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 ✅).

教えてほしいです

13 次の日本文にあう英文になるように, □(1) 眠気を感じて, 私は草の上に横になった。 に適する語を書きなさい。 □(2) ロンドンで生まれたので、彼女は英語を流暢に話す。 sleepy, I laid myself on the grass. in London, she speaks English fluently. □(3) 彼女は泳げないので、海が好きではない。 to swim, she doesn't like the sea. ike the sea □ (4) 昨年に比べて,今年は冬の訪れが早い。 Winter has come early I love with last year. □(5) 率直に言えば、私は彼の歌はあまり好きではない。 Frankly door food. other food, I don't like his songs very much. it was totally con 4 次の文中の下線部を ( 内の指示にしたがって書きかえなさい。 □(1) As I have never read the book, I can't criticize it. (分詞構文に) Oine plane □ (2) Properly trained, dogs can do some tricks. (副詞節に) thers, I can't criticize it. dogs can do some tricks. □ (3) Since their conversation was in Chinese, I didn't understand a word. (分詞構文に) を用いて) □ (4) Lost in thought, he didn't notice us. (副詞節に) I didn't understand a word. □ (5) She knelt on the floor, and she closed her eyes. (with を用いて) She knelt on the floor, □ (6) Printed in haste, the book had many misprints. (副詞節に) he didn't notice us. the book had many misprints.

教えてほしいです

① 次の各組の英文がほぼ同じ内容を表すように,に適する語を書きなさい。 (1) Looking at his alarm clock, he turned pale. he at his alarm clock, he turned pale. (2) As I hadn't heard from him, I decided to visit him. (3) If other things are equal, I will take this job. Other things ding heard from him, I decided to visit him. I will take this job. arrived at Naha around noon. reading, he listened to music. (4) Our plane left Haneda at ten, arriving at Naha around noon. Our plane left Haneda at ten, (5) Since he was tired of reading, he listened to music. ever h (6) My father sat on the sofa, and his legs were crossed. My father sat on the sofa, his legs (7) After the sun had set, the boys started for home. The sun the boys started for home. (8) If we speak generally, Japan has a mild climate. Japan has a mild climate. (9) Her mother having left her room, she began to cry bitterly. her mother (10) Unable to swim well, he doesn't like the sea very much. he left her room, she began to cry bitterly. swim well, he doesn't like the sea very much. (11) Turning to the left, you'll find a tall building on your right. you do/nothing) to the left, you'll find a tall building on your right.

教えてほしいです (画像横になっているかもしれません)

・次の文を分詞構文に書きかえなさい。 また, 英文全体を日本語に直しなさい。 (1) As I had never eaten that fish, I didn't want to eat it. ( (2) He doesn't know how to play the game, so he won't join us. ( (3) She has never been to Europe. Therefore she really wants to visit there. (4) Because nobody came to our house, we had a lot of free time. ike tomor (5) As we haven't done the work yet, we can't go skiing. )

訳について質問なのですが、 「by 場所」を「〈場所〉から」と訳せるとどれだけbyについて調べても書いていなかったのですが、こう訳すのは自然なことなのでしょうか？？

1. Less than 40 percent of all physicians in Wichita are employed as full-time staff by the hospitals where ------- practice. (A) they (B) their (C) them (D) themselves bisq sd bloode JS Virom sted 88W) or (8) CEO of Ad S NE

教えてほしいです

2 次の文の( )内の語を適する形になおしなさい。 (1) My father repaired the (break) wall. (2) She came to school without (have) breakfast. (3) The boy looks (please) with magic tricks. (4) Do you know the woman (wait) for a bus? (5) The letter (write) by Meg has just arrived. (6) I couldn't make myself (understand) in Spanish. (2点×6)

教えてほしいです

次の各文の( )に最も適するものを選び、記号を○で囲みなさい。 (1) Ben has a lot of things ( ) about. ア talk 1 talked talking I to talk (2)( ) near the park, they often take a walk there. ア To live 1 Living ウ Lived I Live (3) Do you mind ( ア to close ) the door for me? 1 to closing ウ close I closing (4) I heard my name ( ア called ) somewhere. 1 calling ウ to call I call (5) Yuki and I went ( ) at the mall last week. 7 to shopping 1 shopped ウ shopping I shop (6) I don't feel ( ) playing soccer today. ア for イ like I on to

➄がわからないので教えてください。 主な問題文は4枚目、答えは5枚目です。

4MさんとNさんは、化学変化と質量の変化を調べる実験を行いました。 問1 ~ 問6に答えなさい。(19点) 実験 1 (1) 図1のように銅の粉末をステンレス皿に広げてのせ, かき混ぜなステンレス皿II がらガスバーナーで十分に加熱し酸化銅にした。 このとき,加熱す る前の銅の質量と加熱してできる酸化銅の質量を調べた。 (2)次の表は、その結果をまとめたものである。 銅の粉末 ガス 図 1 表 銅の質量[g] 酸化銅の質量 〔g] 0.20 0.40 0.60 0.80 0.25 0.50 0.75 1.00 1.25 1.00 Mさん 銅の質量に比べてできた酸化銅の質量が大きいのは、酸化銅が、 銅と空気中の酸素が結 びついてできたからですね。 Sk ス Nさん「化学変化の前後で、その反応に関係している物質全体の質量は変わらない」という① 質量 保存の法則は、この実験でも成り立っていると考えられるから、銅の質量とできた酸化銅の 質量の差は、銅に結びついた酸素の質量ということになりますね。 Mさん そうすると, 銅の質量と銅に結びついた酸素の質量とは比例の関係にあり、反応する銅の 質量と酸素の質量の比は、つねに41になっていることがわかります。 問1 実験1でできた酸化銅の色として最も適切なものを、次のア~エの中 書きなさい。 (2点) (2) (3

（3）の解説の赤線引いたところの分母ってなぜそうなるんですか？🙇‍♂️

= 〔I〕 正三角形A,B,C。において,A,B,=p, AoCo = g とおく。辺ABo Back CAを2:1に内分する点をそれぞれ A, B, C, とする。さらに△ABCに ついて,辺 A, B, B, C, CA, を2:1に内分する点をそれぞれ A2, B2, C2と する。この操作を回繰り返したときに得られる点を An, B, Cn とするとき, 次の問いに答えよ。 . (1) A,Bをとで表し, 内積 A Bo A,BI の値を求めよ。 (2) 自然数に対して, Ax+1Bkt1=-1/3A-1Bk-1 が成立することを示せ。 (3) Ao Aznをとんで表せ。 AR-1613 Ck Ak/ Ck-1 Bk-1 Bk

至急どの選択肢が正解なのか教えてください

stranoP (12) A shopping mall stands (1. be 2. rice fields 3. to 4. used 5. where).