В.Ф. Бутузов - Лекции по математическому анализу, страница 12

Íàéäåì d y:20d20 (sin x) = (sin x)(20) · (dx)(20) = sin(x + 20 · π/2)(dx)20 == sin x · (dx)20 .2. Ïóñòü òåïåðü x ôóíêöèÿ íåêîòîðîé íåçàâèñèìîé ïåðåìåííîé t: x = ϕ(t).  ýòîì ñëó÷àådx = ϕ0 (t)dt, dy = f 0 (ϕ(t)) · ϕ0 (t)dt,è äàëåå ïîëó÷àåì:00000d y = d(dy) = dt · d f (ϕ(t)) · ϕ (t) = dt · f (ϕ(t)) · ϕ (t) dt =2= f (ϕ(t)) · (ϕ (t)) + f (ϕ(t)) · ϕ (t) dt2 =0002000= f 00 (ϕ(t)) · (ϕ0 (t)dt)2 + f 0 (ϕ(t)) · ϕ00 (t)dt2 = f 00 (x)(dx)2 + f 0 (x)d2 x.Èòàê,d2 y = f 00 (x)(dx)2 + f 0 (x)d2 x.Òàêèì îáðàçîì, ôîðìà âòîðîãî äèôôåðåíöèàëà íå èíâàðèàíòíà. Ýòî æå îòíîñèòñÿ ê äèôôåðåíöèàëàì áîëåå âûñîêîãî ïîðÿäêà.Ÿ 10.

Ïðîèçâîäíûå âåêòîð-ôóíêöèèÅñëè êàæäîìó ÷èñëó t èç ìíîæåñòâà T ïîñòàâëåí â ñîîòâåòñòâèå íåêîòîðûé âåêòîð ~r, òî ãîâîðÿò, ÷òî íà ìíîæåñòâå T çàäàíàâåêòîðíàÿ ôóíêöèÿ (èëè âåêòîð-ôóíêöèÿ) ~r = ~r(t).Ìîäóëü âåêòîðà ~r(t) áóäåì îáîçíà÷àòü, êàê îáû÷íî, |~r(t)|.Îòìåòèì, ÷òî |~r(t)| ñêàëÿðíàÿ ôóíêöèÿ àðãóìåíòà t.Îïðåäåëåíèå. Âåêòîð ~a íàçûâàåòñÿ ïðåäåëîì âåêòîðôóíêöèè ~r(t) ïðè t → t , åñëè0lim |~r(t) − ~a| = 0.t→t0Ãë. 4. Ïðîèçâîäíûå è äèôôåðåíöèàëû.76Îáîçíà÷åíèå:lim ~r(t) = ~a èëè ~r(t) → ~a ïðèt → t0 .t→t0Çàôèêñèðóåì çíà÷åíèå àðãóìåíòà t è äàäèì åìó ïðèðàùåíèå∆t 6= 0. Âåêòîð-ôóíêöèÿ ïîëó÷èò ïðèðàùåíèå∆~r = ~r(t + ∆t) − ~r(t).Îïðåäåëåíèå. Åñëè ñóùåñòâóåòlim∆t→0∆~r,∆tòî îí íàçûâàåòñÿ ïðîèçâîäíîé âåêòîð-ôóíêöèè ~r(t) â òî÷êå t.Îáîçíà÷åíèå: r~0 (t) èëè d~r/dt.Çàäàäèì ïðÿìîóãîëüíóþ ñèñòåìó êîîðäèíàò Oxyz è ââåäåìáàçèñ {~i, ~j , ~k}.

Ðàçëîæèì âåêòîð ~r(t) ïî ýòîìó áàçèñó:~r(t) = x(t)~i + y(t)~j + z(t)~k = {x(t), y(t), z(t)};q|~r(t)| = x2 (t) + y 2 (t) + z 2 (t) .Óòâåðæäåíèå. Äëÿ òîãî, ÷òîáû ~r(t) → ~a = {a , a , a } ïðè t → t ,123íåîáõîäèìî è äîñòàòî÷íî, ÷òîáûx(t) → a , y(t) → a , z(t) → a ïðè t → t .Äîêàçàòåëüñòâî íåñëîæíî ïðîâåñòè, èñïîëüçóÿ ðàâåíñòâî12300q|~r(t) − ~a| = (x(t) − a1 )2 + (y(t) − a2 )2 + (z(t) − a3 )2 .Èç ýòîãî óòâåðæäåíèÿ, â ÷àñòíîñòè, ñëåäóåò, ÷òîd~r= x0 (t)~i + y 0 (t)~j + z 0 (t)~k = {x0 (t), y 0 (t), z 0 (t)},dtò.å. âû÷èñëåíèå ïðîèçâîäíîé âåêòîð-ôóíêöèè ñâîäèòñÿ ê âû÷èñëåíèþ ïðîèçâîäíûõ åå êîîðäèíàò.Îïðåäåëåíèå. Ìíîæåñòâî êîíöîâ âñåõ âåêòîðîâ ~r(t) (t ∈ T ),îòëîæåííûõ îò íà÷àëà êîîðäèíàò (òî÷êè Î), íàçûâàåòñÿ ãîäîãðàôîì âåêòîð-ôóíêöèè ~r = ~r(t) (ðèñ. 4.8).Ôèçè÷åñêèé ñìûñë ãîäîãðàôà ýòî òðàåêòîðèÿ −òî÷êè, äâè−→æåíèå êîòîðîé â ïðîñòðàíñòâå çàäàíî óðàâíåíèåì ~r = OM = ~r(t).Ôèçè÷åñêèé ñìûñë ïðîèçâîäíîé d~r/dt ýòî ñêîðîñòü òî÷êè.

Ìîæíî äîêàçàòü, ÷òî âåêòîð d~r/dt ÿâëÿåòñÿ êàñàòåëüíûì êãîäîãðàôó.10. Ïðîèçâîäíûå âåêòîð-ôóíêöèè77zM (x(t ), y (t ), z (t ) )rr (t )rkr Oirr ¢(t )rjyxÐèñ. 4.8.Ïðàâèëà äèôôåðåíöèðîâàíèÿ âåêòîð-ôóíêöèé1)2)3)0~r1 (t) ± ~r2 (t) = r~0 1 (t) ± r~0 2 (t);0f (t) · ~r(t) = f 0 (t) · ~r(t) + f (t) · r~0 (t);0~r1 (t) · ~r2 (t) = r~0 1 (t) · ~r2 (t) + ~r1 (t) · r~0 2 (t);0 ~r1 (t) × ~r2 (t) = r~0 1 (t) × ~r2 (t) + ~r1 (t) × r~0 2 (t) .4)Çäåñü ~r · ~r ñêàëÿðíîå ïðîèçâåäåíèå, à [~r × ~r ] âåêòîðíîåïðîèçâåäåíèå âåêòîðîâ ~r è ~r .Ýòè ïðàâèëà íåòðóäíî îáîñíîâàòü, èñïîëüçóÿ âûðàæåíèÿ äëÿ ~r ±± ~r , ..., [~r × ~r ] â êîîðäèíàòàõ.Ïðîèçâîäíûå âûñøèõ ïîðÿäêîâ âåêòîð-ôóíêöèè ââîäÿòñÿ,êàê è äëÿ ñêàëÿðíîé ôóíêöèè:1121221212d2~rd=2dtdtd~rdt, ...

, ~r(n)0(n−1)(t) = ~r(t) . êîîðäèíàòàõ:d2~r= {x00 (t), y 00 (t), z 00 (t)}.2dtÏðèìåð 1. Ðàññìîòðèì ïðÿìóþ (îñü âðàùåíèÿ) è âåêòîð ~a ñíà÷àëîì íà ýòîé ïðÿìîé, ñîñòàâëÿþùèé óãîë α ñ ïðÿìîé. Ïóñòüâåêòîð ~a âðàùàåòñÿ âîêðóã ïðÿìîé ñ ïîñòîÿííîé óãëîâîé ñêîðîñòüþ ω , ïðè÷åì óãîë α è äëèíà âåêòîðà ~a îñòàþòñÿ íåèçìåííûìè.Ãë. 4. Ïðîèçâîäíûå è äèôôåðåíöèàëû.78Ïîëîæèì |~a| = a è ââåäåì âåêòîð ω~, óêîòîðîãî |~ω | = ω , à íàïðàâëåíèå ïîêàçàíîíà ðèñ. 4.9.Âåêòîð ~a çàâèñèò îò âðåìåíè: ~a = ~a(t).Äîêàæåì, ÷òîzrωra (0)ra (t )φd~a= [~ω × ~a].dtαφ=ωtxyÐèñ.

4.9.Ââåäåì ïðÿìîóãîëüíóþ ñèñòåìó êîîðäèíàòOxyz òàê, ÷òîáû ïîëîæèòåëüíîå íàïðàâëåíèå îñè Oz ñîâïàëî ñ íàïðàâëåíèåì âåêòîðà ω~ , è çàïèøåì êîîðäèíàòû âåêòîðà ~a(t):~a(t) = {a sin α · cos ωt, a sin α · sin ωt, a cos α}.Ââåäåì îáîçíà÷åíèÿ: a sin α = b, a cos α = c, òîãäàd~a= {−bω sin ωt, bω cos ωt, 0},dt~i~j[~ω × ~a(t)] = 00 b cos ωt b sin ωtîòêóäà ñëåäóåò, ÷òîrωra1ra2ra~kωc = {−bω sin ωt, bω cos ωt, 0},d~a= [~ω × ~a].dtÅñëè íà÷àëî âåêòîðà ~a íå ëåæèò íà îñè âðàùåíèÿ, òî äîêàçàííàÿ ôîðìóëà îñòàåòñÿ â ñèëå,ïîñêîëüêó òàêîé âåêòîð ìîæíî ïðåäñòàâèòü â âèäåðàçíîñòè äâóõ âåêòîðîâ ñ íà÷àëàìè íà îñè âðàùåíèÿ (ðèñ.

4.10):~a = ~a1 − ~a2 ⇒d~ad~ad~a= 1− 2 =dtdtdt= [~ω × ~a1 ] − [~ω × ~a2 ] = [~ω × (~a1 − ~a2 )] = [~ω × ~a] .Ðèñ. 4.10.Ïðèìåð 2. Ðàññìîòðèì òâåðäîå òåëî (íàïðèìåð, Çåìíîé øàð), âðàùàþùååñÿ ñ ïîñòîÿííîé óãëîâîé ñêîðîñòüþ ω~ îòíîñèòåëüíî íåïîäâèæíîé ñèñòåìû êîîðäèíàò ñ áàçèñîì{i~ , j~ , k~ } (ðèñ. 4.11).

Ââåäåì íà ýòîì òâåðäîì òåëå ñâîé áàçèñ00010. Ïðîèçâîäíûå âåêòîð-ôóíêöèè79{~i, ~j , ~k}. Îí âðàùàåòñÿ âìåñòå ñ òâåðäûì òåëîì ñ óãëîâîé ñêîðîñòüþ ω~ , ïîýòîìó~i = ~i(t), ~j = ~j(t), ~k = ~k(t),d~jd~i= ω~ × ~i ,= ω~ × ~j ,dtdthid~k= ω~ × ~k .dtÐàññìîòðèì òî÷êó M , äâèæóùóþñÿ âíóòðè òåëà èëè ïîåãî ïîâåðõíîñòè. Åå ïîëîæåíèå îòíîñèòåëüíî íåïîäâèæíîé ñèñòåìû êîîðäèíàò â êàæäûé ìîìåíò âðåìåíè ìîæíî çàäàòü~ , êîòîðûé îáîçíà÷èì ~r(t). Âûâåäåì ôîðìóëóðàäèóñ-âåêòîðîì OMñêîðîñòè òî÷êè M îòíîñèòåëüíî íåïîäâèæíîé ñèñòåìû êîîðäèíàò, ò.å.

ôîðìóëó äëÿ ~v = d~r/dt. Ñ ýòîé öåëüþ ðàçëîæèì âåêòîð~r(t) ïî (âðàùàþùåìóñÿ) áàçèñó {~i, ~j , ~k}:~r(t) = x(t)~i + y(t)~j + z(t)~k.Îòñþäà ñëåäóåò, ÷òîd~rdx ~d~idy ~d~jdz ~d~k=·i+x· +·j+y·+·k+z·=dtdtdtdtdtdtdt=dx ~ dy ~ dz ~·i+·j+·kdtdtdthi+x· ω~ × ~i + y · ω~ × ~j + z · ω~ × ~k .Âûðàæåíèå â ñêîáêàõ ïðåäñòàâëÿåò ñîáîé ñêîðîñòü òî÷êè îòíîñèòåëüíî ñâÿçàííîãî ñ òåëîì áàçèñà {~i, ~j , ~k}.

Íàçîâåì åå îòíîñèòåëüíîé ñêîðîñòüþ è îáîçíà÷èì ~v . Òàêèì îáðàçîì,îòí.d~r= ~vîòí. + ω~ × x~i + y~j + z~kdthi= ~vîòí. + [~ω × ~r] . ïîëó÷åííîì ðàâåíñòâå âåêòîðíîå ïðîèçâåäåíèå [~ω × ~r] ïðåäñòàâëÿåò ñîáîé êîìïîíåíòó ñêîðîñòè, îáóñëîâëåííóþ âðàùåíèåìòåëà. Íàçîâåì åå ïåðåíîñíîé ñêîðîñòüþ è îáîçíà÷èì ~v .Èòàê, àáñîëþòíàÿ ñêîðîñòü òî÷êè M (ò.å. ñêîðîñòü îòíîñèòåëüíî íåïîäâèæíîé ñèñòåìû êîîðäèíàò) ðàâíà ñóììå ïåðåíîñíîé è îòíîñèòåëüíîé ñêîðîñòåé:ïåð.~v =d~r= ~vïåð. + ~vîòí. .dtÏîëó÷èì òåïåðü ôîðìóëó äëÿ óñêîðåíèÿ òî÷êè M . Èìååì:d~vïåð.d~vd~vd~rd~a ==+ îòí. = ω~×+dtdtdtdtdthidx ~ dy ~ dz ~·i+·j+·kdtdtdt=Ãë. 4.

Ïðîèçâîäíûå è äèôôåðåíöèàëû.80rωrk0rj0Ori0rkrr (t ) MrjriÐèñ. 4.11.= [~ω × (~vïåð. + ~vîòí. )] +d2 x ~ d2 y ~d2 z ~·i+·j+·k +dt2dt2dt2i dy dz hdx ~· ω~ × ~i +· ω~ × ~j +· ω~ ×k=+dtdtdt= [~ω × ~vïåð. ] + [~ω × ~vîòí. ] + ~aîòí. + [~ω × ~vîòí. ] == ~aïåð. + ~aîòí. + 2 [~ω × ~vîòí. ] . ýòîì ðàâåíñòâå ñëàãàåìûå[~ω × ~vïåð. ] =: ~aïåð.èd2 zd2 x ~ d2 y ~· i + 2 · j + 2 · ~k =: ~aîòí.2dtdtdtÿâëÿþòñÿ ñîîòâåòñòâåííî ïåðåíîñíûì è îòíîñèòåëüíûì óñêîðåíèÿìè, à ñëàãàåìîå 2 [~ω × ~v ] òàê íàçûâàåìûì êîðèîëèñîâûìóñêîðåíèåì ~aÈòàê, àáñîëþòíîå óñêîðåíèå ðàâíî ñóììå ïåðåíîñíîãî, îòíîñèòåëüíîãî è êîðèîëèñîâà óñêîðåíèé:îòí.êîð.~a = ~aïåð.

+ ~aîòí. + ~aêîð.Îòìåòèì, ÷òî ~aïåð. = [~ω × ~vïåð. ] = ω~ × [~ω × ~r] ïðåäñòàâëÿåòñîáîé äâîéíîå âåêòîðíîå ïðîèçâåäåíèå.Ãëàâà 5ÈÍÒÅÃÐÀË۟ 1. Ïåðâîîáðàçíàÿ è íåîïðåäåëåííûé èíòåãðàëÊ ïîíÿòèþ ïåðâîîáðàçíîé ïðèâîäèò ñëåäóþùàÿ ôèçè÷åñêàÿçàäà÷à. Ïóñòü x âðåìÿ, y = f (x) êîîðäèíàòà òî÷êè, äâèæóùåéñÿ ïî îñè y , â ìîìåíò âðåìåíè x. Òîãäà f 0 (x) = v(x) ñêîðîñòü òî÷êè â ìîìåíò âðåìåíè x. Åñëè èçâåñòíà çàâèñèìîñòüêîîðäèíàòû îò âðåìåíè, ò.å.

èçâåñòíà ôóíêöèÿ f (x), òî äëÿíàõîæäåíèÿ ñêîðîñòè v(x) íóæíî âûïîëíèòü îïåðàöèþ äèôôåðåíöèðîâàíèÿ.Ïóñòü òåïåðü íàîáîðîò èçâåñòíà çàâèñèìîñòü ñêîðîñòè îòâðåìåíè, ò.å. èçâåñòíà ôóíêöèÿ v(x), à òðåáóåòñÿ íàéòè çàâèñèìîñòü êîîðäèíàòû îò âðåìåíè, ò.å. ôóíêöèþ f (x), òàêóþ, ïðîèçâîäíàÿ f 0 (x) êîòîðîé ðàâíà çàäàííîé ôóíêöèè v(x): f 0 (x) = v(x).Òåì ñàìûì âîçíèêàåò çàäà÷à, îáðàòíàÿ äèôôåðåíöèðîâàíèþ.Ïóñòü ôóíêöèÿ y = f (x) îïðåäåëåíà íà ïðîìåæóòêå X .Îïðåäåëåíèå.

Ôóíêöèÿ F (x) íàçûâàåòñÿ ïåðâîîáðàçíîé äëÿôóíêöèè f (x) íà ïðîìåæóòêå X , åñëè ∀x ∈ X : F 0 (x) = f (x). ñîîòâåòñòâèè ñ ýòèì îïðåäåëåíèåì ôóíêöèÿ f (x), çàäàþùàÿ êîîðäèíàòó òî÷êè íà îñè â ìîìåíò âðåìåíè x, ÿâëÿåòñÿïåðâîîáðàçíîé äëÿ ôóíêöèè v(x), çàäàþùåé ñêîðîñòü òî÷êè:f 0 (x) = v(x).Ïðèìåðû.1) F (x) = ln x ïåðâîîáðàçíàÿ äëÿ f (x) = 1/x íà ïîëóïðÿìîé X+ = (0; +∞), ò.ê. (ln x)0 = 1/x ∀x ∈ (0; +∞).F (x) = ln(−x) ïåðâîîáðàçíàÿ äëÿ f (x) = 1/x íà ïîëóïðÿìîé X− = (−∞; 0), ò.ê.0ln(−x) = −1/(−x) = 1/x ∀x ∈ (−∞; 0).Îòñþäà ñëåäóåò, ÷òî F (x) = ln |x| ïåðâîîáðàçíàÿ äëÿ f (x) == 1/x íà ïîëóïðÿìûõ X+ è X− .2) Äëÿ ôóíêöèèf (x) = |x| =x, åñëè x > 0,−x, åñëè x < 0Ãë. 5. Èíòåãðàëû82ïåðâîîáðàçíîé íà (−∞; +∞) ÿâëÿåòñÿ ôóíêöèÿF (x) = ñàìîì äåëå,x2 /2, åñëè x > 0,−x2 /2, åñëè x < 0.∀x > 0 : F 0 (x) = x = f (x); ∀x < 0 : F 0 (x) = −x = f (x).Îñòàåòñÿ äîêàçàòü (ñäåëàéòå ýòî ñàìîñòîÿòåëüíî), ÷òî∃F 0 (0) = 0 = f (0).3) Ðàññìîòðèì ôóíêöèþåñëè x > 0,0, åñëè x = 0,−1, åñëè x < 0.Îíà èìååò ïåðâîîáðàçíóþ F (x) = x íà ïîëóïðÿìîé (0; +∞),èìååò ïåðâîîáðàçíóþ F (x) = −x íà ïîëóïðÿìîé (−∞; 0), íî íåèìååò ïåðâîîáðàçíîé íà âñåé ÷èñëîâîé ïðÿìîé (−∞; +∞).Îòìåòèì, ÷òî åñëè F (x) ïåðâîîáðàçíàÿ äëÿ f (x) íà ïðîìåæóòêå X , òî åñòü ∀x ∈ X : F 0 (x) = f (x), òî F (x) + C , ãäå C ëþáîå ÷èñëî, òàêæå ïåðâîîáðàçíàÿ äëÿ f (x) íà ïðîìåæóòêå X ,òàê êàê F (x) + C 0 = F 0 (x) + C 0 = f (x) + 0 = f (x).

Ñïðàâåäëèâîè îáðàòíîå.(f (x) = sgn x =+1,Òåîðåìà 1 (îñíîâíàÿ òåîðåìà èíòåãðàëüíîãî èñ÷èñëåíèÿ).Åñëè F (x) è F (x) ëþáûå äâå ïåðâîîáðàçíûå äëÿ ôóíêöèèf (x) íà ïðîìåæóòêå X , òî F (x) − F (x) = C = const íà ýòîìïðîìåæóòêå.Äîêàçàòåëüñòâî.Ïîëîæèì F (x) = F (x) − F (x). Íóæíî äîêàçàòü, ÷òî F (x) == const íà ïðîìåæóòêå X . Èìååì121122∀x ∈ X : F 0 (x) = F10 (x) − F20 (x) = f (x) − f (x) = 0.Òàêèì îáðàçîì, íóæíî äîêàçàòü, ÷òî åñëè F 0 (x) ≡ 0 íà ïðîìåæóòêå X , òî F (x) = C = const íà X . Ïðè âñåé î÷åâèäíîñòè ýòîãîóòâåðæäåíèÿ ìû ïîêà íå ìîæåì åãî äîêàçàòü.