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数学 大学生・専門学校生・社会人

多様体の接空間に関する基底定理の証明です。g(q)=∫〜と定義した関数を微積分学の基本定理を用いながら変形してg(q)=g(0)+∑gᵢuⁱと導出するのですが、これがうまくいきません。 自分は、g(q)の式をまず両辺tで微分して、次に両辺uⁱで積分して、最後に両辺tで積分... 続きを読む

12. Theorem.If{ = (x', , x") is a coordinate system in M at p, then its coordinate vectors d, lp, …… 0,l, forma basis for the tangent space T,(M); and D= E(x) 。 i=1 for all ve T(M). Proof. By the preceding remarks we can work solely on the coordinate neighborhood of G. Since u(c) = Othere is no loss of generality in assuming ど(p) = 0eR". Shrinking W if necessary gives E(W) = {qe R":|q| < } for some 8. Ifg is a smooth function on E(W) then for each 1 <isndefine og (tq) dt du g(9) = for all qe {(W). It follows using the fundamental theorem of calculus that g= g(0) + E&,u' on (W). Thus if fe &(M), setting g = f。' yields f= f(P) + Ex on U. Applying d/ax' gives f(p) = (f /0x)(P). Thus applying the tangent vector e to the formula gives (f) = 0+ E(x'(p) + E Ap)u(x) = E(Px). ず ax Since this holds for all f e &(M), the tangent vectors v and Z Ux') d,l, are equal. It remains to show that the coordinate vectors are linearly independent. But if ) a, o.l, = 0, then application to x' yields dxi 0=24 (P) = 2q d」= 4. In particular the (vector space) dimension of T,(M) is the same as the dimension of M.

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英語 高校生

見辛かったらすみませんが、問2の英作文の添削をしていただきたいです🤲

2015年度 英語 13 They should not have to lose their livelihoods. Secondlv. although I agree 次の英文は、英語のディベートの授業での対話です。この対話を読み,下の問い に答えなさい。 with Keiko that climate change is a serious problem, I think air travel is a much less damaging source of carbon pollution than, for example, burnimg MR. ROBERTS: Good morning, everybody. Today we have the first of our Coal. We should address these more serious causes first. Finally, the most student debates. The topic this morning is air travel, and the argument is. 法 important reason is the issue of freedom. People in a democracy expect the “Domestic air travel in Japan should be discouraged." Keiko will first argue right to travel as they wish. It has made our lives more convenient and in favor of the proposal and Junko will speak against. Over to you, Keiko. created more opportunities for everyone. We should not reduce our quality of life. Thank you for listening. KEIKO: Good morning, everybody. I am going to speak today in favor of the proposal. Limitless air travel inside Japan is a luxury that we can no longer MR. ROBERTS: Thank you to our two speakers. Now, are there any afford. The main reason is that it is extremely damaging to the questions from the audience? environment. The carbon gases produced by aircraft are very high and are a 頭人 proven cause of global warming. Therefore, we should try and reduce air 問1 次の(1)~(5)の文について, 本文の内容と一致するものには○.一致しないも travel as much as possible to protect the environment, not only for Japan but のには×を,それぞれ解答欄に記入しなさい。 also for the whole world. Secondly, air travel is actually unnecessary within the four islands of Japan. Now, while I appreciate that travel to Okinawa is (1) Keiko's main reason for opposing domestic air travel is the damage it a different case, there are efficient train services throughout Japan that are causes to the environment. h fast and environmentally friendly. These make an excellent alternative to air (2) Keiko thinks that air travel is cheap and efficient. flights, so there is really no need to use airplanes in most cases. Finally, (3) Keiko recommends that people travel between Okinawa and mainland there is an economic reason. Air travel is expensive for passengers because Japan by ship. the price of fuel keeps increasing, but airline companies are still struggling (4) Junko thinks that climate change is not an important problem. to make profits. Moreover, regional airports often cannot attract enough (5) Junko thinks that a lot of jobs will be lost if air travel is reduced. customers, and, overall, the industry is not cost-effective, It's time to admit that air travel is too damaging and expensive to be sustainable. Thank you. 問2 次の質問に英語で答えなさい。 Which of the two speakers do you more agree with? Give at least one JUNKO: Thank you, Keiko. I am going to disagree with the proposal for three reason for your answer. (解答欄:13.8cm×4行) reasons. Firstly, I think there is a strong economic case to support flying. Many jobs depend on air travel, not just the pilots and cabin staff, but also people who work in airports or for airlines, or who rely on fast transport.

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英語 高校生

見辛かったらすみませんが、問2の英作文の添削をしていただきたいです🤲

2015年度 英語 13 They should not have to lose their livelihoods. Secondlv. although I agree 次の英文は、英語のディベートの授業での対話です。この対話を読み,下の問い に答えなさい。 with Keiko that climate change is a serious problem, I think air travel is a much less damaging source of carbon pollution than, for example, burnimg MR. ROBERTS: Good morning, everybody. Today we have the first of our Coal. We should address these more serious causes first. Finally, the most student debates. The topic this morning is air travel, and the argument is. 法 important reason is the issue of freedom. People in a democracy expect the “Domestic air travel in Japan should be discouraged." Keiko will first argue right to travel as they wish. It has made our lives more convenient and in favor of the proposal and Junko will speak against. Over to you, Keiko. created more opportunities for everyone. We should not reduce our quality of life. Thank you for listening. KEIKO: Good morning, everybody. I am going to speak today in favor of the proposal. Limitless air travel inside Japan is a luxury that we can no longer MR. ROBERTS: Thank you to our two speakers. Now, are there any afford. The main reason is that it is extremely damaging to the questions from the audience? environment. The carbon gases produced by aircraft are very high and are a 頭人 proven cause of global warming. Therefore, we should try and reduce air 問1 次の(1)~(5)の文について, 本文の内容と一致するものには○.一致しないも travel as much as possible to protect the environment, not only for Japan but のには×を,それぞれ解答欄に記入しなさい。 also for the whole world. Secondly, air travel is actually unnecessary within the four islands of Japan. Now, while I appreciate that travel to Okinawa is (1) Keiko's main reason for opposing domestic air travel is the damage it a different case, there are efficient train services throughout Japan that are causes to the environment. h fast and environmentally friendly. These make an excellent alternative to air (2) Keiko thinks that air travel is cheap and efficient. flights, so there is really no need to use airplanes in most cases. Finally, (3) Keiko recommends that people travel between Okinawa and mainland there is an economic reason. Air travel is expensive for passengers because Japan by ship. the price of fuel keeps increasing, but airline companies are still struggling (4) Junko thinks that climate change is not an important problem. to make profits. Moreover, regional airports often cannot attract enough (5) Junko thinks that a lot of jobs will be lost if air travel is reduced. customers, and, overall, the industry is not cost-effective, It's time to admit that air travel is too damaging and expensive to be sustainable. Thank you. 問2 次の質問に英語で答えなさい。 Which of the two speakers do you more agree with? Give at least one JUNKO: Thank you, Keiko. I am going to disagree with the proposal for three reason for your answer. (解答欄:13.8cm×4行) reasons. Firstly, I think there is a strong economic case to support flying. Many jobs depend on air travel, not just the pilots and cabin staff, but also people who work in airports or for airlines, or who rely on fast transport.

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物理 高校生

物体が壁に衝突したり、物体が衝突しあった時、衝突面に垂直でない速度ベクトルの大きさは変わらないので、鉛直方向に上がる時間と下る時間は変わらないのですか? (5)番です

27 (1) Aを原点として鉛直上向きにy軸をとる。落下するのは y=0 のとき だから,求める時間をもとして公式②を用いると も=22こうすればいづれ っかる (2) 鉛直方向の初速度を同じにする必要がある(するとAとBはいつも同じ高 0=ut, +;(-g)t? 1 g さにいる)。そこで ひ . sin a == V Vsin a =u (3) 最高点に達するまでの時間をむとすると, 公式①より t。= もとして ひ 0=u+(-g)t。 : =2 t2= g 求めると早い この間にBは右へ1の距離を動けばよいので Vu COS a = g Vu 1- 2 U Vu ひ 2 1= (Vcos a)t。=- V1- sin' a g ア V- 三 2 9 V (4) 求める水平成分を v.とする。水平方向での運 衝突直前 MV cos a = (M + m)u。 M, m 動量保存則より 止 Vcosa MV ひ,ミ M+m M M+m V-U' らで い。 COS a = Vx 直後 M+m O 0 の避 鈴直成分は A, B共に衝突前が0なので なので た 水平方向は外力がないので運動量保存は厳密に成りたつ。一方, 鉛直方向は重力が かかっているが,瞬間的な衝突では、(重力の力積が無視できるため)近似的に適用し てよい。問題文にとくに断りがなければ,瞬間衝突と思ってよい。 (5)初速ひ。での水平投射に入る。落下時間はtななので (鈴直方向によがる時間 と下りる時間は等しい) Mu (M+m)g V-u x= Uxt2=-

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