. If you follow the directions carefully, you can create a beautiful paper flower or animal. However, origami is more than folding paper. First, origami is an important part of Japanese life. For example, nature is important in Japan. In Japan, people care about the seasons, weather, water, or other things in nature. Origami is also a part of nature. That is why the most popular origami shapes are things like animals. Birds, fish, flowers, and stars are all popular shapes. It is a quiet activity, and can calm the mind and body. People who do origami like the activity as much as the art. They like it because origami demands a lot of attention. When people think hard about creating something, they forget about their problems. This allows them ④ to calm down. ⑤ Origami is also good for teaching children. They also learn to work carefully. Also, origami has squares and triangles. These shapes are important in all kinds of learning. Origami helps children to learn about these shapes. Maybe you can try to do origami yourself. You only need some paper and a book of instructions. You can find instructions for many origami shapes on the Internet. instruction direction (1)①,②の( )内の語を適する形にかえなさい。 (2) 30( に適するものを, ア~エから1つ選びなさい。 (3) (4) 7 to paint with 1 to talk with 1 (2) making. to play with I to help with (イ) ④に適するものを,ア~エから1つ選びなさい。 Origami is also easy to learn. 1 Origami is also good for your imagination. Origami is also difficult to learn. I Origami is also good for your mind. ⑤にはA~Cの文が入ります。 自然な流れの文章になる配列を, ア~エから1つ選びなさい。 A Children must follow these steps exactly. B First, origami has many steps. C This way, children can learn to follow instructions. ア A-B-C イ A-C-B B-A-C I B-C-A (5)本文の内容にあうものを, ア~エから1つ選びなさい。 If you follow some instructions for paper folds, you can enjoy many different origami shapes. Learning origami gives us a good chance to help animals on the earth. You may feel tired if you try hard to do origami carefully. I The most important thing for children's education is origami. (土)

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