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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

③に適する英語を書きなさいという問題で、 答えはBecauseなのですが、 「Becauseを文頭に置く場合は前に情報がないといけない」と言う部分が引っかかっています 解説お願いします🙇🏻‍♀️⸒⸒

通りです。 en you take pictures. afe One la [2] 次の英文は千葉県に住んでいる中学生の恵子が, 長野県に住む祖母の家にでかけたときの 体験をまとめ、 夏休み明けの英語の授業で発表したものです。 この文章を読んで問いに答え なさい。 I have a grandmother in Nagano. She lives alone in a small village. I have been But this summer was ( ①) to her house with my family in summer many times. because only my brother, Akira, and I went there by train. saw our grandmother at the station. We felt relieved when we One Our We stayed at her house for two weeks and we had a lot of fun. In the morning, we often went to catch insects after we helped our grandmother with the housework. We tried to catch fish in the river in the afternoon, but we didn't catch many. day, we went to a mountain near her house and we found some mushrooms. grandmother said, "These will be good for dinner." "Can we really eat those?" I asked. She answered, "Yes." And later, we cooked them together. sometimes told us some interesting old stories of Nagano. We enjoyed ( At night, she ② ) to her. Two weeks was very short. ( ③ ) we were having a very good time, we wanted to stay longer. But the end of August came, and we had to come back to Chiba for school. On the morning of the last day, we got up at 5:00 to make breakfast for our grandmother with the mushrooms. When she came into the kitchen, she said, "This looks very good. Thank you so much." We said to her, "We have been really happy with you. Can we come in winter next time?" You She answered, “Sure. can ski here in winter." When we eat mushrooms, we Akira and I still talk about the summer in Nagano. think about our grandmother. Now, everyone, look at these mushrooms. This evening, we're going to cook them for our parents. If you like, I'll tell you how. Why don't you try at your home? (注) grandmother alone ひとりで village : * felt relieved : ほっとした catch : 捕まえる insect: housework F : parents: mushroom : きのこ end : 終わり kitchen: <-4-

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