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 ✅).

内容的には間違ってないか。文法は合っているか。の2点で英文を見てもらいたいです。全部で5文で、対話の穴埋め問題です。 ⤵︎ ⤵︎私が描きたかったことです。 1、電気を変えるのを手伝って欲しい 2、あなたの誕生日は2月25じゃなかった？(2月のスペルが間違ってます🙇‍♂️)... 続きを読む

II. 以下に指示された二人の対話を完成させるのに, 最もふさわしいと考えられる 英文を6語以上で書きなさい。 1) A: I'm thinking about changing the design of my bedroom. B: What were you thinking of doing? A: ( ) B: That will really brighten the atmosphere of the room. Let me know if you need a hand. : 2) A Hi, George. Happy birthday! B: Huh? What do you mean? It's not my birthday today. A: ( ) B: No, it's the 25th of March. But, that's okay. You can say it to me again next month. 3) A Did you hear that Tracey and Belinda decided to get married? B Yes, Belinda called me last night. It's wonderful news. We need to think about a present. A: ( ) B: That's a great idea; they both love entertaining at home. 4) A Why were you late this morning? B Well, there was no room to leave my bicycle at the station. A Really? Were all the spaces taken? B: Yes. I think people should be able to leave their bicycles anywhere. A: ( ) 5) A Don't you think John did really well in the debate contest? B: Yes, I was surprised. He is usually quite shy.

代名詞の使い方を解説してほしいです。

1・2年 129 単元別入試対策 ●基本例文 ● 1 She told me an interesting story. ③ That is her pen and this is mine. 彼女は私におもしろい話をしてくれました。 あれは彼女のペンで,これは私のものです。 I lost my pen, so I have to buy a new one. 私はペンをなくしたので、 新しいものを買わなければなりません。 1人称代名詞所有代名詞・再帰代名詞 単数 複数 ~を/に〜のもの 再帰代名詞 ~自身 主格 所有格 ~は/が 〜の ~を/に 〜のもの 目的格所有代名詞 再帰代名詞 ~自身 主格 所有格 目的格所有代名詞 ~は/が 〜の 1人称 I my me mine 2人称 you your you yours he his him his myself yourself you himself we our us ours ourselves your you yours yourselves 3人称 she her her hers it its it herself they itself their theirs them themselves adol 人称代名詞は,主格所有格・目的格の3つの形があり、人称と数(単数・複数)によって変化する。所有代名 詞は「~のもの」という意味で1語で〈所有格名詞> を表す。 再帰代名詞は「~自身」という意味を表す。 例 She told me an interesting story. (彼女は私におもしろい話をしてくれました。)→1989 2 指示代名詞 this that these those 単数形 this (これは,これ,この) 近くのもの人を指す 遠くのもの人を指す that (あれはあれ、あの) 複数形 blis these (これらは,これら,これらの) those (あれらは, あれら, あれらの ) 例 That is her pen and this is mine (= my pen). (あれは彼女のペンで,これは私のもの(=私のペン)です。)→② til vehave 101 190g nist 3 不定代名詞:some・any (every ・no) 不特定のものを漠然と指す すべてのものを指す some/any (いくつか、いくらか) (肯・疑), (何も~ない) [杏] every (すべての〜 存在しないことを表す *no (少しも 全く~ない) something/anything (何か) 〔昔・疑), (何も~ない) [杏] everything (すべてのもの) nothing (何も~ない) someone/anyone (だれか) 〔青・疑〕, (だれも~ない) [杏] everyone (だれでも,みな) no one [nobody] ( だれも~ない) 不特定の人やものおよび一定でない数量を表す代名詞を不定代名詞という。 表中で some および any は,単数・ 複数両方扱い。 その他の語はふつう単数扱いである。 ※ every, no は代名詞ではなく形容詞。 肯定文中の some, something, someone は, 否定文 疑問文中ではふつう any, anything, anyone になる 4 it と不定代名詞 one it 前に出た名詞と同一の I lost my watch, so I'm looking for it now. ものを指す ( (私は時計をなくしたので, それ(=私の時計) を今, 探しているところです。) 前に出た名詞と同じ種類の I lost my pen, so I have to buy a new one. one 不特定のもの(人) を指す (私はペンをなくしたので、 新しいもの (=ペン) を買わなければなりません。) 不定代名詞の one (もの, 人)は、 前に1度出た名詞がくり返されるとき その名詞の代わりに用いる。 one, it ともに単数 の名詞を受ける。 複数の名詞を受ける場合はそれぞれ ones, they を用いる。 5 その他の不定代名詞 each (それぞれ) 単数扱い Each of us has an idea. (私たちのそれぞれが考えを持っています。) | another (もう1人 [1つ] 別の人 [もの]) This is another of his cars. (これは彼のもう1台の車です。) others (他人 [他のもの] 複数扱い both (両方(とも)) Some are kind and others are not. (親切な人もいればそうでない人もいる。) 単数・複数 両方扱い several (何人か [いくつか] all (みな [すべて]) |Both of them are kind. (彼らは両方とも親切です。) | Several of them were absent. (彼らの何人かは欠席でした。) All of the boys are from Tokyo. (その少年たちはみな東京出身です。) All of the money is hers. (そのお金はすべて彼女のものです。)

英検2級のWritingです。 採点お願いします。

◆以下のTOPICについて、 あなたの意見とその理由を2つ書きなさい。 ●POINTSは理由を書く際の参考となる観点を示したものです。 ただし, これら以外の観点から理由 を書いてもかまいません。 語数の目安は80語~100語です。 freezes. W oin vd of jo TOPIC pank Pish take acob woll I What happ doed so an if S ewolle i & particles of These days, many university students have part-time jobs. Do you think it is a good idea for students to work while attending university? POINTS Fee ●Time • Experience 北

英検2級のWritingです。 採点(添削)お願いします🙇‍♂️🙏

以下のTOPICについて, あなたの意見とその理由を2つ書きなさい。 ●POINTSは理由を書く際の参考となる観点を示したものです。 ただし, これら以外の観点から理由 を書いてもかまいません。 bib vil 語数の目安は80語~100語です。 bool yari aliriw slime just by li OT $ TOPIC of 8 The vol OTA dless of Today, some kindergartens have started teaching English to their children. Do you think it is a good idea to teach English to children in kindergartens? POINTS Career ●Japanese language Effectiveness IT e rtion syd ti essio the speaker yd boau sd. I

②の最後の文　just create 〜　environment that のthatは関係代名詞の主格用法？ですか？

5 1 1 百倍へ How to Avoid Being Late 急用を思い出して相 You never want to be late for class. But once in a while, 249 words you stay up late at night, losing yourself in a TV show or video game. Or maybe you simply find that you are incapable of waking up early. You to be on time, but you just can't. You're at your wit's end, and decide that you're just not cut out for mornings. you want may ② Don't worry! There are solutions. Experts say that going to sleep earlier is the first step. Getting more sleep at night will makea 遅刻しないようにす ① 誰だって自ら望ん かしをして テレビ番組 に早起きができない い でもどうしてもそ contribution to earlier and more productive mornings. Some people turn to sleeping pills, but these should be a last resort. Just create a 10 bedroom environment that is suitable for a good night's sleep. ③ Another good idea is to develop a new routine for the mornings. If you have some delicious tea or breakfast treats to enjoy in the course of getting ready for school, you'll look forward to starting your day. And if you feel like you're always on the go, set aside a few minutes to breathe 15 quietly and relax. ④ What if you're still late? If the class is already in progress when you get there, after class, get in contact with a friend and find out what you missed in the class. And even if you fail at first, don't lose sight of your goal. Just keep trying, and you'll become a more は朝型に向いていない ② 大丈夫! 解決策 夜の睡眠時間を のにするのに役立つ すべきだ。 とにかく う。 ③ 朝に新しい習 ている間に、おいし 1日を始めるのが うなら.2.3分時 う。 ○ それでもや もう授業が進行 聞き逃したこと

読み方をカタガナで教えて欲しいです。

? What animals are in danger now? Meg: Have you ever heard of the IUCN Red List? Kaito: No, I've never heard of it. What is it? Meg: The Red List gives us information about endangered animals, birds, plants, and so on. Kaito: What animals are on the list? Meg: Pandas, cheetahs, and gorillas, for example. Kaito: Really? I didn't know that. Meg: Some animals and birds in Japan are also on it. I want everyone to know that. Kaito: Why don't we write an article for our class newspaper? Meg: That's a good idea. [79 words] 5

英文の添削お願いします🙏🏻 英検のエッセイ練習で書きました。

Topic Agree or disagree: People will stop subscribing to printed newspapers in the future. I agree with the idea that people will stop subscribing to printed newspapers in the future. I have two opinions why I think so. First of all, to print newspapers uses a lot of papers. They are made by wood that is important nature material. The kind of nature things has limit to use, so we should not use them too much. The plants also play role in the keeping air clean. This is the reason why I think it is bad for environment. Secondly, the printed newspapers are too big to carry. Thesedays we can get it online and we cand read it all the time wherever you are. For example on the train, in office, in café and so on. For these reasons, I think people will stop subscribing printed newspapers.

教えて欲しいです。単元は動詞の語法です

1. When I realized I was wrong, I apologized ( Dhim my mistake 3 to him my mistake him for my mistake to him for my mistake 2. I graduated ( ) high school in 2015.m 1 from duo@for 3. We ( ) the problem of poverty. ①discussed 3 discussed of インター bist 4. Do you think I ( ) my elder sister? Dresemble resemble with (立正大) 4 of ③at de (酪農学園大) discussed about the discussed with して楽しめる b biabidea (東洋大) 3 resemble for b①am resembling 5. Kate () her baby boy on the bed after dinner. bid oa ed or blu ①laid lain give/tal 3lay 9131 (京都産業大) lie blow- em evig bnim LOV (広島修道大) ①raise ②rise wwo no alim to no 6. Please ( ) your hand if you want to ask a question. to baim w prey set up hard ratake up Sunda tegral 'nob 98819 7. Because of the bad weather, I decided ( ) out. ①not to go vud king/from home last night, to not going ③not going ④going not yud o (清泉女子大) guinisque beringsr 8. The repair shop failed ( ) my car. (東北芸術工科大) ①to fix 2 to affix ③fixed ④fix Ieva un baimer) Botong sandT 9. There was a big fire yesterday. Fortunately, everybody managed (). 011 (亜細亜大) Descape escaped Bevig 3 would escape aDob to escape 10. I closed my eyes and pretended ( ) asleep. o be Last wee toob edT 880 of Obe evig(玉川大) being 3 of being to be toy mailliW bus arrotiv 280 (to) during the exam.

教えてください

A. Complete the paragraph with words from the box. One word is used twice. beat energy range seems steady Much of the research done on music and exercise focuses on what the ideal BPM is for running. But it also 2 studies that having a predictable, 3 music with sudden changes in the 4- from recent rhythm is important. Generally, like free-form jazz, is less suitable. Why? If the music speed changes while we run, we tend to adjust our movements to match the 5 Each time we change our steps to move faster or slower, we lose 6