ÌÈÍÎÁÐÍÀÓÊÈ ÐÎÑÑÈÈ
ôåäåðàëüíîå ãîñóäàðñòâåííîå áþäæåòíîå îáðàçîâàòåëüíîå ó÷ðåæäåíèå
âûñøåãî ïðîôåññèîíàëüíîãî îáðàçîâàíèÿ
¾Ñàíêò-Ïåòåðáóðãñêèé ãîñóäàðñòâåííûé òåõíîëîãè÷åñêèé èíñòèòóò
(òåõíè÷åñêèé óíèâåðñèòåò)¿
(ÑÏáÃÒÈ(ÒÓ))
Êàôåäðà ìàòåìàòèêè
Ò. Â. Ñëîáîäèíñêàÿ, À. À. Ãðóçäêîâ, Ò. Â. Âèííèê
Ìàòåìàòèêà
(âòîðîé ñåìåñòð)
Ó÷åáíîå ïîñîáèå äëÿ ñòóäåíòîâ çàî÷íîé ôîðìû
îáó÷åíèÿ
Ñàíêò-Ïåòåðáóðã
2016
ÓÄÊ 512.64, 514.123.1, 517.1, 517.2, 517.3
Ñëîáîäèíñêàÿ, Ò. Â. Ìàòåìàòèêà (âòîðîé ñåìåñòð): ó÷åáíîå ïîñîáèå
äëÿ ñòóäåíòîâ çàî÷íîé ôîðìû îáó÷åíèÿ [Òåêñò]: / Ò. Â. Ñëîáîäèíñêàÿ,
À. À. Ãðóçäêîâ, Ò. Â. Âèííèê. ÑÏá.: ÑÏáÃÒÈ(ÒÓ), 2016. 85 ñ.
Ó÷åáíîå ïîñîáèå ñîäåðæèò êðàòêîå èçëîæåíèå òåîðåòè÷åñêîãî ìàòåðèàëà íåîáõîäèìîãî äëÿ âûïîëíåíèÿ êîíòðîëüíûõ ðàáîò, çàäàíèÿ êîíòðîëüíûõ ðàáîò è ïðèìåðû ðåøåíèÿ òèïîâûõ âàðèàíòîâ. Ïðåäíàçíà÷åíî äëÿ
ñòóäåíòîâ ïåðâîãî êóðñà çàî÷íîé ôîðìû îáó÷åíèÿ. Ïîñîáèå ñîñòàâëåíî â
ñîîòâåòñòâèè ñ ó÷åáíîé ïðîãðàììîé ïî äèñöèïëèíàì ¾Ìàòåìàòèêà¿, ¾Âûñøàÿ ìàòåìàòèêà¿, ¾Ìàòåìàòè÷åñêèé àíàëèç¿.
Ó÷åáíîå ïîñîáèå ñîîòâåòñòâóþò ñëåäóþùèì êîìïåòåíöèÿì ïîäãîòîâêè
áàêàëàâðîâ: ÎÊ-10 íàïðàâëåíèÿ 09.03.01 ¾Èíôîðìàòèêà è âû÷èñëèòåëüíàÿ òåõíèêà¿; ÏÊ-19 íàïðàâëåíèÿ 15.03.04 ¾Àâòîìàòèçàöèÿ òåõíîëîãè÷åñêèõ ïðîöåññîâ è ïðîèçâîäñòâ¿; ÏÊ-2 íàïðàâëåíèÿ 15.03.02 ¾Òåõíîëîãè÷åñêèå ìàøèíû è îáîðóäîâàíèå¿; ÎÏÊ-1 íàïðàâëåíèÿ 08.03.01 ¾Ñòðîèòåëüñòâî¿.
Ðèñ. 18, áèáëèîãð. 11 íàçâ.
Ðåöåíçåíòû:
1. Áàëòèéñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò,
ïðîôåññîð êàôåäðû âûñøåé ìàòåìòàèêè, êàíäèäàò ôèçèêîìàòåìàòè÷åñêèõ íàóê, äîöåíò Á. Ï. Ðîäèí
2.Ðîññèéñêèé ãîñóäàðñòâåííûé ãèäðîìåòåîðîëîãè÷åñêèé óíèâåðñèòåò, äîöåíò êàôåäðû ìàòåìàòè÷åñêîãî ìîäåëèðîâàíèÿ,
êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, äîöåíò Â. Ã. Íèêèòåíêî
Èçäàíèå ïîäãîòîâëåíî â ðàìêàõ âûïîëíåíèÿ âíóòðèâóçîâñêîãî çàäàíèÿ
ïî îêàçàíèþ îáðàçîâàòåëüíûõ óñëóã
Óòâåðæäåíî íà çàñåäàíèè ó÷åáíî-ìåòîäè÷åñêîé êîìèññèè ôàêóëüòåòà
èíôîðìàöèîííûõ òåõíîëîãèé è óïðàâëåíèÿ 05.02.2016.
Ðåêîìåíäîâàíî ê èçäàíèþ ÐÈÑ ÑÏáÃÒÈ(ÒÓ)
Ââåäåíèå
Äèñöèïëèíà ¾Ìàòåìàòèêà¿ îòíîñèòñÿ ê öèêëó îáùåíàó÷íûõ äèñöèïëèí.
Öåëü êóðñà ôîðìèðîâàíèå íàó÷íîãî ìèðîâîççðåíèÿ ó ñòóäåíòîâ, ïðèîáðåòåíèå èìè ìàòåìàòè÷åñêèõ çíàíèé, óìåíèé è íàâûêîâ, íåîáõîäèìûõ äëÿ
èçó÷åíèÿ äðóãèõ îáùåíàó÷íûõ è ñïåöèàëüíûõ äèñöèïëèí, à òàêæå ñàìîñòîÿòåëüíîãî èçó÷åíèÿ ñïåöèàëüíîé ëèòåðàòóðû. Èçó÷åíèå êóðñà íåîáõîäèìî
äëÿ ôîðìèðîâàíèÿ ñïîñîáíîñòè ìàòåìàòè÷åñêîãî èññëåäîâàíèÿ ïðèêëàäíûõ çàäà÷, ïðàâèëüíîãî èñòîëêîâàíèÿ è îöåíêè ïîëó÷àåìûõ ðåçóëüòàòîâ,
à òàêæå ôîðìèðîâàíèÿ íàâûêîâ ñàìîñòîÿòåëüíîé èññëåäîâàòåëüñêîé ðàáîòû.
 äàííîì ó÷åáíîì ïîñîáèè ïðåäñòàâëåíû ÷åòûðå êîíòðîëüíûå ðàáîòû
âòîðîãî ñåìåñòðà ïî ñëåäóþùèì ðàçäåëàì:
• Äèôôåðåíöèàëüíîå èñ÷èñëåíèå ôóíêöèè íåñêîëüêèõ ïåðåìåííûõ;
• Âû÷èñëåíèå è ïðèëîæåíèÿ äâîéíûõ è òðîéíûõ èíòåãðàëîâ;
• Êðèâîëèíåéíûå è ïîâåðõíîñòíûå èíòåãðàëû;
• Ýëåìåíòû òåîðèè ïîëÿ.
Äëÿ êàæäîé ðàáîòû óêàçûâàåòñÿ ñîäåðæàíèå äàííîé ðàáîòû, âàðèàíòû
çàäàíèé è ïðèìåðû ðåøåíèÿ.
Óêàçàíèÿ ïî âûïîëíåíèþ êîíòðîëüíûõ ðàáîò
Êîíòðîëüíàÿ ðàáîòà ìîæåò áûòü íàïèñàíà îò ðóêè íà ëèñòàõ ôîðìàòà
À4 èëè ïðåäñòàâëåíà â ðàñïå÷àòàííîì âèäå. Ëèñòû äîëæíû áûòü ñêðåïëåíû ñòåïëåðîì, ïðè÷åì êàæäàÿ êîíòðîëüíàÿ ðàáîòà ñäàåòñÿ îòäåëüíî. Ðàáîòà ìîæåò áûòü íàïèñàíà îò ðóêè â òåòðàäè.  ýòîì ñëó÷àå êàæäàÿ ðàáîòà
ñäàåòñÿ â îòäåëüíîé òåòðàäè.
Íà òèòóëüíîì ëèñòå óêàçûâàåòñÿ ïîëíîå íàçâàíèå óíèâåðñèòåòà, ôàêóëüòåò, êàôåäðà, ôàìèëèÿ, èìÿ, îò÷åñòâî ñòóäåíòà, íîìåð ó÷åáíîé ãðóïïû,
íîìåð êîíòðîëüíîé ðàáîòû, íîìåð âàðèàíòà, ôàìèëèÿ è èíèöèàëû ïðåïîäàâàòåëÿ, ïðîâåðÿþùåãî ðàáîòó, ãîä è ñòàâèòñÿ ëè÷íàÿ ïîäïèñü ñòóäåíòà.
Ðàáîòà ñ÷èòàåòñÿ âûïîëíåííîé, åñëè âñå çàäà÷è ðåøåíû âåðíî. Åñëè â
ðåøåíèè êàêîé-ëèáî çàäà÷è äîïóùåíà îøèáêà, òî ñòóäåíò äîëæåí ñäåëàòü
ðàáîòó íàä îøèáêàìè (çàíîâî ðåøèòü çàäà÷ó). Ðàáîòà íàä îøèáêàìè äîëæíà ðàñïîëàãàòüñÿ ïîñëå çàïèñè ðåøåíèÿ ïîñëåäíåé çàäà÷è êîíòðîëüíîé ðàáîòû.
3
Ñòóäåíò ñàìîñòîÿòåëüíî âûáèðàåò âàðèàíò êîíòðîëüíîé ðàáîòû â ñîîòâåòñòâèè ñ íà÷àëüíîé áóêâîé ñâîåé ôàìèëèè.
Áóêâà Íîìåð âàðèàíòà
À
1
Á
2
Â
3
Ã
4
Ä
5
Å, œ
6
Æ
7
Ç
8
È, É
9
Ê
10
Ë
11
Ì
12
Í
13
Î
14
Ï
15
Ð
16
Ñ
17
Ò
18
Ó
19
Ô
20
Õ
21
Ö, Þ
22
×
23
Ø,Ù
24
Ý, ß
25
4
Êîíòðîëüíàÿ ðàáîòà  5
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  5
Çàäàíèå  1
Íàéäèòå ïîëíûé äèôôåðåíöèàë ôóíêöèè.
Çàäàíèå  2
Íàéäèòå ïðîèçâîäíûå ñëîæíîé ôóíêöèè.
Çàäàíèå  3
Èññëåäóéòå ôóíêöèþ íà ýêñòðåìóì.
Çàäàíèå  4
Íàéäèòå íàèáîëüøåå è íàèìåíüøåå çíà÷åíèå ôóíêöèè â çàìêíóòîé îáëàñòè D, îãðàíè÷åííîé çàäàííûìè ëèíèÿìè.
Óêàçàíèå.
Ïåðåä ðåøåíèåì çàäà÷ êîíòðîëüíîé ðàáîòû ðåêîìåíäóåòñÿ îçíàêîìèòüñÿ ñî ñëåäóþùèìè ìåòîäè÷åñêèìè óêàçàíèÿìè:
1. Äèôôåðåíöèàëüíîå èñ÷èñëåíèå ôóíêöèé íåñêîëüêèõ ïåðåìåííûõ: ìåòîäè÷åñêèå óêàçàíèÿ  924. Ñîñòàâèòåëè: Áåðåçíèêîâà Â.Â., Ïàóëüñåí À.Í., Ðîìàíîâñêàÿ Ë.Í. ÑÏá.: ÑÏáÃÒÈ(ÒÓ), 2002.
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû
Ïðîèçâîäíûå è äèôôåðåíöèàëû ôóíêöèé íåñêîëüêèõ ïåðåìåííûõ
Îòîáðàæåíèå
f : D → R,
ãäå D ⊂ R2 , íàçûâàåòñÿ ôóíêöèåé äâóõ ïåðåìåííûõ, ïðè ýòîì ïèøóò
z = f (x, y).
Àíàëîãè÷íî îïðåäåëÿåòñÿ ôóíêöèÿ n ïåðåìåííûõ, åñëè D ⊂ Rn .
5
Âåëè÷èíû
∆x z = f (x + ∆x, y) − f (x, y)
è
∆y z = f (x, y + ∆y) − f (x, y)
íàçûâàþòñÿ ÷àñòíûìè ïðèðàùåíèÿìè ïî ïåðåìåííûì x è y ñîîòâåòñòâåííî.
Åñëè ñóùåñòâóåò ïðåäåë
∆x z
,
∆x→0 ∆x
lim
òî îí íàçûâàåòñÿ ÷àñòíîé ïðîèçâîäíîé ôóíêöèè z ïî ïåðåìåííîé x è
îáîçíà÷àåòñÿ îäíèì èç ñèìâîëîâ
∂z
∂f (x, y)
′
′
, zx , fx (x, y),
.
∂x
∂x
Àíàëîãè÷íî
∂f (x, y)
∆y z
∂z
′
′
=
= zy = fy (x, y) =
.
∆y→0 ∆y
∂y
∂y
lim
Ïóñòü ôóíêöèÿ äâóõ ïåðåìåííûõ z = f (x, y) èìååò íåïðåðûâíûå ÷àñòíûå ïðîèçâîäíûå â íåêîòîðîé îáëàñòè. Òîãäà ïîëíîå ïðèðàùåíèå ôóíêöèè
∆z = f (x + ∆x, y + ∆y) − f (x, y)
ìîæíî ïðåäñòàâèòü â âèäå
∆z =
∂z
∂z
∆x + ∆y + α∆x + β∆y,
∂x
∂y
ãäå α → 0 è β → 0 ïðè ∆x → 0, ∆y → 0.
Ëèíåéíàÿ ÷àñòü ïîëíîãî ïðèðàùåíèÿ ôóíêöèè íàçûâàåòñÿ åå ïîëíûì
äèôôåðåíöèàëîì è îáîçíà÷àåòñÿ ÷åðåç dz . Ïîëàãàÿ ïî îïðåäåëåíèþ
∆x = dx,
∆y = dy,
èìååì
dz =
∂z
∂z
dx +
dy.
∂x
∂y
(1)
Àíàëîãè÷íî îïðåäåëÿåòñÿ ïîëíûé äèôôåðåíöèàë ôóíêöèè n ïåðåìåííûõ.
Åñëè ÷àñòíûå ïðîèçâîäíûå ôóíêöèè äâóõ ïåðåìåííûõ
z = f (x, y) :
∂z
∂z
′
′
= fx (x, y),
= fy (x, y),
∂x
∂y
6
ñóùåñòâóþò â íåêîòîðîé îáëàñòè, òî èõ òîæå ìîæíî ðàññìàòðèâàòü, êàê
ôóíêöèè äâóõ ïåðåìåííûõ, ïîýòîìó îò íèõ ìîæíî ñíîâà íàõîäèòü ÷àñòíûå ïðîèçâîäíûå, êîòîðûå íàçûâàþòñÿ ÷àñòíûìè ïðîèçâîäíûìè âòîðîãî
ïîðÿäêà:
)
∂z
′′
= fxx (x, y),
∂x
( )
∂ 2z
∂ ∂z
′′
=
= fxy (x, y),
∂x∂y
∂y ∂x
( )
∂ 2z
∂ ∂z
′′
=
= fyx (x, y),
∂y∂x ∂x ∂y
( )
∂ 2z
∂ ∂z
′′
=
f
=
yy (x, y).
∂y 2
∂y ∂y
Åñëè ôóíêöèÿ z = f (x, y) è åå ÷àñòíûå ïðîèçâîäíûå îïðåäåëåíû è
íåïðåðûâíû â íåêîòîðîé îêðåñòíîñòè òî÷êè M (x, y), òî â ýòîé òî÷êå ñìå′′
′′
øàííûå ïðîèçâîäíûå ðàâíû, ò. å. fyx = fxy .
∂ 2z
∂
=
∂x2
∂x
(
Ïðîèçâîäíàÿ ñëîæíîé ôóíêöèè
Äëÿ íàõîæäåíèÿ ÷àñòíûõ ïðîèçâîäíûõ ñëîæíîé ôóíêöèè
z = f (x, y),
ãäå x = x(u, v),
y = y(u, v).
èñïîëüçóþòñÿ ôîðìóëû
∂z
∂z ∂x ∂z ∂y
=
+
,
∂u ∂x ∂u ∂y ∂u
(2)
∂z
∂z ∂x ∂z ∂y
=
+
.
∂v
∂x ∂v ∂y ∂v
 ñëó÷àå, êîãäà
(3)
z = f (x, y),
ãäå x = x(t), y = y(t),
ïðèìåíÿåòñÿ ôîðìóëà
∂z dx ∂z dy
dz
=
+
.
dt
∂x dt ∂y dt
(4)
Åñëè æå
z = f (x, y),
ãäå y = y(x),
7
òî
dz
∂z ∂z dy
=
+
.
dx ∂x ∂y dx
Âåëè÷èíû
(5)
dz
dz
è
â ôîðìóëàõ (4) è (5), ñîîòâåòñòâåííî, íàçûâàþòñÿ
dt
dx
ïîëíûìè ïðîèçâîäíûìè.
Ýêñòðåìóì ôóíêöèé äâóõ ïåðåìåííûõ
Òî÷êà M0 (x0 , y0 ) íàçûâàåòñÿ òî÷êîé ëîêàëüíîãî ìàêñèìóìà (ìèíèìóìà) ôóíêöèè z = f (x, y), åñëè ôóíêöèÿ íåïðåðûâíà â ýòîé òî÷êå è äëÿ
âñåõ òî÷åê M (x, y) èç íåêîòîðîé îêðåñòíîñòè òî÷êè M0 (x0 , y0 ) âûïîëíÿåòñÿ
íåðàâåíñòâî
f (x0 , y0 ) ≥ f (x, y)
(f (x0 , y0 ) ≤ f (x, y)) .
Ìàêñèìóì è ìèíèìóì ôóíêöèè íàçûâàþòñÿ åå ýêñòðåìóìàìè. Åñëè
â îêðåñòíîñòè òî÷êè ýêñòðåìóìà ñóùåñòâóþò è íåïðåðûâíû ÷àñòíûå ïðîèçâîäíûå ïåðâîãî ïîðÿäêà, òî â òî÷êå ýêñòðåìóìà îíè îáðàùàþòñÿ â íîëü:
′
′
(6)
fx (x0 , y0 ) = fy (x0 , y0 ) = 0,
(íåîáõîäèìîå óñëîâèå ýêñòðåìóìà ). Òî÷êè ôóíêöèè z = f (x, y), äëÿ
êîòîðûõ âûïîëíåíî íåîáõîäèìîå óñëîâèå ýêñòðåìóìà (6) íàçûâàþòñÿ ñòàöèîíàðíûìè òî÷êàìè ôóíêöèè. Ñòàöèîíàðíûå òî÷êè ìîãóò íå ÿâëÿòüñÿ òî÷êàìè ýêñòðåìóìà, ïîýòîìó äëÿ íèõ ñëåäóåò ïðîâåðèòü äîñòàòî÷íîå
óñëîâèå ýêñòðåìóìà.
Ïóñòü M0 (x0 , y0 ) ñòàöèîíàðíàÿ òî÷êà ôóíêöèè è â íåêîòîðîé îêðåñòíîñòè ýòîé òî÷êè ôóíêöèÿ èìååò íåïðåðûâíûå âòîðûå ïðîèçâîäíûå. Ââåäåì îáîçíà÷åíèÿ
′′
′′
′′
fxx (x0 , y0 ) = A, fxy (x0 , y0 ) = B, fyy (x0 , y0 ) = C,
è ïóñòü
∆=
A B
B C
= AC − B 2 .
Òîãäà
1) åñëè ∆ > 0, òî òî÷êà M0 (x0 , y0 ) ÿâëÿåòñÿ òî÷êîé ýêñòðåìóìà äàííîé
ôóíêöèè, ïðè÷åì ýòî òî÷êà ëîêàëüíîãî ìàêñèìóìà ïðè A < 0 è òî÷êà
ëîêàëüíîãî ìèíèìóìà ïðè A > 0;
2) åñëè ∆ < 0, òî ýêñòðåìóìà â òî÷êå M0 (x0 , y0 ) ôóíêöèÿ íå èìååò;
3) åñëè ∆ = 0, òî òðåáóþòñÿ äîïîëíèòåëüíûå èññëåäîâàíèÿ.
8
Íàõîæäåíèå íàèáîëüøåãî è íàèìåíüøåãî çíà÷åíèÿ ôóíêöèè
Ïóñòü ôóíêöèÿ z = f (x, y) îïðåäåëåíà è íåïðåðûâíà â íåêîòîðîé îáëàñòè D è íà åå ãðàíèöå, êîòîðàÿ çàäàåòñÿ ëèíèÿìè Γi , i = 1, 2, . . . , m,
ò. å. â çàìêíóòîé îáëàñòè D̄. Òîãäà ôóíêöèÿ z = f (x, y) äîñòèãàåò â íåêîòîðûõ òî÷êàõ ýòîé îáëàñòè ñâîåãî íàèáîëüøåãî è íàèìåíüøåãî çíà÷åíèé.
Ýòè òî÷êè è ñàìè çíà÷åíèÿ ìîæíî íàéòè ñëåäóþùèì îáðàçîì.
1. Èùåì ñòàöèîíàðíûå òî÷êè ôóíêöèè, ïðèðàâíèâàÿ ê íóëþ åå ÷àñòíûå
ïðîèçâîäíûå.
2. Îòáèðàåì òå ñòàöèîíàðíûå òî÷êè, êîòîðûå ïðèíàäëåæàò îáëàñòè D.
3. Ïîäñòàâëÿåì â ôóíêöèþ z = f (x, y) óðàâíåíèå ãðàíè÷íîé ëèíèè Γ1
è ïîëó÷àåì ôóíêöèþ îäíîé ïåðåìåííîé. Èùåì ñòàöèîíàðíûå òî÷êè ïîëó÷åííîé ôóíêöèè, ïðèíàäëåæàùèå òîìó èíòåðâàëó èçìåíåíèÿ ïåðåìåííîé,
êîòîðûé îïðåäåëÿåò ÷àñòü ãðàíèöû îáëàñòè D. Ýòó ïîñëåäîâàòåëüíîñòü
äåéñòâèé ïîâòîðÿåì è äëÿ ëèíèé Γ2 , . . . , Γm .
4. Âûïèñûâàåì óãëîâûå òî÷êè, ò. å. òå òî÷êè, â êîòîðûõ ïðîèñõîäèò
ïåðåñå÷åíèå ãðàíè÷íûõ ëèíèé.
5. Âû÷èñëÿåì çíà÷åíèÿ ôóíêöèè â òî÷êàõ, âûáðàííûõ â ïóíêòàõ 2, 3, 4.
Âûáèðàåì íàèáîëüøåå è íàèìåíüøåå èç âû÷èñëåííûõ çíà÷åíèé.
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  5
Âàðèàíò  1.
1. z = 2x3 y − 4xy 3 .
√
2. z = x2 + y 2 + 3, x = ln t,
√
3. z = y x − 2y 2 − x + 14y.
4. z = 3x + y − xy,
y=
√
3
t.
D : y = x, y = 4, x = 0.
Âàðèàíò  2.
1. z = arctg x +
2. z = x2 e−y ,
√
y.
u
x = cos(u − v), y = sin .
v
3. z = x3 + 8y 3 − 6xy + 5.
4. z = xy − x − 2y,
D : x = 3, y = x, y = 0.
9
Âàðèàíò  3.
1. z = x2 y sin x − 3y.
(
)
2. z = ln ex + e−y , x = t3 , y = t2 .
3. z = 1 + 15x − 2x2 − xy − 2y 2 .
4. z = x2 + 2xy − 4x + 8y,
D : x = 0, y = 0, x = 1, y = 2.
Âàðèàíò  4.
1. z = arcsin xy − 3xy 2 .
√
(
)
2
2. z = sin x cos y, x = ln u + v , y = v − u2 .
3. z = 1 + 6x − x2 − xy − y 2 .
4. z = 5x2 − 3xy + y 2 ,
D : x = 0, x = 1, y = 0, y = 1.
Âàðèàíò  5.
1. z = 5xy 4 + 2x2 y 7 .
x
2. z =
, x = eu−v , y = sin(u + v).
x−y
3. z = 2x3 + 2y 3 − 6xy + 5.
4. z = x2 + 2xy − y 2 − 4x,
D : x − y + 1 = 0, x = 3, y = 0.
Âàðèàíò  6.
1.
(
)
cos x2 − y 2 + x3 .
2. z = xy ,
u
x = ev ,
y = ln(v − u).
3. z = 3x3 + 3y 3 − 9xy + 10.
4. z = x2 + y 2 − 2x − 2y + 8,
D : x + y = 1, y = 0, x = 0.
Âàðèàíò  7.
(
)
1. z = ln 3x2 − 2y 2 .
2. z = x2 ey ,
x = sin(u − v),
y = cos uv.
3. z = x2 + xy + y 2 + x − y + 1.
10
4. z = 2x3 − xy 2 + y 2 ,
D : x = 0, x = 1, y = 0, y = 6.
Âàðèàíò  8.
1. z = 5xy 2 − 3x3 y 4 .
2. z = x sin xy,
√
y = − 1 − v2.
(
)
x = ln u2 − 1 ,
3. z = 4(x − y) − x2 − y 2 .
4. z = 3x + 6y − x2 − xy − y 2 ,
D : x = 0, x = 1, y = 0, y = 1.
Âàðèàíò  9.
1. z = arcsin(x + y).
2. z = xy ,
u
x = ln(u − v),
y = ev .
3. z = 6(x − y) − 3x2 − 3y 2 .
4. z = x2 − 2y 2 + 4xy − 6x − 1,
D : x = 0, y = 0, x + y = 3.
Âàðèàíò  10.
1. z = arctg(2x − y).
x
2. z = arcsin ,
y
y=
√
x2 + 1.
3. z = x2 + xy + y 2 − 6x − 9y.
4. z = x2 + 2xy − 10,
D : y = 0, y = x2 − 4.
Âàðèàíò  11.
1. z = 7x3 y −
√
xy.
u
, y = 3u − 2v.
v
3. z = (x − 2)2 + 2y 2 − 10.
2. z = x2 ln y,
x=
4. z = xy − 2x − y,
D : x = 0, x = 3, y = 0, y = 4.
Âàðèàíò  12.
√
1. z = x2 + y 2 + 2xy + 1.
11
2. z = ey−2x ,
x = u3 ,
y = u sin v.
3. z = (x − 5)2 + y 2 + 1.
4. z = 0, 5x2 − xy,
D : y = 8, y = 2x2 .
Âàðèàíò  13.
1. z = ex+y−4 .
2. z = arccos
2x
,
y
x = sin t,
y = cos2 t.
3. z = x3 + y 3 − 3xy.
4. z = 3x2 + 3y 2 − 2x − 2y + 2,
D : x = 0, y = 0, x + y = 1.
Âàðèàíò  14.
1. z = cos(3x + y) − x2 .
2. z = arcsin
x
,
2y
x = sin t,
y = cos2 t.
3. z = 2xy − 2x2 − 4y 2 .
4. z = 2x2 + 3y 2 + 1,
√
9
9 − x2 , y = 0.
4
D: y=
Âàðèàíò  15.
1. z = tg
x+y
.
x−y
2. z = ey−2x−1 , x = cos t, y = sin t.
√
3. z = x y − x2 − y + 6x + 3.
4. z = x2 − 2xy − y 2 + 4x + 1,
D : x = −3, y = 0, x + y = −1.
Âàðèàíò  16.
y
1. z = ctg .
x
(
)
2. z = ln e−x + ey ,
x = t2 ,
y = t3 .
3. z = 2xy − 5x2 − 3y 2 + 2.
12
4. z = 3x2 + 3y 2 − x − y + 1,
D : x = 5, y = 0, x − y = 1.
Âàðèàíò  17.
1. z = xy 4 − 3x2 y + 1.
2. z = x2 e−y ,
x = sin t,
y = sin2 t.
3. z = xy (12 − x − y) .
4. z = 2x2 + 2xy − 0, 5y 2 − 4x,
D : y = 2x, y = 2, x = 0.
Âàðèàíò  18.
(
)
1. z = ln x + xy − y 2 .
2. z = ey−2x ,
x = sin t,
y = t3 .
3. z = xy − x2 − y 2 + 9.
5
4. z = x2 − 2xy + y 2 − 2x,
2
D : x = 0, x = 2, y = 0, y = 2.
Âàðèàíò  19.
1. z = 2x2 y 2 + x3 − y 3 .
2. z = xy ,
x = et ,
y = ln t.
3. z = 2xy − 3x2 − 2y 2 + 10.
4. z = xy − 3x − 2y,
D : x = 0, x = 4, y = 0, y = 4.
Âàðèàíò  20.
1. z =
√
3x2 − 2y 2 + 5 .
2. z = ln (ex + ey ) ,
x = t2 ,
y = t3 .
3. z = x3 + 8y 3 − 6xy + 1.
4. z = x2 + xy − 2,
D : y = 4x2 − 4, y = 0.
Âàðèàíò  21.
1. z = arcsin
x+y
.
x
13
2. z = x2 ey , x = cos t, y = sin t.
√
3. z = y x − y 2 − x + 6y.
4. z = x2 y(4 − x − y),
D : x = 0, y = 0, y = 6 − x.
Âàðèàíò  22.
1. z = arctg(x − y).
2. z = ey−2x+2 ,
x = sin t,
y = cos t.
3. z = xy(6 − x − y).
D : x = 0, x = 2, y = −1, y = 4.
4. z = x3 + y 3 − 3xy,
Âàðèàíò  23.
√
1. z = 3x2 − y 2 + x .
2. z = y x ,
x = ln(t − 1),
t
y = e2 .
3. z = x2 + y 2 − xy + x + y.
4. z = 4(x − y) − x2 − y 2 ,
D : x + 2y = 4, x − 2y = 4, x = 0.
Âàðèàíò  24.
1. z = y 2 + 3xy − x4 .
(
)
2. z = ln e2x + e−y ,
x = t4 ,
y = t3 .
3. z = 2x3 − xy 2 + 5x2 + y 2 .
4. z = 6xy − 9x2 − 9y 2 + 4x + 4y,
D : x = 0, y = 0, x = 1, y = 2.
Âàðèàíò  25.
(
)
1. z = arcsin x2 + y 3 .
2. z = x2 + (x + y)2 ,
x = et ,
y = cos t.
3. z = 3x2 − x3 + 3y 2 + 4y.
4. z = x4 − y 4 ,
D : x2 + y 2 = 1.
14
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ
Âàðèàíò 1.
Çàäàíèå 1.
Íàéäèòå ïîëíûé äèôôåðåíöèàë ôóíêöèè z = x2 y 3 cos x − 4x.
Ðåøåíèå.
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè:
(
)
∂z
= y 3 2x cos x − x2 sin x − 4 = 2xy 3 cos x − x2 y 3 sin x − 4,
∂x
∂z
= 3y 2 x2 cos x.
∂y
Òîãäà ïîëíûé äèôôåðåíöèàë ôóíêöèè â ñîîòâåòñòâèè ñ ôîðìóëîé (1) ðàâåí
(
)
dz = 2xy 3 cos x − x2 y 3 sin x − 4 dx + 3y 2 x2 cos xdy.
Çàäàíèå 2.
Íàéäèòå ïðîèçâîäíûå ñëîæíîé ôóíêöèè.
u
z = x3 e−3y , x = sin uv, y = cos .
v
Ðåøåíèå.
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå ýòèõ ôóíêöèé:
∂z
∂z
= 3x2 e−3y ,
= −3x3 e−3y ,
∂x
∂y
∂x
∂x
= v cos uv,
= u cos uv,
∂u
∂v
1
u ∂y
u
u
∂y
= − sin ,
= 2 sin .
∂u
v
v ∂v
v
v
Âîñïîëüçóåìñÿ ôîðìóëàìè (2) è (3):
)
(
∂z ∂x ∂z ∂y
u
∂z
1
2 −3y
3 −3y
=
+
= 3x e v cos uv − 3x e
=
− sin
∂u ∂x ∂u ∂y ∂u
v
v
(
x
u)
u
2 −3y
= 3x e
v cos uv + sin
, ãäå x = sin uv, y = cos .
v
v
v
∂z
∂z ∂x ∂z ∂y
u
u
=
+
= 3x2 e−3y u cos uv − 3x3 e−3y 2 sin =
∂v
∂x ∂v ∂y ∂v
v
v
15
2 −3y
= 3x e
(
x
u)
u cos uv − 2 sin
,
v
v
u
v
ãäå x = sin uv, y = cos .
Çàäàíèå 3.
Èññëåäóéòå ôóíêöèþ z = x3 + 3xy 2 − 15x − 12y íà ýêñòðåìóì.
Ðåøåíèå.
Íàéäåì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè.
∂z
= 3x2 + 3y 2 − 15,
∂x
∂z
= 6xy − 12.
∂y
Ýòè ïðîèçâîäíûå îïðåäåëåíû è íåïðåðûâíû íà âñåé ïëîñêîñòè. Äëÿ íàõîæäåíèÿ ñòàöèîíàðíûõ òî÷åê ïðèðàâíÿåì ÷àñòíûå ïðîèçâîäíûå ê íóëþ è
ïîëó÷èì ñèñòåìó óðàâíåíèé:
{ 2
3x + 3y 2 − 15 = 0,
6xy − 12 = 0.
Îòêóäà èìååì
{
x2 + y 2 = 5,
xy = 2.
Óìíîæèì âòîðîå óðàâíåíèå íà 2 è ïðèáàâèì ê ïåðâîìó, ïîëó÷èì
x2 + 2xy + y 2 = 9, ò. å. (x + y)2 = 9.
Ñëåäîâàòåëüíî x + y = −3 èëè x + y = 3. Èìååì äâå ñèñòåìû:
{
x + y = −3,
xy = 2.
{
x + y = 3,
xy = 2.
Ïîëó÷àåì ÷åòûðå ðåøåíèÿ, ò. å. ÷åòûðå ñòàöèîíàðíûå òî÷êè M1 (−2; −1),
M2 (−1; −2), M3 (2; 1), M4 (1; 2).
Íàõîäèì âòîðûå ÷àñòíûå ïðîèçâîäíûå:
∂ 2z
∂ 2z
∂ 2z
= 6x,
= 6y,
= 6x.
∂x2
∂x∂y
∂y 2
Ïðîâåðÿåì âûïîëíåíèå äîñòàòî÷íûõ óñëîâèé ñóùåñòâîâàíèÿ ýêñòðåìóìà
äëÿ êàæäîé èç ñòàöèîíàðíûõ òî÷åê.
Äëÿ M1 (−2; −1) :
A1 = 6 · (−2) = −12, B1 = 6 · (−1) = −6, C1 = 6 · (−2) = −12,
16
∆1 =
−12 −6
= 144 − 36 = 108 > 0,
−6 −12
ñëåäîâàòåëüíî, â òî÷êå M1 (−2; −1) åñòü ýêñòðåìóì è, ò.ê. A1 = −12 < 0,
òî ýòî ëîêàëüíûé ìàêñèìóì, ïðè÷åì
zmax (−2; −1) = (−2)3 + 3(−2)(−1)2 − 15(−2) − 12(−1) = 28.
Äëÿ M2 (−1; −2) :
A2 = −6, B2 = −12, C2 = −6,
∆2 =
−6 −12
= 36 − 144 =< 0,
−12 −6
ñëåäîâàòåëüíî, â òî÷êå M2 (−1; −2) ýêñòðåìóìà íåò.
Äëÿ M3 (2; 1) :
A3 = 12, B3 = 6, C3 = 12,
∆3 =
12 6
= 144 − 36 = 108 > 0,
6 12
ñëåäîâàòåëüíî, â òî÷êå M3 (2; 1) åñòü ýêñòðåìóì è, ïîñêîëüêó A3 = 12 > 0,
òî ýòî ëîêàëüíûé ìèíèìóì, ïðè÷åì
zmin (2; 1) = 23 + 3 · 2 · 12 − 15 · 2 − 12 · 1 = −28.
Äëÿ M4 (1; 2) :
A4 = 6, B4 = 12, C4 = 6,
∆4 =
6 12
= 36 − 144 =< 0,
12 6
ñëåäîâàòåëüíî, â òî÷êå M4 (1; 2) ýêñòðåìóìà íåò.
Çàäàíèå 4.
Íàéäèòå íàèáîëüøåå è íàèìåíüøåå çíà÷åíèÿ ôóíêöèè
z = x2 − xy + y 2 − 4x
â çàìêíóòîé îáëàñòè D̄, îãðàíè÷åííîé ëèíèÿìè
x = 0, y = 0, 2x + 3y − 12 = 0.
Ðåøåíèå.
Èçîáðàçèì îáëàñòü D̄ íà ÷åðòåæå (ñì. ðèñ. 1).
17
Ðèñóíîê 1 ×åðòåæ îáëàñòè D (ê çàäàíèþ 4 ïåðâîãî âàðèàíòà).
Íàéäåì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè
∂z
= 2x − y − 4,
∂x
∂z
= −x + 2y.
∂y
Ïðèðàâíÿåì ÷àñòíûå ïðîèçâîäíûå ê íóëþ è íàéäåì ñòàöèîíàðíûå òî÷êè:
{
2x − y − 4 = 0,
−x + 2y = 0.
Ðåøàÿ ñèñòåìó, ïîëó÷èì òî÷êó M1 (2 23 ; 1 13 ). Íàíåñåì åå íà ÷åðòåæ 1, âèäèì,
÷òî M1 ∈ D̄.
Ãðàíèöà îáëàñòè D̄ ñîñòîèò èç òðåõ îòðåçêîâ ïðÿìûõ. Èññëåäóåì ôóíêöèþ íà êàæäîì èç ýòèõ îòðåçêîâ.
Íà îòðåçêå OA, ïðÿìîé y = 0, èìååì
z = x2 − 4x, x ∈ [0; 6].
Íàõîäèì ñòàöèîíàðíóþ òî÷êó:
z ′ = 2x − 4, 2x − 4 = 0, x = 2.
Ïîëó÷àåì òî÷êó M2 (2; 0).
Íà îòðåçêå AB ïðÿìîé 2x + 3y − 12 = 0 :
y=
12 − 2x
, ïîäñòàâëÿåì â ôóíêöèþ z :
3
18
12 − 2x
z =x −x
+
3
2
(
12 − 2x
3
)2
− 4x =
19 2 40
x − x + 16, x ∈ [0; 6].
9
3
Íàõîäèì:
zx′ =
38
40
x− .
9
3
Ïðèðàâíèâàÿ zx′ = 0, ïîëó÷àåì
3
∈ (0; 6),
19
è, ñëåäîâàòåëüíî,
x=3
3
12 − 2 · 3 19
17
=1 .
y=
3
19
)
(
3
; 1 17
Ïîëó÷èëè ñòàöèîíàðíóþ òî÷êó M3 3 19
19 .
Íà îòðåçêå OB ïðÿìîé x = 0 èìååì z = y 2 . Òîãäà z ′ = 2y è z ′ = 0 ïðè
y = 0, ò. å. íà ãðàíèöå îòðåçêà [0; 4].
Äîáàâèì ê íàéäåííûì ðàíåå òî÷êàì
(
)
(
)
2 1
3 17
M1 2 ; 1
, M2 (2; 0), M3 3 ; 1
3 3
19 19
óãëîâûå òî÷êè
O(0; 0), A(6; 0), B(0; 4).
Âû÷èñëèì çíà÷åíèÿ ôóíêöèè âî âñåõ âûáðàííûõ òî÷êàõ:
2 1
1
z(2 ; 1 ) = −5 ;
3 3
3
z(2; 0) = −4;
(
)
3 17
z 3 ;1
≈ −5, 053;
19 19
z(0; 0) = 0; z(6; 0) = 12; z(0; 4) = 16.
Òàêèì îáðàçîì, íàèáîëüøåå çíà÷åíèå ôóíêöèè z(0; 4) = 16 äîñòèãàåòñÿ â óãëîâîé ãðàíè÷íîé òî÷êå îáëàñòè, à íàèìåíüøåå çíà÷åíèå ôóíêöèè
z(2 32 ; 1 13 ) = −5 13 âî âíóòðåííåé òî÷êå îáëàñòè.
Âàðèàíò 2.
Çàäàíèå 1.
Íàéäèòå ïîëíûé äèôôåðåíöèàë ôóíêöèè z = arctg xy −
19
√
3
x.
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè:
Ðåøåíèå.
1
∂z
y
=
− √
,
3
2
2
∂x 1 + x y
3 x2
∂z
x
=
.
∂y
1 + x2 y 2
Òîãäà ïîëíûé äèôôåðåíöèàë ôóíêöèè â ñîîòâåòñòâèè ñ ôîðìóëîé (1) ðàâåí
(
dz =
1
y
√
−
1 + x2 y 2 3 3 x2
)
dx +
x
dy.
1 + x2 y 2
Çàäàíèå 2.
Íàéäèòå ïðîèçâîäíóþ ñëîæíîé ôóíêöèè:
z = ln(y 2 − 3x + 4), ãäå x = e2t , y = sin t.
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè z = f (x, y) è ïðîèçâîäíûå ôóíêöèé x = x(t) è y = y(t) :
Ðåøåíèå.
∂z
−3
∂z
2y
= 2
,
= 2
,
∂x y − 3x + 4 ∂y
y − 3x + 4
dx
dy
= 2e2t ,
= cos t.
dt
dt
Âîñïîëüçóåìñÿ ôîðìóëîé (4):
dz
∂z dx ∂z dy
6e2t
2y cos t
=
+
=− 2
+ 2
,
dt
∂x dt ∂y dt
y − 3x + 4 y − 3x + 4
ãäå x = e2t , y = sin t.
Çàäàíèå 3.
Èññëåäóéòå ôóíêöèþ z = x2 − xy + y 2 + 9x − 6y + 10 íà ýêñòðåìóì.
Ðåøåíèå.
Íàéäåì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè:
∂z
= 2x − y + 9,
∂x
∂z
= −x + 2y − 6.
∂y
20
Ýòè ïðîèçâîäíûå îïðåäåëåíû è íåïðåðûâíû íà âñåé ïëîñêîñòè. Äëÿ íàõîæäåíèÿ ñòàöèîíàðíûõ òî÷åê ïðèðàâíÿåì ÷àñòíûå ïðîèçâîäíûå ê íóëþ è
ïîëó÷èì ñèñòåìó óðàâíåíèé:
{
2x − y + 9 = 0,
−x + 2y − 6 = 0.
Ðåøàÿ ñèñòåìó, ïîëó÷àåì îäíó ñòàöèîíàðíóþ òî÷êó M0 (−4; 1).
Íàõîäèì âòîðûå ÷àñòíûå ïðîèçâîäíûå:
∂ 2z
∂ 2z
∂ 2z
= 2,
= −1,
= 2.
∂x2
∂x∂y
∂y 2
Âòîðûå ÷àñòíûå ïðîèçâîäíûå ïîñòîÿííûå, ïîýòîìó âû÷èñëÿòü çíà÷åíèå
â ñòàöèîíàðíîé òî÷êå íå íàäî. Ïðîâåðÿåì âûïîëíåíèå äîñòàòî÷íûõ óñëîâèé
ñóùåñòâîâàíèÿ ýêñòðåìóìà.
A = 2, B = −1, C = 2,
∆=
2 −1
= 4 − 1 = 3 > 0,
−1 2
ñëåäîâàòåëüíî, â òî÷êå M0 (−4; 1) ôóíêöèÿ èìååò ýêñòðåìóì è, ïîñêîëüêó
A = 2 > 0, òî ýòî ëîêàëüíûé ìèíèìóì, ïðè÷åì
zmin (−4; 1) = (−4)2 − (−4) · 1 + 12 + 9(−4) − 6 · 1 + 10 = −11.
Çàäàíèå 4.
Íàéäèòå íàèáîëüøåå è íàèìåíüøåå çíà÷åíèÿ ôóíêöèè
z = x2 − xy + 4
â çàìêíóòîé îáëàñòè D̄, îãðàíè÷åííîé ëèíèÿìè
y = x2 + 1, y = 5.
Èçîáðàçèì îáëàñòü D̄ íà ÷åðòåæå (ñì. ðèñóíîê 2). Íàéäåì ÷àñòíûå ïðîèçâîäíûå ôóíêöèè:
Ðåøåíèå.
∂z
= 2x − y,
∂x
∂z
= −x.
∂y
21
Ðèñóíîê 2 ×åðòåæ îáëàñòè D ê çàäà÷å 4 (âòîðîé âàðèàíò).
Ïðèðàâíÿåì ÷àñòíûå ïðîèçâîäíûå ê íóëþ è íàéäåì ñòàöèîíàðíûå òî÷êè:
{
2x − y = 0,
−x = 0.
Ïîëó÷àåì îäíó ñòàöèîíàðíóþ òî÷êó O(0; 0), íî O(0; 0) ∈
/ D̄.
Ãðàíèöà îáëàñòè D̄ ñîñòîèò èç ïàðàáîëû è îòðåçêà ïðÿìîé. Èññëåäóåì
ôóíêöèþ íà êàæäîì ó÷àñòêå ãðàíèöû.
Íà ïàðàáîëå y = x2 + 1, x ∈ [−2; 2] èìååì:
(
)
z = x2 − x x2 + 1 + 4 = x2 − x3 − x + 4, x ∈ [−2; 2].
Íàõîäèì ñòàöèîíàðíûå òî÷êè:
z ′ = 2x − 3x2 − 1, z ′ = 0 ïðè 2x − 3x2 − 1 = 0, ò. å. 3x2 − 2x + 1 = 0.
Äàííîå êâàäðàòíîå óðàâíåíèå íå èìååò âåùåñòâåííûõ êîðíåé, ñëåäîâàòåëüíî, ñòàöèîíàðíûõ òî÷åê ôóíêöèÿ íà ýòîé ãðàíèöå íå èìååò.
Íà îòðåçêå ïðÿìîé y = 5 ïîëó÷àåì:
z = x2 − 5x + 4, x ∈ [−2; 2].
Íàõîäèì ñòàöèîíàðíûå òî÷êè:
z ′ = 2x − 5,
1
z ′ = 0 ïðè 2x − 5 = 0, ò. å. x = 2 ∈
/ [−2; 2].
2
22
Ïîëó÷èëè, ÷òî è íà äàííîì ó÷àñòêå ãðàíèöû ôóíêöèÿ íå èìååò ñòàöèîíàðíûõ òî÷åê.
Èòàê, íàèáîëüøåå è íàèìåíüøåå çíà÷åíèÿ ôóíêöèÿ ìîæåò ïðèíèìàòü
òîëüêî â óãëîâûõ òî÷êàõ (òî÷êàõ ïåðåñå÷åíèÿ ëèíèé, îïðåäåëÿþùèõ ãðàíèöó) A(−2; 5) è B(2; 5). Âû÷èñëèì çíà÷åíèÿ ôóíêöèè â ýòèõ òî÷êàõ.
z(−2; 5) = (−2)2 − (−2) · 5 + 4 = 18,
z(2; 5) = 22 − 2 · 5 + 4 = −2.
Òàêèì îáðàçîì, íàèìåíüøåå çíà÷åíèå ôóíêöèÿ äîñòèãàåò â òî÷êå B(2; 5),
à íàèáîëüøåå â òî÷êå A(−2; 5).
23
Êîíòðîëüíàÿ ðàáîòà  6
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  6
Çàäàíèå  1
Èçìåíèòå ïîðÿäîê èíòåãðèðîâàíèÿ.
Çàäàíèå  2
Âû÷èñëèòå äâîéíîé èíòåãðàë.
Çàäàíèå  3
Âû÷èñëèòå ïëîùàäü ôèãóðû, îãðàíè÷åííîé çàäàííûìè ëèíèÿìè.
Çàäàíèå  4
Âû÷èñëèòå îáú¼ì òåëà, îãðàíè÷åííîãî äàííûìè ïîâåðõíîñòÿìè.
Óêàçàíèå.
Ïåðåä ðåøåíèåì çàäà÷ êîíòðîëüíîé ðàáîòû ðåêîìåíäóåòñÿ îçíàêîìèòüñÿ ñî ñëåäóþùèìè ìåòîäè÷åñêèìè óêàçàíèÿìè:
1. Ãðóçäêîâ, À.À. Âû÷èñëåíèå è ïðèëîæåíèÿ äâîéíûõ èíòåãðàëîâ: ìåòîäè÷åñêèå óêàçàíèÿ / À. À. Ãðóçäêîâ, Ì. Á. Êóï÷èíåíêî. ÑÏá.:
ÑÏáÃÒÈ(ÒÓ),- 2013. 58 c.
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû
Äâîéíîé èíòåãðàë îò ôóíêöèè äâóõ ïåðåìåííûõ z = f (x, y) ïî çàìêíóòîé îãðàíè÷åííîé îáëàñòè D îáîçíà÷àåòñÿ ñëåäóþùèì îáðàçîì:
∫∫
f (x, y) dx dy
D
Åñëè ôóíêöèÿ z = f (x, y) îïðåäåëåíà è íåïðåðûâíà â îáëàñòè D, à îáëàñòü
D ýëåìåíòàðíà îòíîñèòåëüíî êîîðäèíàòíûõ îñåé, ò. å. ìîæåò áûòü çàäàíà
îäíèì èç äâóõ ñïîñîáîâ:
D = {(x, y)| a ≤ x ≤ b, φ1 (x) ≤ y ≤ φ2 (x)}
èëè
D = {(x, y)| c ≤ y ≤ d, ψ1 (y) ≤ x ≤ ψ2 (y)},
24
ãäå ôóíêöèè y = φ1 (x), y = φ2 (x), x = ψ1 (y), x = ψ2 (y) íåïðåðûâíû, òî
äâîéíîé èíòåãðàë âû÷èñëÿåòñÿ ïåðåõîäîì ê ïîâòîðíîìó èíòåãðàëó. Ýòîò
ïåðåõîä çàäàåòñÿ ôîðìóëàìè:
)
∫b ( φ∫2 (x)
f (x, y) dx dy =
f (x, y)dy dx,
∫∫
a
D
φ1 (x)
)
∫d ( ψ∫2 (y)
f (x, y) dx dy =
f (x, y)dx dy,
∫∫
c
D
(7)
(8)
ψ1 (y)
Çíà÷åíèå èíòåãðàëà ïðè ýòîì íå çàâèñèò îò ïîðÿäêà èíòåãðèðîâàíèÿ. Åñëè
îáëàñòü D íå ÿâëÿåòñÿ ýëåìåíòàðíîé îòíîñèòåëüíî êîîðäèíàòíûõ îñåé, òî
åå ðàçáèâàþò íà ýëåìåíòàðíûå ÷àñòè, äâîéíîé èíòåãðàë òîãäà áóäåò ðàâåí
ñóììå èíòåãðàëîâ ïî ýëåìåíòàðíûì ÷àñòÿì, è â êàæäîì èç ñëàãàåìûõ ýòîé
ñóììû ïåðåõîäÿò îò äâîéíîãî èíòåãðàëà ê ïîâòîðíîìó. Ïåðåõîä îò âû÷èñëåíèÿ èíòåãðàëà ïî ôîðìóëå (7) ê âû÷èñëåíèþ èíòåãðàëà ïî ôîðìóëå (8) è
íàîáîðîò íàçûâàåòñÿ èçìåíåíèåì ïîðÿäêà èíòåãðèðîâàíèÿ. Ïðè èçìåíåíèè
ïîðÿäêà èíòåãðèðîâàíèÿ ìîæåò îêàçàòüñÿ, ÷òî âìåñòî îäíîãî èíòåãðàëà ïîÿâèòñÿ ñóììà èíòåãðàëîâ, èëè ñóììà èíòåãðàëîâ ìîæåò ïðåîáðàçîâàòüñÿ â
îäèí èíòåãðàë.
Äëÿ êðàòêîñòè îáû÷íî ïèøóò:
φ∫2 (x)
)
∫b ( φ∫2 (x)
∫b
f (x, y)dy dx = dx
f (x, y)dy,
a
a
φ1 (x)
φ1 (x)
ψ∫2 (y)
)
∫d ( ψ∫2 (y)
∫d
f (x, y)dx dy = dy
f (x, y)dx,
c
c
ψ1 (y)
ψ1 (y)
íî ýòè çàïèñè íå èçìåíÿþò ñïîñîáà âû÷èñëåíèÿ.
Ïëîùàäü ïëîñêîé îáëàñòè D ÷èñëåííî ðàâíà äâîéíîìó èíòåãðàëó, îò
ôóíêöèè, òîæäåñòâåííî ðàâíîé åäèíèöå, ò. å.,
∫∫
S=
(9)
dx dy
D
Îáúåì öèëèíäðè÷åñêîãî òåëà, ò. å. ìíîæåñòâà òî÷åê G :
G = {(x, y, z)| (x, y) ∈ D, 0 ≤ z ≤ f (x, y)},
25
ãäå f (x, y) íåïðåðûâíàÿ ôóíêöèÿ, ìîæåò áûòü âû÷èñëåí ïî ôîðìóëå:
∫∫
V (G) =
(10)
f (x, y) dx dy.
D
Îáúåì ïðîèçâîëüíî îãðàíè÷åííîãî òåëà G ⊂ R3 ìîæíî âû÷èñëèòü ñ
ïîìîùüþ òðîéíîãî èíòåãðàëà:
∫∫∫
V (G) =
dx dy dz.
G
Ñàì òðîéíîé èíòåãðàë âû÷èñëÿåòñÿ ïåðåõîäîì ê ïîâòîðíîìó èíòåãðàëó, êîòîðûé äëÿ ýëåìåíòàðíîé îòíîñèòåëüíîé êîîðäèíàòíûõ îñåé ôóíêöèè
ìîæåò áûòü ðàçëè÷íûìè ñïîñîáàìè ñâåäåí ê ïîâòîðíîìó èíòåãðàëó, íàïðèìåð, òàê:
φ∫2 (x)
ψ∫
2 (x,y)
2 (x,y)
) )
∫b ( φ∫2 (x)( ψ∫
∫b
dx dy dz =
dz dy dx = dx
dy
dz.
∫∫∫
a
G
φ1 (x)
a
ψ1 (x,y)
φ1 (x)
ψ1 (x,y)
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  6
Âàðèàíò  1.
∫1
1.
1−x
∫ 2
f (x, y) dy.
dx
−1
∫∫
2.
√
− 1−x2
(
)
12x2 y 2 + 16x3 y 3 dxdy,
√
D : x = 1, y = x2 , y = − x.
D
3. x = 4y − y 2 , x + y = 6.
√
√
4. y = 16 2x, y = 2x, z = 0, x + z = 2.
Âàðèàíò  2.
∫2
1.
∫x
f (x, y) dy.
dx
1
1
x
26
∫∫
2.
(
)
18x2 y 2 + 32x3 y 3 dxdy,
√
D : x = 1, y = x3 , y = − 3 x.
D
3. x = y 2 − 2y, y = −x.
√
√
5x
5 x
4. y = 5 x, y = , z = 0, z = 5 +
.
3
3
Âàðèàíò  3.
3
∫2
1.
∫y+3
dy f (x, y) dx.
0
2y 2
∫∫
2.
(
)
12xy + 9x2 y 2 dxdy,
D : x = 1, y =
√
x, y = −x2 .
D
3. y 2 = 4x + 4, y = 2 − x.
√
4. x2 + y 2 = 2, y = x, z = 0, y = 0, z = 15x.
Âàðèàíò  4.
∫4
1.
√
25−x2
∫
dx
f (x, y) dy.
0
∫∫
2.
3x
4
(
)
8xy + 9x2 y 2 dxdy,
D : x = 1, y = −x2 , y =
√
3
x.
D
3. 3y 2 = 25x, 5x2 = 9y.
√
4. x + y = 2, y = x, z = 0, z = 12y.
Âàðèàíò  5.
∫1
1.
0
∫x+3
dx f (x, y) dy.
∫∫
2.
2x2
(
)
18x2 y 2 + 32x3 y 3 dxdy,
D
27
D : x = 1, y = −x2 , y =
√
3
x.
3. y = x2 , 4y = x2 , y = 4.
√
√
1
4. x = 20 2y, x = 5 2y, z = 0, z + y = .
2
Âàðèàíò  6.
5y
∫0
1.
∫4
dy
−4
∫∫
2.
−
f (x, y) dx.
√
9+y 2
(
)
27x2 y 2 + 48x3 y 3 dxdy,
D : x = 1, y = −x3 , y =
√
x.
D
3. xy = 4, y = x, x = 4.
√
5 y
5y
5
√
4. y =
, x = , z = 0, z = (3 + y) .
2
6
6
Âàðèàíò  7.
∫0
1.
∫x+3
dx f (x, y) dy.
−1
∫∫
2.
2x2
(
)
4xy + 3x2 y 2 dxdy,
√
D : x = 1, y = x2 , y = − x.
D
3. x = 4 − y 2 , x + 2y − 4 = 0.
√
4. x2 + y 2 = 2, x = y, x = 0, z = 0, z = 30y.
Âàðèàíò  8.
∫1
1.
x∫2 +1
dx
0
∫∫
2.
f (x, y) dy.
−1
(
)
8xy + 18x2 y 2 dxdy,
D : x = 1, y = −x2 , y =
D
3. y = x2 , 4y = x2 , x = 2, x = −2.
12x
√
4. x + y = 2, x = y, z = 0, z =
.
5
28
√
3
x.
Âàðèàíò  9.
∫1
1.
∫3−y
dy f (x, y) dx.
0
2y 2
∫∫ (
2.
)
4
xy + 9x2 y 2 dxdy,
5
D : x = 1, y = −x3 , y =
√
x.
D
3. x + 4 = y 2 , x + 3y = 0.
√
√
1
4. y = 17 2x, y = 2 2x, z = 0, x + z = .
2
Âàðèàíò  10.
∫4
1.
√
2
∫25−y
f (x, y) dx.
dy
3y
4
0
∫∫
2.
(
)
6xy + 24x3 y 3 dxdy,
D : x = 1, y = −x2 , y =
√
x.
D
3. y = x2 , y = 6 − x, y = 0.
√
√
5 x
5x
5 (3 + x)
4. y =
, y = , z = 0, z =
.
3
9
9
Âàðèàíò  11.
∫0
1.
∫3−x
dx f (x, y) dy.
3
−2
∫∫
2.
2x2
(
)
4xy + 16x3 y 3 dxdy,
√
D : x = 1, y = x3 , y = − 3 x.
D
3. y = 4 − x2 , y = x + 2.
√
15x
.
4. x2 + y 2 = 8, y = 2x, y = 0, z = 0, z =
11
29
Âàðèàíò  12.
∫4
1.
√ 2
∫9+y
dy
f (x, y) dx.
0
2.
5y
4
)
∫∫ (
25 4 4
2 2
6x y + x y dxdy,
3
√
D : x = 1, y = x2 , y = − x.
D
5
.
x
√
4. x + y = 4, y = 2x, z = 0, z = 3y.
3. y = 6 − x, y =
Âàðèàíò  13.
∫1
1.
∫2−x
dx f (x, y) dy.
x
0
∫∫
2.
(
)
xy − 4x3 y 3 dxdy,
√
D : x = 1, y = x3 , y = − x.
D
3. y = x3 , y = 4x.
(
√ )
√
5 3+ y
5 y
5y
4. x =
, x = , z = 0, z =
.
6
18
18
Âàðèàíò  14.
∫2
1.
0
2.
∫x+2
dx f (x, y) dy.
x2
)
∫∫ (
50 4 4
2 2
3x y + x y dxdy,
3
D : x = 1, y = −x3 , y =
D
3. y 2 = 9x, y = x + 2.
√
√
4. x = 19 2y, x = 4 2y, z = 0, y + z = 2.
30
√
3
x.
Âàðèàíò  15.
∫1
1.
∫x2
dx
0
∫∫
2.
f (x, y) dy.
−x2
(
)
44xy + 16x3 y 3 dxdy,
√
D : x = 1, y = x2 , y = − 3 x.
D
2
3. y = x + 1, y 2 = 9 − x.
√
30y
2
2
.
4. x + y = 8, x = 2y, x = 0, z = 0, z =
11
Âàðèàíò  16.
∫1
1.
4−x
∫ 2
f (x, y) dy.
dx
0
∫∫
2.
2x+1
y cos xy dxdy,
D : x = 1, x = 2, y =
π
, y = π.
2
D
3. xy = 4, x + y − 5 = 0.
√
3x
4. x + y = 4, x = 2y, z = 0, z = .
5
Âàðèàíò  17.
∫1
1.
∫x
dx
0
∫∫
f (x, y) dy.
−x
y 2 sin
2.
xy
dxdy,
2
D : x = 0, y =
√
x
, y = π.
2
D
3. x − y + 1 = 0, y = cos x, y = 0.
√
√
4. y = 6 3x, y = 3x, z = 0, x + z = 3.
Âàðèàíò  18.
√
x
∫2
∫2
1.
f (x, y) dy.
dx
0
x2
4
31
∫∫
−
2
2.
y e
xy
4
dxdy,
D : x = 0, y = 2, y = x.
D
3. y = 2x − x2 , y = x.
√
√
5 x
5x
5 (3 + x)
4. y =
, y = , z = 0, z =
.
6
18
18
Âàðèàíò  19.
∫3
1.
√
25−x2
∫
dx
f (x, y) dy.
0
0
∫∫
4ye2xy dxdy,
2.
1
D : x = 1, x = , y = ln 3, y = ln 4.
2
D
√
x, xy = 1, x = 2, y = 0.
√
5x
4. x2 + y 2 = 18, y = 3x, y = 0, z = 0, z = .
11
3. y =
Âàðèàíò  20.
∫4
1.
√
25−x2
∫
f (x, y) dy.
dx
0
0
√
∫∫
2
2.
4y sin xy dxdy,
D : x = 0, y = x, y =
π
.
2
D
3. y = −x2 + 4, 2x + y − 4 = 0.
√
4. x + y = 6, y = 3x, z = 0, z = 4y.
Âàðèàíò  21.
∫1
1.
0
∫2−y
dy f (x, y) dx.
∫∫
2.
0
y sin xy dxdy,
D : x = 1, x = 2, y =
D
32
π
, y = π.
2
3. y = x2 + 2, x + y = 4.
√
√
4. x = 7 3y, x = 2 3y, z = 0, x + z = 3.
Âàðèàíò  22.
∫4
1.
√
∫x
dx f (x, y) dy.
1
0
∫∫
2
2.
−
y e
xy
2
dxdy,
D : x = 0, y = x, y =
√
2.
D
3. y = −x2 + 8, y = x2 .
4. z = x2 + y 2 , y = x2 , z = 0, y = 1.
Âàðèàíò  23.
∫2
1.
∫3−x
dx f (x, y) dy.
0
0
∫∫
2.
2y cos 2xy dxdy,
D : x = 1, x = 2, y =
π
π
, y=y= .
4
2
D
√
√
3. y = 2 x + 1, y = 4 − 2x, y = 0.
4. y = x2 , z + y = 2, x = 0, z = 0.
Âàðèàíò  24.
3
∫4
1.
∫x
dx
0
∫∫
f (x, y) dy.
x2
8ye4xy dxdy,
2.
1
1
D : x = , x = , y = ln 3, y = ln 4.
4
2
D
3. y = (x + 1)2 , y 2 = x + 1.
4. y + z = 1, x = y 2 + 1, x = 0, y = 0, z = 0.
33
Âàðèàíò  25.
1
∫2
1.
√
∫y
dx f (x, y) dy.
y
0
∫∫
2.
xy
dxdy,
3y 2 sin
2
2x
D : x = 0, y = , y =
3
√
4π
.
3
D
3. y = (x − 2)3 , y = 4x − 8.
√
4. z = 1 − y, x2 = y, z = 0.
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ
Âàðèàíò 1.
Çàäàíèå 1.
Èçìåíèòå ïîðÿäîê èíòåãðèðîâàíèÿ
∫y2
∫2
f (x, y)dx.
dy
1
0
Ðåøåíèå.
Èçîáðàçèì íà ÷åðòåæå îáëàñòü èíòåãðèðîâàíèÿ (ñì. ðèñóíîê 3).
Ðèñóíîê 3 Îáëàñòü èíòåãðèðîâàíèÿ D ê çàäà÷å 1 (âàðèàíò 1)
34
D = {(x, y)| 0 ≤ x ≤ y 2 , 1 ≤ y ≤ 2} :
Çàìåòèì, ÷òî äàííàÿ îáëàñòü D ýëåìåíòàðíà îòíîñèòåëüíî îñè Ox, íî
íåýëåìåíòàðíà îòíîñèòåëüíî îñè Oy , è, ñëåäîâàòåëüíî, ïðè èçìåíåíèè ïîðÿäêà èíòåãðèðîâàíèÿ åå íåîáõîäèìî ðàçáèòü íà äâå ÷àñòè. Èíòåãðàë ïðè
ýòîì áóäåò ðàâåí ñóììå äâóõ èíòåãðàëîâ ïî ýëåìåíòàðíûì îòíîñèòåëüíî
îñè Oy îáëàñòÿì D1 è D2 (ñì. ðèñ. 4).
Ðèñóíîê 4 Ðàçáèåíèå îáëàñòè èíòåãðèðîâàíèÿ íà ÷àñòè (ê çàäà÷å 1 âàðèàíòà 1).
Ó÷èòûâàÿ, ÷òî y > 0, óðàâíåíèå ëèíèè x = y 2 , îãðàíè÷èâàþùåé îáëàñòü
√
D, ïåðåïèøåì â âèäå y = x.
Èìååì:
∫y2
∫2
f (x, y)dx =
dy
1
∫1
0
∫2
dx
0
∫4
f (x, y)dy +
1
∫2
dx
1
f (x, y)dy.
√
x
Çàäàíèå 2.
Âû÷èñëèòü äâîéíîé èíòåãðàë
∫∫
xy
y 2 e− 2
{
dx dy; ãäå D :
D
x = 0, y = 1,
y = x2 .
Èçîáðàçèì íà ÷åðòåæå îáëàñòü D (ñì.ðèñ. 5).
Ïåðåéäåì îò äâîéíîãî èíòåãðàëà ê ïîâòîðíîìó è âû÷èñëèì åãî.
Ðåøåíèå.
∫∫
D
xy
y 2 e− 2
∫1
dx dy =
∫2y
dy
0
xy
y 2 e− 2 dx
0
35
[
]
2 ( xy )
= dx = − d −
=
y
2
Ðèñóíîê 5 Îáëàñòü èíòåãðèðîâàíèÿ D ê çàäà÷å 2 (âàðèàíò 1)
∫2y
∫2y xy (
∫1
xy ( xy )
xy )
− 2
− 2
dy (−2y)e
d −
= −2 ydy e
d −
=
2
2
∫1
=
0
0
∫1
= −2
0
xy 2y
− 2
dy
ye
∫1 (
∫1
∫1
)
2
2
= −2 y e−y − 1 dy = −2 ye−y dy + 2 dy =
0
0
0
0
0
[
] ∫1
(
)
1 ( 2)
2
= ydy = − d −y
= e−y d −y 2 + 2y
2
0
= e−1 − 1 + 2 − 0 =
1
0
−y 2
1
=e
0
1
+ 2y
0
=
0
1
+ 1.
e
Çàäàíèå 3.
Âû÷èñëèòü ïëîùàäü ôèãóðû, îãðàíè÷åííîé ëèíèÿìè y = (x − 1)2 ,
y 2 = x − 1.
Èçîáðàçèì íà ÷åðòåæå îáëàñòü D, ïëîùàäü êîòîðîé íàäî âû÷èñëèòü, (ñì. ðèñ. 6). Ïëîùàäü îáëàñòè D, (ñì. ôîðìóëó(9)) ðàâíà
Ðåøåíèå.
∫∫
S=
dx dy =
D
√
∫x−1
∫2
dx
1
(x−1)2
√
x−1
∫2
dy =
dx =
y
1
∫2
(x−1)2
36
1
)
(√
x − 1 − (x − 1)2 dx =
Ðèñóíîê 6 Îáëàñòü èíòåãðèðîâàíèÿ D â çàäà÷å 3 (âàðèàíò 1).
∫2
=
)
(√
x − 1 − (x − 1)2 d(x − 1) =
(
3
2
(x − 1)3
(x − 1) 2 −
3
3
1
=
2 1 1
− = (êâ.
3 3 3
)
2
=
1
åä.).
Çàäàíèå 4.
Âû÷èñëèòü îáúåì òåëà, îãðàíè÷åííîãî ïîâåðõíîñòÿìè
z = x2 + 3y 2 , x + y = 1, x = 0, y = 0, z = 0.
Òåëî G, îáúåì êîòîðîãî íàäî âû÷èñëèòü, ïðåäñòàâëÿåò ñîáîé
öèëèíäðè÷åñêîå òåëî, îãðàíè÷åííîå ïàðàáîëîèäîì z = x2 + 3y 2 , âåðòèêàëüíîé ïëîñêîñòüþ x + y = 1 è êîîðäèíàòíûìè ïëîñêîñòÿìè (ñì. ðèñ. 7).
Îáúåì òåëà G ìîæíî âû÷èñëèòü ñ ïîìîùüþ äâîéíîãî èíòåãðàëà ïî ïðîåêöèè òåëà â ïëîñêîñòü XOY . Ïðîåêöèÿ (D) èçîáðàæåíà íà ðèñóíêå 8.
Òàêèì îáðàçîì,
Ðåøåíèå.
G = {(x, y, z)| (x, y) ∈ D, 0 ≤ z ≤ x2 + 3y 2 }.
Òîãäà ñîãëàñíî ôîðìóëå (10)
∫∫
∫1
(x2 + 3y 2 ) dx dy =
V (G) =
D
∫1−x
dx (x2 + 3y 2 )dy =
0
0
37
Ðèñóíîê 7 Ðèñóíîê îáëàñòè G (ê çàäà÷å 4 âàðèàíòà 1)
∫1
(x2 y + y 3 )
=
dx =
0
0
∫1
=
∫1
1−x
(
x2 − x
)
3
=
0
∫1
dx −
0
(
( 2
)
x (1 − x) + (1 − x)3 dx =
(1 − x)3 d(1 − x) =
0
x3 x4 (1 − x)4
−
−
3
4
4
)
1
=
0
1 1 1 1
− + =
3 4 4 3
Âàðèàíò 2.
Çàäàíèå 1.
Èçìåíèòå ïîðÿäîê èíòåãðèðîâàíèÿ
∫1
x∫2 +1
dx
0
f (x, y)dy.
0
38
(êóá.åä.).
Ðèñóíîê 8 Ïðîåêöèÿ îáëàñòè G â ïëîñêîñòü XOY
(ê çàäà÷å 4 âàðèàíòà 1)
Èçîáðàçèì íà ÷åðòåæå îáëàñòü èíòåãðèðîâàíèÿ (ñì.ðèñ. 9).
Ðåøåíèå.
D = {(x, y)| 0 ≤ x ≤ 1, 0 ≤ y ≤ x2 + 1}
Äàííàÿ îáëàñòü ýëåìåíòàðíà îòíîñèòåëüíî îñè Oy, íî íåýëåìåíòàðíà îòíîñèòåëüíî îñè Ox. Ïðè èçìåíåíèè ïîðÿäêà èíòåãðèðîâàíèÿ åå íåîáõîäèìî
ðàçáèòü íà äâå ÷àñòè. Èíòåãðàë ïðè ýòîì áóäåò ðàâåí ñóììå äâóõ èíòåãðàëîâ ïî ýëåìåíòàðíûì îòíîñèòåëüíî îñè Ox îáëàñòÿì D1 è D2 (ñì.ðèñ. 10).
Ó÷èòûâàÿ, ÷òî x ≥ 0, óðàâíåíèå êðèâîé y = x2 + 1, îãðàíè÷èâàþùåé
√
îáëàñòü D, ïåðåïèøåì â âèäå x = y − 1. Òîãäà
∫1
x∫2 +1
dx
0
∫1
f (x, y)dy =
0
∫1
dy
0
∫2
f (x, y)dx +
0
∫1
dy
1
√
f (x, y)dx.
y−1
Çàäàíèå 2.
Âû÷èñëèòå äâîéíîé èíòåãðàë
{
∫∫
(9x2 y 2 + 48x3 y 3 ) dx dy, ãäå D :
x = 1, y =
y = −x2 .
√
x,
D
Èçîáðàçèì íà ÷åðòåæå îáëàñòü D (ñì.ðèñ. 11). Ïåðåéäåì îò
äâîéíîãî èíòåãðàëà ê ïîâòîðíîìó è âû÷èñëèì åãî.
Ðåøåíèå.
∫∫
∫1
(9x2 y 2 + 48x3 y 3 ) dx dy =
D
√
∫x
dx (9x2 y 2 + 48x3 y 3 )dy =
−x2
0
39
Ðèñóíîê 9 Ðèñóíîê îáëàñòè èíòåãðèðîâàíèÿ D (ê çàäà÷å 1 âàðèàíòà 2)
Ðèñóíîê 10 Ðàçáèåíèå îáëàñòè èíòåãðèðîâàíèÿ íà ÷àñòè (ê çàäà÷å 1 âàðèàíòà 2)
40
Ðèñóíîê 11 Ðèñóíîê îáëàñòè D (ê çàäà÷å 2 âàðèàíòà 2)
√
x
∫1
∫1
(3x2 y 3 + 12x3 y 4 )
=
−x2
0
(
=
2 9
1
x 2 + 2x6 + x9 − x12
3
3
)
7
(3x 2 + 12x5 + 3x8 − 12x11 )dx =
dx =
0
1
=
0
1
2
+ 2 + − 1 = 2.
3
3
Çàäàíèå 3.
Âû÷èñëèòå ïëîùàäü ôèãóðû, îãðàíè÷åííîé ëèíèÿìè
√
3
y = 3 x, y = , x = 9.
x
Èçîáðàçèì íà ÷åðòåæå îáëàñòü D, ïëîùàäü êîòîðîé íàäî âû÷èñëèòü, (ñì.ðèñ. 12). Äëÿ òîãî, ÷òîáû íàéòè òî÷êó ïåðåñå÷åíèÿ êðèâûõ
√
y = 3 x è y = x3 , ðåøèì óðàâíåíèå:
Ðåøåíèå.
√
√
3
1
3 x = ⇔ x = ⇔ x = 1. Òîãäà y = 3.
x
x
Ñîãëàñíî ôîðìóëå (9) ïëîùàäü îáëàñòè D áóäåò ðàâíà:
∫∫
S=
dx dy =
D
√
x
∫3
∫9
dy =
dx
3
x
1
√
3 x
∫9
y
1
(
3
x
dx = 3
∫9 (
√
)
1
dx =
x−
x
1
)9
√
= 2x x − 3 ln x = 54 − 3 ln 9 − 2 = 52 − 6 ln 3 (êâ.
1
41
åä.).
Ðèñóíîê 12 Ðèñóíîê îáëàñòè D (ê çàäà÷å 3 âàðèàíòà 2)
Çàäàíèå 4.
Âû÷èñëèòå îáúåì òåëà, îãðàíè÷åííîãî ïîâåðõíîñòÿìè
√
√
x = 3 2y, x = 2 2y, y + z = 2, z = 0.
Òåëî G, îáúåì êîòîðîãî íàäî âû÷èñëèòü, ýòî öèëèíäðè÷åñêîå
√
òåëî, îãðàíè÷åííîå äâóìÿ ïîëóïàðàáîëè÷åñêèìè öèëèíäðàìè x = 3 2y è
√
x = 2 2y, ïëîñêîñòüþ y +z = 2, ïàðàëëåëüíîé îñè Ox è ïëîñêîñòüþ XOY .
Èçîáðàçèì òåëî G (ñì.ðèñ. 13), à òàêæå åãî ïðîåêöèþ D íà ïëîñêîñòü XOY
(ñì. ðèñ. 14).
Òàêèì îáðàçîì,
Ðåøåíèå.
G = {(x, y, z)| (x, y) ∈ D, 0 ≤ z ≤ 2 − y}.
Òîãäà ñîãëàñíî ôîðìóëå (10)
∫∫
(2 − y) dx dy =
V (G) =
dy
=
√
√ ∫
(2 − y) 2y dy = 2 ·
0
=
16 16 32
−
=
= (êóá.
3
5
15
√
2 2y
0
D
∫2
√
3∫ 2y
∫2
2
0
√
3 2y
∫2
(2 − y)dx =
(2 − y)x
0
√
2 2y
dy =
(
(
)
)2
√
3
1
4 3 2 5
y2 − y2
2y 2 − y 2 dy = 2 ·
=
3
5
0
åä.).
42
Ðèñóíîê 13 Òåëî G â çàäà÷å 4 âàðèàíòà 2
Ðèñóíîê 14 Ïðîåêöèÿ òåëà G â ïëîñêîñòü XOY
43
(ê çàäà÷å 4 âàðèàíòà 2)
Êîíòðîëüíàÿ ðàáîòà  7
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  7
Çàäàíèå  1
Âû÷èñëèòå êðèâîëèíåéíûé èíòåãðàë ïåðâîãî ðîäà ïî äàííîé ëèíèè.
Çàäàíèå  2
⃗ (x, y) ïðè ïåðåìåùåíèè âäîëü ëèíèè L îò òî÷Âû÷èñëèòå ðàáîòó ñèëû F
êè A äî òî÷êè B .
Çàäàíèå  3
Âû÷èñëèòå ïîâåðõíîñòíûé èíòåãðàë ïåðâîãî ðîäà ïî ïîâåðõíîñòè S , ãäå
S ÷àñòü ïëîñêîñòè π , îòñå÷¼ííàÿ êîîðäèíàòíûìè ïëîñêîñòÿìè.
Çàäàíèå  4
Âû÷èñëèòå ïîâåðõíîñòíûé èíòåãðàë âòîðîãî ðîäà ïî ïîâåðõíîñòè S , ãäå
S ÷àñòü ïëîñêîñòè π , îòñå÷¼ííàÿ êîîðäèíàòíûìè ïëîñêîñòÿìè, â íàïðàâëåíèè íîðìàëè, îáðàçóþùåé îñòðûé óãîë ñ îñüþ Oz . .
Óêàçàíèå.
Ïåðåä ðåøåíèåì çàäà÷ êîíòðîëüíîé ðàáîòû ðåêîìåíäóåòñÿ îçíàêîìèòüñÿ ñî ñëåäóþùèìè ìåòîäè÷åñêèìè óêàçàíèÿìè:
1. Ôàòòàõîâà Ì.Â., Êóï÷èíåíêî Ì.Á. Êðèâîëèíåéíûå èíòåãðàëû. Ðåøåíèå òèïîâûõ çàäà÷: Ìåòîäè÷åñêèå óêàçàíèÿ. ÑÏá.: ÑÏáÃÒÈ(ÒÓ), 2008. 32 c.
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû
Êðèâîëèíåéíûé èíòåãðàë ïåðâîãî ðîäà
Êðèâîëèíåéíûé èíòåãðàë ïåðâîãî (I) ðîäà (ïî äëèíå äóãè êðèâîé) ïî
ïðîñòðàíñòâåííîé êðèâîé L îò ôóíêöèè f = f (x, y, z) îáîçíà÷àåòñÿ
∫
f (x, y, z) dl.
L
44
Åñëè êðèâàÿ ïëîñêàÿ, òî ïîäûíòåãðàëüíàÿ ôóíêöèÿ åñòü ôóíêöèÿ äâóõ
ïåðåìåííûõ è èíòåãðàë èìååò âèä
∫
f (x, y) dl.
L
Åñëè ïîäûíòåãðàëüíàÿ ôóíêöèÿ ïðåäñòàâëÿåò ñîáîé ïëîòíîñòü êðèâîé,
òî êðèâîëèíåéíûé èíòåãðàë I ðîäà çàäàåò ìàññó êðèâîé.
Âû÷èñëåíèå êðèâîëèíåéíîãî èíòåãðàëà ïî äëèíå äóãè çàâèñèò îò ñïîñîáà çàäàíèÿ êðèâîé è ñâîäèòñÿ ê âû÷èñëåíèþ îïðåäåëåííîãî èíòåãðàëà.
1. Åñëè êðèâàÿ L ïðîñòðàíñòâåííàÿ è çàäàíà ïàðàìåòðè÷åñêè:
x = x(t), y = y(t), z = z(t), t ∈ [α; β],
òî äèôôåðåíöèàë äëèíû äóãè
√
dl =
(x′ (t))2 + (y ′ (t))2 + (z ′ (t))2 dt.
(11)
Ñëåäîâàòåëüíî
∫β
∫
f (x, y, z) dl =
√
f (x(t), y(t), z(t)) (x′ (t))2 + (y ′ (t))2 + (z ′ (t))2 dt. (12)
α
L
2. Åñëè êðèâàÿ L ïëîñêàÿ è çàäàíà ïàðàìåòðè÷åñêè:
x = x(t), y = y(t), t ∈ [α; β], òî
∫β
∫
f (x, y) dl =
√
f (x(t), y(t)) (x′ (t))2 + (y ′ (t))2 dt.
α
L
3. Åñëè êðèâàÿ L ïëîñêàÿ è çàäàíà ÿâíûì óðàâíåíèåì:
y = y(x), x ∈ [a; b],
òî äèôôåðåíöèàë äëèíû äóãè ðàâåí
√
dl = 1 + (y ′ (x))2 dx
è
∫
∫b
f (x, y) dl =
L
√
f (x, y(x)) 1 + (y ′ (x))2 dx.
a
4. Åñëè êðèâàÿ L ïëîñêàÿ è çàäàíà ÿâíûì óðàâíåíèåì:
45
x = x(y), y ∈ [c; d], òî
√
dl = 1 + (x′ (y))2 dy,
∫
∫d
f (x, y) dl =
L
(13)
√
f (x(y), y) 1 + (x′ (y))2 dy.
(14)
c
Êðèâîëèíåéíûé èíòåãðàë âòîðîãî ðîäà
Êðèâîëèíåéíûé èíòåãðàë âòîðîãî (II) ðîäà (ïî êîîðäèíàòàì) çàäàåò
ðàáîòó, ñîâåðøàåìóþ ïåðåìåííîé ñèëîé
F⃗ (x, y, z) = P (x, y, z)⃗i + Q(x, y, z)⃗j + R(x, y, z)⃗k
⌣
ïî ïåðåìåùåíèþ ìàòåðèàëüíîé òî÷êè âäîëü ëèíèè L =AB, ò. å. îò òî÷êè
⌣
⌣
A ê òî÷êå B. Çàìåòèì, ÷òî ðàáîòà ïî äóãå AB ðàâíà ðàáîòå ïî äóãå BA,
âçÿòîé ñ ïðîòèâîïîëîæíûì çíàêîì, ò. å. èçìåíåíèå íàïðàâëåíèÿ äâèæåíèÿ
ìåíÿåò çíàê èíòåãðàëà íà ïðîòèâîïîëîæíûé.
Îáîçíà÷àåòñÿ èíòåãðàë ñëåäóþùèì îáðàçîì:
∫
P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz.
(15)
⌣
AB
Òàêæå êàê è äëÿ êðèâîëèíåéíûõ èíòåãðàëîâ I ðîäà, âû÷èñëåíèå êðèâîëèíåéíûõ èíòåãðàëîâ II ðîäà çàâèñèò îò ñïîñîáà çàäàíèÿ êðèâîé è ñâîäèòñÿ ê âû÷èñëåíèþ îïðåäåëåííîãî èíòåãðàëà.
⌣
1. Åñëè êðèâàÿ AB ïðîñòðàíñòâåííàÿ è çàäàíà ïàðàìåòðè÷åñêè óðàâíåíèÿìè
x = x(t), y = y(t), z = z(t), t ∈ [α; β], òî
F⃗ (x, y, z) = P (x, y, z)⃗i + Q(x, y, z)⃗j + R(x, y, z)⃗k
è
∫
∫β
[
P (x, y, z)dx+Q(x, y, z)dy+R(x, y, z)dz = P (x(t), y(t), z(t))x′ (t)+
⌣
α
AB
]
+ Q(x(t), y(t), z(t))y ′ (t) + R(x(t), y(t), z(t))z ′ (t) dt.
46
⌣
2. Åñëè êðèâàÿ AB ïëîñêàÿ è çàäàíà ïàðàìåòðè÷åñêè óðàâíåíèÿìè
x = x(t), y = y(t), t ∈ [α; β], òî
∫
∫β
P (x, y)dx+Q(x, y)dy =
⌣
[P (x(t), y(t))x′ (t) + Q(x(t), y(t))y ′ (t)] dt. (16)
α
AB
⌣
3. Åñëè ïëîñêàÿ êðèâàÿ AB çàäàíà ÿâíûì óðàâíåíèåì:
y = y(x), x ∈ [a; b], ïðè÷åì A(a; y(a)) è B(b; y(b)), òî
∫
∫b
P (x, y)dx + Q(x, y)dy =
⌣
[P (x, y(x)) + Q(x, y(x))y ′ (x)] dx.
(17)
a
AB
⌣
4. Åñëè ïëîñêàÿ êðèâàÿ AB çàäàíà ÿâíûì óðàâíåíèåì:
x = x(y), y ∈ [c; d], ïðè÷åì A(y(c); c) è B(y(d); d), òî
∫
∫d
P (x, y)dx + Q(x, y)dy =
⌣
[P (x(y), y)x′ (y) + Q(x(y), y)] dy.
c
AB
Ïîâåðõíîñòíûé èíòåãðàë ïåðâîãî ðîäà
Ïîâåðõíîñòíûé èíòåãðàë ïåðâîãî (I) ðîäà (ïî ïëîùàäè ïîâåðõíîñòè) îò
ôóíêöèè f = f (x, y, z) ïî ïîâåðõíîñòè S îáîçíà÷àåòñÿ
∫∫
(18)
f (x, y, z) dS.
S
Åñëè ôóíêöèÿ f = f (x, y, z) ïðåäñòàâëÿåò ñîáîé ïëîòíîñòü ïîâåðõíîñòè,
òî ïîâåðõíîñòíûé èíòåãðàë I ðîäà çàäàåò ìàññó ýòîé ïîâåðõíîñòè.
Âû÷èñëåíèå èíòåãðàëà (18) ñâîäèòñÿ ê âû÷èñëåíèþ äâîéíîãî èíòåãðàëà.
Åñëè ïîâåðõíîñòü S çàäàåòñÿ óðàâíåíèåì:
z = z(x, y, ), (x, y) ∈ D ⊂ R2 ,
òî äèôôåðåíöèàë ïëîùàäè ïîâåðõíîñòè
√
dS =
è
( )2
1 + (zx′ )2 + zy′ dxdy
∫∫
∫∫
f (x, y, z) dS =
S
(19)
√
f (x, y, z(x, y))
D
47
( )2
1 + (zx′ )2 + zy′ dxdy.
(20)
Ïîâåðõíîñòíûé èíòåãðàë âòîðîãî ðîäà
Ïîâåðõíîñòíûé èíòåãðàë âòîðîãî (II) ðîäà (ïî êîîðäèíàòàì) ïðåäñòàâëÿåò ñîáîé ïîòîê âåêòîðíîãî ïîëÿ
F⃗ (x, y, z) = P (x, y, z)⃗i + Q(x, y, z)⃗j + R(x, y, z)⃗k
÷åðåç ïîâåðõíîñòü S â íàïðàâëåíèè íîðìàëè ⃗n ê ïîâåðõíîñòè. Íàïðèìåð,
åñëè âçÿòü â êà÷åñòâå âåêòîðíîãî ïîëÿ ïîëå ñêîðîñòåé ÷àñòèö äâèæóùåéñÿ
æèäêîñòè, òî ïîâåðõíîñòíûé èíòåãðàë II ðîäà çàäàåò îáúåì æèäêîñòè,
ïðîòåêàþùèé ÷åðåç ïîâåðõíîñòü S â íàïðàâëåíèè, óêàçàííîì íîðìàëüþ ⃗n.
Îáîçíà÷àåòñÿ èíòåãðàë ñëåäóþùèì îáðàçîì
∫∫
P (x, y, z)dydz + Q(x, y, z)dxdz + R(x, y, z)dxdy.
S
Âû÷èñëåíèå ïîâåðõíîñòíîãî èíòåãðàëà II ðîäà ñâîäèòñÿ ê âû÷èñëåíèþ
äâîéíîãî èíòåãðàëà. Åñëè ïîâåðõíîñòü S çàäàåòñÿ óðàâíåíèåì
z = z(x, y), (x, y) ∈ D ⊂ R2 ,
òî íîðìàëü ê ïîâåðõíîñòè áóäåò ëèáî
⃗n = {−zx′ (x, y); −zy′ (x, y); 1},
(21)
⃗n = {zx′ (x, y); zy′ (x, y); −1},
(22)
ëèáî
ò. å. íîðìàëè (21) è (22) èìåþò ïðîòèâîïîëîæíûå íàïðàâëåíèÿ. Âûáîð íîðìàëè (21) èëè (22) äîëæåí áûòü óêàçàí â óñëîâèè çàäà÷è. Íàïðèìåð, åñëè
óêàçàíî íàïðàâëåíèå íîðìàëè, îáðàçóþùåé îñòðûé óãîë ñ îñüþ Oz, òî ýòî
íîðìàëü (21).
Ïóñòü âûáðàíà íîðìàëü (21). Òîãäà
∫∫
P (x, y, z)dydz + Q(x, y, z)dxdz + R(x, y, z)dxdy =
S
∫∫
=
[
−P (x, y, z(x, y))zx′ (x, y) − Q(x, y, z(x, y))zy′ (x, y)+
D
]
+ R(x, y, z(x, y)) dxdy.
48
(23)
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  7
Âàðèàíò  1.
1.
∫ √
2+
z2
(
2z −
√
x2
+
y2
)
dl,
L
L : x = t cos t, y = t sin t, z = t, 0 6 t 6 2π.
(
)
(
)
2. F⃗ = x2 − 2y ⃗ı + y 2 − 2x ⃗ȷ,
L îòðåçîê ïðÿìîé AB, A(−4; 0), B(0; 2).
∫∫
3.
(2x + 3y + 2z)dS, π : x + 3y + z = 3.
S
∫∫
3xdydz + (y + z)dxdz + (x − z)dxdy,
4.
π : x + 3y + z = 3.
S
Âàðèàíò  2.
∫
1.
(
)
x2 + y 2 dl,
L : x2 + y 2 = 4.
L
(
)
(
)
2. F⃗ = x2 + 2y ⃗ı + y 2 + 2x ⃗ȷ,
L îòðåçîê ïðÿìîé AB, A(−4; 0), B(0; 2).
∫∫
(2 + y − 7x + 9z)dS, π : 2x − y − 2z = −2.
3.
S
∫∫
(3x − 1)dydz + (y − x + z)dxdz + 4zdxdy,
4.
π : 2x − y − 2z = −2.
S
Âàðèàíò  3.
∫
√
1.
L
dl
8 − x2 − y 2
,
L îòðåçîê ïðÿìîé AB, A(0; 0), B(2; 2).
(
)
(
)
x2
= y, A(−4; 0), B(0; 2).
2. F⃗ = x2 + 2y ⃗ı + y 2 + 2x ⃗ȷ, L : 2 −
2
∫∫
3.
(6x + y + 4z)dS, π : 3x + 3y + z = 3.
S
49
∫∫
4.
xdydz + (x + z)dxdz + (y + z)dxdy,
π : 3x + 3y + z = 3.
S
Âàðèàíò  4.
∫
1.
( √
√ )
4 3 x − 3 y dl,
L
L îòðåçîê ïðÿìîé AB, A(−1; 0), B(0; 1).
2. F⃗ = (x + y)⃗ı + 2x⃗ı, L : x2 + y 2 = 4 (y > 0) , A(2; 0), B(−2; 0).
∫∫
(x + 2y + 3z)dS, π : x + y + z = 2.
3.
S
∫∫
(x + z)dydz + (z − x)dxdz + (x + 2y + z)dxdy,
4.
S
π : x + y + z = 2.
Âàðèàíò  5.
∫
1.
L
dl
√
,
5(x − y)
L îòðåçîê ïðÿìîé AB, A(0; 4), B(4; 0).
2. F⃗ = x3⃗ı − y 3⃗ȷ, L : x2 + y 2 = 4 (x > 0, y > 0) , A(2; 0), B(0; 2).
∫∫
3.
(3x − 2y + 6z)dS, π : 2x + y + 2z = 2.
S
∫∫
(y + 2z)dydz + (x + 2z)dxdz + (x − 2y)dxdy,
4.
S
π : 2x + y + 2z = 2.
Âàðèàíò  6.
∫
√
1.
L
y
x2 + y 2
dl,
L : x2 + y 2 = 9 (y > 0) , A(3; 0), B(0; 3).
2. F⃗ = (x + y)⃗ı + (x − y) ⃗ȷ,
L : y = x2 , A(−1; 1), B(1; 1).
50
∫∫
(2x + 5y − z)dS,
3.
π : x + 2y + z = 2.
S
∫∫
(x + z)dydz + 2ydxdz + (x + y − z)dxdy,
4.
π : x + 2y + z = 2.
S
Âàðèàíò  7.
∫
1.
ydl,
L : x = cos3 t, y = sin3 t, A(1; 0), B(0; 1).
L
2. F⃗ = x2 y⃗ı − y⃗ȷ, L îòðåçîê ïðÿìîé AB, A(−1; 0), B(0; 1).
∫∫
3.
(5x − 8y + z)dS, π : 2x − 3y + z = 6.
∫∫
S
(3x − y)dydz + (2y + z)dxdz + (2z − x)dxdy,
4.
π : 2x − 3y + z = 6.
S
Âàðèàíò  8.
∫
1.
ydl,
2
L : y = x,
3
2
(
A(0; 0), B
35
6,
√ )
35
.
3
L
(
)
2. F⃗ = (2xy − y)⃗ı + x2 + x ⃗ȷ, L : x2 + y 2 = 9, A(3; 0), B(−3; 0).
∫∫
3.
(3y − x − z)dS, π : x − y + z = 2.
S
∫∫
(2y + z)dydz + (x − y)dxdz − 2zdxdy,
4.
π : x − y + z = 2.
S
Âàðèàíò  9.
∫
1.
(
)
x2 + y 2 + z 2 dl,
L : x = cos t, y = sin t, z =
L
2. F⃗ = (x + y)⃗ı + (x − y) ⃗ȷ,
y2
= 1 (x > 0, y > 0) , A(1; 0), B(0; 3).
L: x +
9
2
51
√
3t, 0 6 t 6 π.
∫∫
(3y − 2x − 2z)dS,
3.
π : 2x − y − 2z = −2.
S
∫∫
(x + y)dydz + 3ydxdz + (y − z)dxdy,
4.
π : 2x − y − 2z = −2.
S
Âàðèàíò  10.
∫
dl
√
,
x2 + y 2 + z 2
1.
L
L îòðåçîê ïðÿìîé AB, A(1; 1; 1), B(2; 2; 2).
2. F⃗ = y⃗ı − x⃗ȷ, L : x2 + y 2 = 1 (y > 0) , A(1; 0), B(−1; 0).
∫∫
(2x − 3y + z)dS, π : x + 2y + z = 2.
3.
S
∫∫
(x + y − z)dydz − ydxdz + (x + 2z)dxdy,
4.
π : x + 2y + z = 2.
S
Âàðèàíò  11.
1.
∫ √
2y dl,
L : x = 2 (t − sin t) , y = 2 (1 − cos t) , 0 6 t 6 2π.
L
√
√
2. F⃗ = y⃗ı − x⃗ȷ, L : x2 + y 2 = 2 (y > 0) , A( 2; 0), B(− 2; 0).
∫∫
3.
(5x + y − z)dS, π : x + 2y + 2z = 2.
S
∫∫
xdydz + (y − 2z)dxdz + (2x − y + 2z)dxdy,
4.
π : x + 2y + 2z = 2.
S
Âàðèàíò  12.
∫
√
1.
L
dl
x2 + y 2 + 4
,
L îòðåçîê ïðÿìîé AB, A(0; 0), B(1; 2).
2. F⃗ = xy⃗ı + 2y⃗ȷ, L : x2 + y 2 = 1 (x > 0, y > 0) , A(1; 0), B(0; 1).
∫∫
3.
(3x + 2y + 2z)dS, π : 3x + 2y + 2z = 6.
S
52
∫∫
(x + 2z)dydz + (y − 3z)dxdz + zdxdy,
4.
π : 3x + 2y + 2z = 6.
S
Âàðèàíò  13.
∫
dl
,
x−y
1.
L
L îòðåçîê ïðÿìîé AB, A(4; 0), B(6; 1).
2. F⃗ = y⃗ı − x⃗ȷ, L : 2x2 + y 2 = 1 (y > 0) , A
∫∫
3.
(2x + 3y − z)dS, π : 2x + y + z = 2.
(
√1 ; 0
2
)
(
, B
− √12 ; 0
)
.
S
∫∫
(y − z)dydz + (2x + y)dxdz + zdxdy,
4.
π : 2x + y + z = 2.
S
Âàðèàíò  14.
∫
xydl,
1.
L îòðåçîê ïðÿìîé AB, A(4; 0), B(4; 2).
L
(
)
2. F⃗ = x2 + y 2 (⃗ı + 2⃗ȷ) , L : x2 + y 2 = 9 (y > 0) , A(3; 0), B(−3; 0).
∫∫
3.
(9x + 2y + z)dS, π : 2x + y + z = 4.
S
∫∫
4xdydz + (x − y − z)dxdz + (3y + 2z)dxdy,
4.
π : 2x + y + z = 4.
S
Âàðèàíò  15.
∫
1.
(x + y) dl,
L îòðåçîê ïðÿìîé AB, A(1; 0), B(0; 1).
L
)
(
)
(
√
√
2
2
2
2
⃗
2. F = x + y x + y ⃗ı + y − x x + y ⃗ȷ,
L : îòðåçîê ïðÿìîé AB, A(1; 0), B(−1; 0).
∫∫
3.
(3x + 8y + 8z)dS, π : x + 4y + 2z = 8.
S
53
∫∫
(2z − x)dydz + (x + 2y)dxdz + 3zdxdy,
4.
π : x + 4y + 2z = 8.
S
Âàðèàíò  16.
∫
1.
z 2 dl
,
x2 + y 2
L : x = 2 cos t, y = 2 sin t, z = 2t, 0 6 t 6 2π.
L
2. F⃗ = x2 y⃗ı − xy 2⃗ȷ, L : x2 + y 2 = 4 (x > 0, y > 0) , A(2; 0), B(0; 2).
∫∫
3.
(4y − x + 4z)dS, π : x − 2y + 2z = 2.
S
∫∫
4zdydz + (x − y − z)dxdz + (3y + z)dxdy, π : x − 2y + 2z = 2.
4.
S
Âàðèàíò  17.
∫
1.
(x + y) dl,
L îòðåçîê ïðÿìîé AB, A(−1; 0), B(0; 1).
L
(
)
(
)
√
√
2
2
2
2
⃗
2. F = x + y x + y ⃗ı − y − x x + y ⃗ȷ,
L : x2 + y 2 = 16 (x > 0, y > 0) , A(4; 0), B(0; 4).
∫∫
3.
(7x + y + 2z)dS, π : 3x − 2y + 2z = 6.
S
∫∫
(x + y)dydz + (y + z)dxdz + 2(x + z)dxdy, π : 3x − 2y + 2z = 6.
4.
S
Âàðèàíò  18.
∫
1.
xdl,
L : x = 5 cos t, y = 5 sin t, z = t, 0 6 t 6 2π.
L
2. F⃗ = y 2⃗ı − x2⃗ȷ, L : x2 + y 2 = 9 (x > 0, y > 0) , A(3; 0), B(0; 3).
∫∫
3.
(2x + 3y + z)dS, π : 2x + 3y + z = 6.
S
54
∫∫
(x + y + z)dydz + 2zdxdz + (y − 7z)dxdy, π : 2x + 3y + z = 6.
4.
S
Âàðèàíò  19.
∫
xydl,
1.
L îòðåçîê ïðÿìîé AB, A(5; 0), B(0; 3).
L
2. F⃗ = (x − y)⃗ı + ⃗ȷ, L : x2 + y 2 = 4 (y > 0) , A(2; 0), B(−2; 0).
∫∫
(4x − y + z)dS, π : x − y + z = 2.
3.
S
∫∫
(2x − z)dydz + (y − x)dxdz + (x + 2z)dxdy, π : x − y + z = 2.
4.
S
Âàðèàíò  20.
∫
1.
xdl,
L : x = 3 cos t, y = 3 sin t, z = 2t, 0 6 t 6 2π.
L
(
)
2. F⃗ = x2 + y 2 ⃗ı + y 2⃗ȷ, L îòðåçîê ïðÿìîé AB, A(2; 0), B(0; 2).
∫∫
(4x − 4y − z)dS, π : x + 2y + 2z = 4.
3.
S
∫∫
(2y − z)dydz + (x + y)dxdz + xdxdy,
4.
π : x + 2y + 2z = 4.
S
Âàðèàíò  21.
∫
1.
( √
√ )
4 3 x − 3 3 y dl,
L : x = cos3 t, y = sin3 t, z = t, 0 6 t 6
L
(
)
2. F⃗ = y 2 − y ⃗ı + (2xy + x) ⃗ȷ,
L : x2 + y 2 = 9 (y > 0) , A(3; 0), B(−3; 0).
∫∫
3.
(6x − y + 8z)dS, π : x + y + 2z = 2.
S
55
π
.
2
∫∫
4.
(x + z)dydz + (x + 3y)dxdz + ydxdy,
π : x + y + 2z = 2.
S
Âàðèàíò  22.
∫
1.
xydl,
L îòðåçîê ïðÿìîé AB, A(3; 0), B(0; 3).
L
(
)
2. F⃗ = xy − y 2 ⃗ı + x⃗ȷ, L : y = 2x2 , A(0; 0), B(1; 2).
∫∫
(2x + 5y + z)dS, π : x + y + 2z = 2.
3.
S
∫∫
(2z − x)dydz + (x − y)dxdz + (3x + z)dxdy, π : x + y + 2z = 2.
4.
S
Âàðèàíò  23.
∫
1.
xdl,
L : y = −x2 + 2x + 3, A(−1; 0), B(1; 4).
L
2. F⃗ = x⃗ı + y⃗ȷ, L : îòðåçîê ïðÿìîé AB, A(1; 0), B(0; 3).
∫∫
3.
(4x − y + 4z)dS, π : 2x + 2y + z = 4.
S
∫∫
(x + z)dydz + zdxdz + (2x − y)dxdy,
4.
π : 2x + 2y + z = 4.
S
Âàðèàíò  24.
∫
y 2 dl,
1.
L : x = t − sin t, y = 1 − cos t, 0 6 t 6 2π.
L
2. F⃗ = −y⃗ı + x⃗ȷ, L : y = x3 , A(0; 0), B(2; 8).
∫∫
3.
(5x + 2y + 2z)dS, π : x + 2y + z = 2.
S
∫∫
4.
(3x + y)dydz + (x + z)dxdz + ydxdy, π : x + 2y + z = 2.
S
56
Âàðèàíò  25.
∫
ydl,
1.
L : y 2 = 2x, A(0; 0), B(2; 2).
L
y2
2
⃗
2. F = −x⃗ı + y⃗ȷ, L : x +
= 1 (x > 0, y > 0) , A(1; 0), B(0; 3).
9
∫∫
(2x + 5y + 10z)dS, π : 2x + y + 3z = 6.
3.
S
∫∫
(y + z)dydz + (2x − z)dxdz + (y + 3z)dxdy, π : 2x + y + 3z = 6.
4.
S
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ
Âàðèàíò 1.
Çàäàíèå 1
Âû÷èñëèòå èíòåãðàë
∫
(2z −
√
x2 + y 2 ) dl, L : x = t cos t, y = t sin t, z = t, t ∈ [0; 2π].
L
Ðåøåíèå.
Êðèâàÿ çàäàíà ïàðàìåòðè÷åñêè. Âû÷èñëèì ïðîèçâîäíûå:
x′ (t) = cos t − t sin t, y ′ (t) = sin t + t cos t, z ′ (t) = 1.
Òîãäà ñîãëàñíî ôîðìóëå (11)
√
dl =
=
(cos t − t sin t)2 + (sin t + t cos t)2 + 1dt =
√
cos2 t − 2t cos t sin t + t2 sin2 t + sin2 t + 2t cos t sin t + t2 cos2 t + 1dt =
√
√
2
2
2
2
2
= 1 + (cos t + sin t) + t (sin t + cos t)dt = 2 + t2 dt.
Âû÷èñëèì èíòåãðàë, èñïîëüçóÿ ôîðìóëó (12):
∫
L
∫2π (
)√
√
√
2
2
2
2
2
2
2t − t cos t + t sin t
2 + t2 dt =
(2z − x + y ) dl =
0
57
]
[
∫2π √
∫2π √
1
1
= t 2 + t2 dt = tdt = d(2 + t2 ) =
2 + t2 d(2 + t2 ) =
2
2
0
0
√ (
2π
)
2 2
1
2 32
2 32
= (2 + t )
=
(1 + 2π ) − 1 .
3
3
0
Çàäàíèå 2.
Âû÷èñëèòå ðàáîòó ñèëû
F⃗ (x, y) = (x2 + y 2 )⃗i + 2xy⃗j
ïî ïåðåìåùåíèþ ìàòåðèàëüíîé òî÷êè âäîëü äóãè êðèâîé L :
òî÷êè A(0; 0) äî òî÷êè B(1; 1).
y = x3 îò
Ðàáîòó ñèëû ïî ïåðåìåùåíèþ ìàòåðèàëüíîé òî÷êè âû÷èñëÿåì
êàê êðèâîëèíåéíûé èíòåãðàë II ðîäà, èñïîëüçóÿ ôîðìóëó (15):
Ðåøåíèå.
∫
∫
(x2 + y 2 )dx + 2xydy.
P (x, y)dx + Q(x, y)dy =
⌣
⌣
AB
AB
⌣
Äóãà AB çàäàíà ÿâíûì óðàâíåíèåì: y = x3 , x ∈ [0; 1], ñëåäîâàòåëüíî,
ïðèìåíèìà ôîðìóëà (17). Âû÷èñëèì ïðîèçâîäíóþ: y ′ (x) = 3x2 , òîãäà
∫
∫1
(x2 + x6 + 2x · x3 · 3x2 )dx =
(x2 + y 2 )dx + 2xydy =
⌣
0
AB
(
∫1
2
=
6
(x + 7x )dx =
x3
+ x7
3
0
)
1
0
1
=1 .
3
Çàäàíèå 3.
Âû÷èñëèòå èíòåãðàë
∫∫
(1 + x − 2y + 4z) dS
S
ïî ÷àñòè ïëîñêîñòè π :
ïëîñêîñòÿìè.
3x − y − 2z = 2, îòñå÷åííîé êîîðäèíàòíûìè
Èçîáðàçèì íà ÷åðòåæå ÷àñòü ïëîñêîñòè π è åå ïðîåêöèþ íà
ïëîñêîñòü XOY (ñì. ðèñóíîê 15).
Ðåøåíèå.
58
Ðèñóíîê 15 a) ïëîñêîñòü π, b) åå ïðîåêöèÿ íà ïëîñêîñòü XOY
1, çàäà÷è 3 è 4)
Íàïèøåì ÿâíîå óðàâíåíèå ïëîñêîñòè:
1
3
3
1
z = x − y − 1. Òîãäà zx′ = ; zy′ = − ,
2
2
2
2
è, â ñîîòâåòñòâèè ñ ôîðìóëîé (19):
√
dS =
9 1
1 + + dxdy =
4 4
√
14
dxdy.
2
Âû÷èñëèì èíòåãðàë ïî ôîðìóëå (20):
∫∫
(1 + x − 2y + 4z) dS =
S
59
(âàðèàíò
√
14
2
=
∫∫
3
1
(1 + x − 2y + 4( x − y − 1))dxdy =
2
2
D
√
=
14
2
2
∫∫
√ ∫3
∫0
14
(7x − 4y − 3)dxdy =
dx
(7x − 4y − 3)dy =
2
D
0
3x−2
2
√ ∫3
∫0
14
=−
dx
(7x − 4y − 3)d(7x − 4y − 3) =
8
0
3x−2
2
√ ∫3
14
(7x − 4y − 3)2
=−
16
0
2
√
0
dx = −
3x−2
∫3
14
16
(
)
(7x − 3)2 − (5 − 5x)2 dx =
0
2
2
)
√ ( ∫3
∫3
14 1
(7x − 3)2 d(7x − 3) + 25 (1 − x)2 d(1 − x) =
=−
16 7

0
√
14  1
=−
(7x − 3)3
16
21
0
2
3
2
3
+
0

25
(1 − x)3  =
3
0
√ ( (
√
)
(
))
14 1 125
25 1
14 · 137
=−
+ 27 +
+1
=−
.
16 21 27
3 27
216
Çàäàíèå 4.
Âû÷èñëèòå ïîâåðõíîñòíûé èíòåãðàë
∫∫
ydydz + (x − z)dxdz + xdxdy,
S
ãäå S ÷àñòü ïëîñêîñòè π : 3x − y − 2z = 2, îòñå÷åííîé êîîðäèíàòíûìè
ïëîñêîñòÿìè â íàïðàâëåíèè íîðìàëè, îáðàçóþùåé îñòðûé óãîë ñ îñüþ Oz.
Ïî óñëîâèþ çàäà÷è ÿâíîå óðàâíåíèå ïîâåðõíîñòè S ýòî ÿâíîå
óðàâíåíèå ïëîñêîñòè π, ò. å.
Ðåøåíèå.
3
1
z = x − y − 1,
2
2
60
à íîðìàëü:
{
}
⃗n = −zx′ (x, y); −zy′ (x, y); 1 =
{
}
3 1
− ; ;1 .
2 2
Èíòåãðàë áóäåì âû÷èñëÿòü, ñâîäÿ åãî ê äâîéíîìó èíòåãðàëó ïî ôîðìóëå (23), ÷åðòåæ îáëàñòè D óæå èçîáðàæåí íà ðèñ. 15.
∫∫
ydydz + (x − z)dxdz + xdxdy =
S
(
)
)
∫∫ (
3
1
3
1
=
− y+
x − x + y + 1 + x dxdy =
2
2
2
2
D
∫∫ (
=
)
∫∫
5
1
1
3
(3x − 5y + 2) dxdy =
x− y+
dxdy =
4
4
2
4
D
D
2
1
=
4
∫3
∫0
dx
0
2
1
(3x − 5y + 2)dy = −
20
3x−2
∫3
∫0
0
3x−2
2
=−
1
40
2
∫3
0
(3x − 5y + 2)2
dx = −
(3x−2)
0
1
120
=−
(
)
(3x + 2)2 − (12 − 12x)2 dx =
2
∫3
(3x + 2)2 d(3x + 2) −
18
5
0
=−
1
40
∫3
0
2
=−
(3x − 5y + 2)d(3x − 5y + 2) =
dx
1
(3x + 2)3
360
∫3
(1 − x)2 d(1 − x) =
0
2
3
0
6
− − (1 − x)3
5
2
3
0
=−
6 1
1
(64 − 8) − ( − 1) =
360
5 27
7
52
+
= 1.
45 45
Âàðèàíò 2.
Çàäàíèå 1.
Âû÷èñëèòå èíòåãðàë
∫
xy dl, L : y 2 =
x
2
îò òî÷êè A(0; 0) äî òî÷êè B(2; 1).
L
61
Êðèâàÿ çàäàíà L óðàâíåíèåì y 2 = x2 , ðàâíîñèëüíûì óðàâíåíèþ
x = 2y 2 , ò. å. x = x(y), y ∈ [0; 1]. Òîãäà ïî ôîðìóëå (13), ïîñêîëüêó
Ðåøåíèå.
x′ = 4y, èìååì: dl =
√
1 + 16y 2 dy.
Âû÷èñëèì èíòåãðàë, èñïîëüçóÿ ôîðìóëó (14):
∫1
∫
2y 2 · y
xy dl =
√
1 + 16y 2 dy.
0
L
√
2
Ñäåëàåì çàìåíó ïåðåìåííîé: t =
√ 1 + 16y . Òîãäà íîâûå ïðåäåëû èíòåãðèðîâàíèÿ áóäóò: tí = 1 è tâ = 17. Ïðè ýòîì
1√ 2
tdt
y=
t − 1 (y ≥ 0), dy = √
.
4
4 t2 − 1
Ïîäñòàâëÿÿ â èíòåãðàë, ïîëó÷èì:
√
∫
∫ 17
xy dl =
0
L
√
√
∫ 17
1
2(t2 − 1) t2 − 1 · t · tdt
√
=
(t2 − 1)t2 dt =
2
128
16 · 4 · 4 t − 1
0
√
∫ 17
1
(t4 −t2 )dt =
(3t5 −5t3 )
128 · 15
0
√
391 · 17
=
.
960
1
=
128
√
17
=
0
√
√
1
(3( 17)5 −5( 17)3 ) =
128 · 15
Çàäàíèå 2.
⃗ (x, y) = (x − y)⃗i + (x − y)⃗j ïî ïåðåìåùåíèþ
Âû÷èñëèòå ðàáîòó ñèëû F
2
2
ìàòåðèàëüíîé òî÷êè âäîëü äóãè êðèâîé L : x9 + y1 = 1, (x ≥ 0, y ≥ 0)
îò òî÷êè A(3; 0) äî òî÷êè B(0; 1).
Êðèâàÿ L ýëëèïñ. Çàïèøåì ïàðàìåòðè÷åñêîå óðàâíåíèå äàííîãî ýëëèïñà: x = 3 cos t, y = sin t. Òî÷êè A è B ðàñïîëîæåíû íà îñÿõ
êîîðäèíàò, è ïðè ïåðåìåùåíèè îò òî÷êè A ê òî÷êå B ïàðàìåòð t èçìåíÿåòñÿ îò 0 äî π2 ò. å. t ∈ [0; π2 ]. Âûïèøåì ïðîèçâîäíûå:
Ðåøåíèå.
x′ (t) = −3 sin t, y ′ (t) = cos t.
⃗ (x, y) ðàâíà:
Òîãäà ñîãëàñíî ôîðìóëå (16) ðàáîòà ñèëû F
∫
∫
(x − y)dx + (x + y)dy =
P (x, y)dx + Q(x, y)dy =
L
L
62
π
∫2
((3 cos t − sin t) (−3 sin t) + (3 cos t + sin t) cos t) dt =
=
0
π
∫2
=
(
)
−9 cos t sin t + 3 sin2 t + 3 cos2 t + cos t sin t =
0
π
π
[
]
= 3 sin2 t + 3 cos2 t = 3 = 3
∫2
∫2
dt − 8
0
π
2
= [cos tdt = dt sin t] = 3t
=
0
π
∫2
−8
0
cos t sin tdt =
3π
sin td(sin t) =
− 4 sin2 t
2
0
π
2
=
0
3π
− 4.
2
Çàäàíèå 3.
Âû÷èñëèòå èíòåãðàë
∫∫
(2 − x + 2y + z) dS ïî ÷àñòè ïëîñêîñòè
S
π : 2x − y − 2z = −2, îòñå÷åííîé êîîðäèíàòíûìè ïëîñêîñòÿìè.
Èçîáðàçèì íà ÷åðòåæå ÷àñòü ïëîñêîñòè π è åå ïðîåêöèþ íà
ïëîñêîñòü XOY (ñì. ðèñóíîê 16).
Ðåøåíèå.
Íàïèøåì ÿâíîå óðàâíåíèå ïëîñêîñòè: z = x −
1
zy′ = − , è, â ñîîòâåòñòâèè ñ ôîðìóëîé (19):
2
√
√
1
9
3
dS = 1 + 1 + dxdy =
dxdy = dxdy.
4
4
2
1
y + 1. Òîãäà zx′ = 1;
2
Âû÷èñëèì èíòåãðàë ïî ôîðìóëå (20):
∫∫
3
(2 − x + 2y + z) dS =
2
S
9
=
4
∫∫
(2 − x + 2y + x −
y
+ 1)dxdy =
2
D
∫∫
D
9
(2 + y)dxdy =
4
∫0
2(x+1)
∫
9
(2 + y)dy =
8
dx
−1
0
63
∫0
2(x+1)
(2 + y)
−1
2
dx =
0
9
=
8
=
∫0
−1
)
(2 + x)2
−x
3
(
9
2
(
)
9
(4 + 2x)2 − 4 dx =
2
0
=
−1
9
2
(∫0
∫0
(2 + x) dx −
2
(
−1
)
8 1
− − 1 = 6.
3 3
)
dx =
−1
Çàäàíèå 4.
Âû÷èñëèòå ïîâåðõíîñòíûé èíòåãðàë
∫∫
xdydz + (2z − y)dxdz + 2ydxdy,
S
ãäå S ÷àñòü ïëîñêîñòè π : 2x − y − 2z = −2, îòñå÷åííîé êîîðäèíàòíûìè
ïëîñêîñòÿìè â íàïðàâëåíèè íîðìàëè, îáðàçóþùåé îñòðûé óãîë ñ îñüþ Oz.
Ïî óñëîâèþ çàäà÷è ÿâíîå óðàâíåíèå ïîâåðõíîñòè S ýòî ÿâíîå
óðàâíåíèå ïëîñêîñòè π, ò. å.
Ðåøåíèå.
z =x−
y
+ 1,
2
à íîðìàëü:
1
⃗n = {−zx′ (x, y); −zy′ (x, y); 1} = {−1; ; 1}.
2
Èíòåãðàë áóäåì âû÷èñëÿòü, ñâîäÿ åãî ê äâîéíîìó èíòåãðàëó ïî ôîðìóëå (23), ÷åðòåæ îáëàñòè D óæå èçîáðàæåí íà ðèñ. 16.
∫∫
xdydz + (2z − y)dxdz + 2ydxdy =
S
)
∫∫ (
∫∫
1
=
−x + (2x − y + 2 − y) + 2y) dxdy =
(y + 1)dxdy =
2
D
D
∫0
=
2(x+1)
∫
1
(y + 1)dy =
2
dx
−1

0
∫0
∫0
∫0
2(x+1)
(y + 1)
2
dx =
0
−1


∫0
1
1 1
(2x + 3)2 d(2x + 3) − x
(2x + 3)2 dx − dx = 
2
2 2
−1
−1
−1
(
)
(
)
0
3
1 1
1 (2x + 3)3
(27 − 1) − 1 = 2 .
−1 =
=
2
3
2 6
4
−1
0
=
64
−1

=
Ðèñóíîê 16 a) ïëîñêîñòü π, b) åå ïðîåêöèÿ íà ïëîñêîñòü XOY (âàðèàíò
2, çàäà÷è 3 è 4)
65
Êîíòðîëüíàÿ ðàáîòà  8
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  8
Çàäàíèå  1
Âû÷èñëèòå ãðàäèåíò ñêàëÿðíîãî ïîëÿ â çàäàííîé òî÷êå M0 .
Çàäàíèå  2
⃗ (M ).
Ïðîâåðüòå, áóäåò ëè ñîëåíîèäàëüíûì äàííîå âåêòîðíîå ïîëå F
Çàäàíèå  3
⃗ (M ).
Ïðîâåðüòå, áóäåò ëè ïîòåíöèàëüíûì äàííîå âåêòîðíîå ïîëå F
Çàäàíèå  4
Âû÷èñëèòå öèðêóëÿöèþ ïëîñêîãî âåêòîðíîãî ïîëÿ
F⃗ (x, y) = P (x, y)⃗ı + Q(x, y)⃗ȷ
âäîëü çàìêíóòîãî êîíòóðà Γ (â ïîëîæèòåëüíîì íàïðàâëåíèè)
1) íåïîñðåäñòâåííî;
2) èñïîëüçóÿ ôîðìóëó Ãðèíà.
Óêàçàíèå.
Ïåðåä ðåøåíèåì çàäà÷ êîíòðîëüíîé ðàáîòû ðåêîìåíäóåòñÿ îçíàêîìèòüñÿ ñî ñëåäóþùèìè ìåòîäè÷åñêèìè óêàçàíèÿìè:
1. Ãðóçäêîâ, À.À. Ôîðìóëà Ñòîêñà: ìåòîäè÷åñêèå óêàçàíèÿ / À. À. Ãðóçäêîâ, Ì. Á. Êóï÷èíåíêî. ÑÏá.: ÑÏáÃÒÈ(ÒÓ),- 2012. 54 c.
2. Ãðóçäêîâ, À.À. Ôîðìóëà Îñòðîãðàäñêîãî-Ãàóññà: ìåòîäè÷åñêèå óêàçàíèÿ / À. À. Ãðóçäêîâ, Ì. Á. Êóï÷èíåíêî. ÑÏá.: ÑÏáÃÒÈ(ÒÓ),2014. 26 c.
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû
Ïóñòü â îáëàñòè G ⊂ R3 (èëè D ⊂ R2 ) çàäàíà ñêàëÿðíàÿ ôóíêöèÿ òî÷êè
f = f (M ), òîãäà ãîâîðÿò, ÷òî â G (èëè D) çàäàíî ñêàëÿðíîå ïîëå .
Ïóñòü â îáëàñòè G ⊂ R3 (èëè D ⊂ R2 ) çàäàíà âåêòîðíàÿ ôóíêöèÿ òî÷êè
F⃗ = F⃗ (M ), òîãäà ãîâîðÿò, ÷òî â G (èëè D) çàäàíî âåêòîðíîå ïîëå .
66
Åñëè ââåäåíà äåêàðòîâà ïðÿìîóãîëüíàÿ ñèñòåìà êîîðäèíàò, òî ñêàëÿðíîå
ïîëå ìîæåò áûòü çàïèñàíî â âèäå:
u = f (x, y, z),
ò. å. ñêàëÿðíîå ïîëå ýòî ôóíêöèÿ òðåõ (èëè äâóõ) ïåðåìåííûõ.
Âåêòîðíîå ïîëå ìîæåò áûòü çàïèñàíî â âèäå:
F⃗ (x, y, z) = P (x, y, z)⃗i + Q(x, y, z)⃗j + R(x, y, z)⃗k.
Äëÿ ïëîñêîãî âåêòîðíîãî ïîëÿ èìååì:
F⃗ (x, y) = P (x, y)⃗i + Q(x, y)⃗j.
Ãðàäèåíòîì ñêàëÿðíîãî ïîëÿ íàçûâàåòñÿ âåêòîðíàÿ ôóíêöèÿ òî÷êè,
çàäàâàåìàÿ ôîðìóëîé:
grad f (x, y, z) = ∇f (x, y, z) =
∂f ⃗ ∂f ⃗ ∂f ⃗
i+
j+
k.
∂x
∂y
∂z
(24)
Äèâåðãåíöèåé âåêòîðíîãî ïîëÿ íàçûâàåòñÿ ñêàëÿðíàÿ ôóíêöèÿ òî÷-
êè, çàäàâàåìàÿ ôîðìóëîé:
∂Q ∂R
∂P
+
+
.
div F⃗ (x, y, z) = ∇ · F⃗ (x, y, z) =
∂x
∂y
∂z
(25)
Ðîòîðîì âåêòîðíîãî ïîëÿ íàçûâàåòñÿ âåêòîðíàÿ ôóíêöèÿ òî÷êè, çà-
äàâàåìàÿ ôîðìóëîé:
rot F⃗ (x, y, z) = ∇ × F⃗ (x, y, z) =
(
=
Çäåñü ∇ =
⃗i
⃗j
⃗k
∂
∂x
∂
∂y
∂
∂z
=
(26)
P Q R
(
)
(
)
∂P
∂R ⃗
∂Q ∂P ⃗
∂R ∂Q ⃗
−
i+
−
j+
−
k.
∂y
∂z
∂z
∂x
∂x
∂y
)
∂⃗
∂
∂
i + ⃗j + ⃗k îïåðàòîð Ãàìèëüòîíà, ñî÷åòàþùèé
∂x
∂y
∂z
â ñåáå äèôôåðåíöèàëüíóþ è âåêòîðíóþ ïðèðîäû.
Öèðêóëÿöèåé âåêòîðíîãî ïîëÿ âäîëü çàìêíóòîãî êîíòóðà Γ íàçûâàåòñÿ êðèâîëèíåéíûé èíòåãðàë âòîðîãî ðîäà:
I
P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz.
Γ
67
(27)
Äëÿ ïëîñêîãî ïîëÿ öèðêóëÿöèÿ áóäåò ðàâíà:
I
(28)
P (x, y)dx + Q(x, y)dy.
Γ
Ïîòîêîì âåêòîðíîãî ïîëÿ
F⃗ (x, y, z) = P (x, y, z)⃗i + Q(x, y, z)⃗j + R(x, y, z)⃗k
(29)
÷åðåç ïîâåðõíîñòü S â íàïðàâëåíèè íîðìàëè ⃗n íàçûâàåòñÿ ïîâåðõíîñòíûé
èíòåãðàë âòîðîãî ðîäà:
∫∫
P (x, y, z)dydz + Q(x, y, z)dxdz + R(x, y, z)dxdy.
S
Âåêòîðíîå ïîëå íàçûâàåòñÿ ñîëåíîèäàëüíûì , åñëè åãî ïîòîê ÷åðåç ëþáóþ êóñî÷íî-ãëàäêóþ çàìêíóòóþ ïîâåðõíîñòü ðàâåí íóëþ, ò. å.
∫∫
P (x, y, z)dydz + Q(x, y, z)dxdz + R(x, y, z)dxdy = 0, ∀S ⊂ G.
S
Íåïðåðûâíî äèôôåðåíöèðóåìîå âåêòîðíîå ïîëå áóäåò ñîëåíîèäàëüíûì
òîãäà è òîëüêî òîãäà, êîãäà
div F⃗ (x, y, z) = 0.
Âåêòîðíîå ïîëå íàçûâàåòñÿ ïîòåíöèàëüíûì , åñëè åãî öèðêóëÿöèÿ ïî
ëþáîìó êóñî÷íî-ãëàäêîìó çàìêíóòîìó êîíòóðó ðàâíà íóëþ, ò. å.
I
P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz = 0, ∀Γ ⊂ G.
Γ
Íåïðåðûâíî äèôôåðåíöèðóåìîå âåêòîðíîå ïîëå áóäåò ïîòåíöèàëüíûì
òîãäà è òîëüêî òîãäà, êîãäà
rot F⃗ (x, y, z) = 0.
Ñîëåíîèäàëüíûå è ïîòåíöèàëüíûå âåêòîðíûå ïîëÿ îáëàäàþò
ðÿäîì âàæíûõ ñâîéñòâ, ïîýòîìó òàê âàæíî óìåòü îïðåäåëÿòü, áóäåò ëè
âåêòîðíîå ïîëå ñîëåíîèäàëüíûì èëè ïîòåíöèàëüíûì.
⃗ (x, y) = P (x, y)⃗i + Q(x, y)⃗j
Öèðêóëÿöèþ ïëîñêîãî âåêòîðíîãî ïîëÿ F
ìîæíî âû÷èñëÿòü äâóìÿ ñïîñîáàìè, à èìåííî:
Çàìå÷àíèå.
68
1) êàê êðèâîëèíåéíûé èíòåãðàë
I
(30)
P (x, y)dx + Q(x, y)dy,
Γ
èñïîëüçóÿ ôîðìóëû (16)(17);
2) ïî ôîðìóëå Ãðèíà
∫∫ (
I
P (x, y)dx + Q(x, y)dy =
Γ
∂Q ∂P
−
∂x
∂y
)
dxdy,
(31)
D
ãäå D ìíîæåñòâî, îãðàíè÷åííîå êîíòóðîì Γ.  ôîðìóëå Ãðèíà ïðåäïîëàãàåòñÿ, ÷òî êîíòóð îáõîäèòñÿ â ïîëîæèòåëüíîì íàïðàâëåíèè.
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  8
Âàðèàíò  1.
)
(
√ 1 1
yz 2
1. U (x, y, z) = 2 , M0
2; √ ; √ .
x
2 3
(
)
(
)
2. F⃗ (x, y, z) = x2 y + y 3 ⃗ı + zx3 − xy 2 ⃗ȷ + (x − y) ⃗k.
3. F⃗ (x, y, z) = (2x + yz)⃗ı + (2y + xz) ⃗ȷ + (2z + xy) ⃗k.
4. F⃗ (x, y) = (x + 3y)⃗ı + x2⃗ȷ,
L : y = x2 + 5x + 1,
Âàðèàíò  2.
(
1. U (x, y, z) = x2 yz 3 ,
M0
y = x + 1.
√ )
1
3
2; ;
.
3
2
(
)
2. F⃗ (x, y, z) = xy 2⃗ı + x2 y⃗ȷ − x2 + y 2 z⃗k.
3. F⃗ (x, y, z) = (2x − yz)⃗ı + (2y − xz) ⃗ȷ + (2z − xy) ⃗k.
4. F⃗ (x, y) = 2y⃗ı + (x + 3y) ⃗ȷ,
L : y = −x2 + x + 2,
Âàðèàíò  3.
(
3
1. U (x, y, z) =
z
,
xy 2
M0
√ )
1
3
; 2;
.
3
2
69
y = x + 1.
(
)
(
)
2. F⃗ (x, y, z) = y 2⃗ı − x2 + y 3 ⃗ȷ + 3z 3y 2 + 1 ⃗k.
3. F⃗ (x, y, z) = (2x + yz)⃗ı + (2y + xz) ⃗ȷ + (2z + xy) ⃗k.
L : y = −x2 + 2x + 3,
4. F⃗ (x, y) = x⃗ı + (2x + y) ⃗ȷ,
y = 2x + 2.
Âàðèàíò  4.
)
(
z
1
1. U (x, y, z) = 3 2 , M0 1; 2; √ .
xy
6
(
)
(
)
(
)
2. F⃗ (x, y, z) = x z 2 − y 2 ⃗ı + y x2 − z 2 ⃗ȷ + z y 2 − x2 ⃗k.
3. F⃗ (x, y, z) = (2x − 4yz)⃗ı + (2y − 4xz) ⃗ȷ + (2z − 4xy) ⃗k.
(
)
4. F⃗ (x, y) = (3x + y)⃗ı + x2 + 1 ⃗ȷ, L : y = x2 + 3x − 2, y = −x + 3.
Âàðèàíò  5.
(
√
)
1 1
2; √ ; √ .
M0
2 3
(
)
2. F⃗ (x, y, z) = (1 + 2xy)⃗ı − y 2 z⃗ȷ + z 2 y − 2zy + 1 ⃗k.
x2
1. U (x, y, z) = 2 ,
yz
3. F⃗ (x, y, z) = (2x − 3yz)⃗ı + (2y − 3xz) ⃗ȷ + (2z − 3xy) ⃗k.
4. F⃗ (x, y) = (2x + 3y)⃗ı + (x − 3y) ⃗ȷ, L : y = x2 + 3x + 2,
Âàðèàíò  6.
z2
1. U (x, y, z) = 2 ,
xy
(
M0
1
; 2;
3
y = 2x + 2.
√ )
2
.
3
y
(x + y) ln z ⃗
x
k.
2. F⃗ (x, y, z) = ⃗ı + ⃗ȷ −
yz
xz
xy
3. F⃗ (x, y, z) = (−3x + yz)⃗ı + (−3y + xz) ⃗ȷ + (−3z + xy) ⃗k.
4. F⃗ (x, y) = (x − 3y)⃗ı − x2⃗ȷ,
Âàðèàíò  7.
xz 2
1. U (x, y, z) =
,
y
(
M0
L : y = 2x2 + 6x + 1,
)
1 1
√ ; √ ;1 .
6 6
70
y = x − 2.
(
)
(
)
(
)
2. F⃗ (x, y, z) = x2 y − x2 z ⃗ı + z 2 + 2xyz ⃗ȷ + x2 − 2xyz ⃗k.
3. F⃗ (x, y, z) = (2x + 2yz)⃗ı + (2y + 2xz) ⃗ȷ + (2z + 2xy) ⃗k.
(
)
4. F⃗ (x, y) = (y − 3x)⃗ı + 1 − x2 ⃗ȷ, L : y = 2x2 + 6x + 3, y = 3x + 2.
Âàðèàíò  8.
(
)
yz 2
1 1 1
1. U (x, y, z) =
, M0 √ ; √ ; √ .
x
2 2 3
(y
)
(
( 2
)
x)
⃗
⃗ȷ +
− 2xyz ⃗k.
2. F (x, y, z) = x (y − z) + yz ⃗ı + 2xyz +
z
x
3. F⃗ (x, y, z) = (4x + yz)⃗ı + (4y + xz) ⃗ȷ + (4z + xy) ⃗k.
4. F⃗ (x, y) = (2x − 3y)⃗ı + (x + y) ⃗ȷ, L : y = −x2 + 3x + 3, y = 2x + 1.
Âàðèàíò  9.
(
2
1. U (x, y, z) =
xy
,
z2
M0
√ )
1
2
; 2;
.
3
3
(
)
(
)
(
)
2. F⃗ (x, y, z) = x2 z − x2 y + 1 ⃗ı + x2 − 2xyz ⃗ȷ + y 2 + 2xyz ⃗k.
3. F⃗ (x, y, z) = (2x + 5yz)⃗ı + (2y + 5xz) ⃗ȷ + (2z + 5xy) ⃗k.
(
)
4. F⃗ (x, y) = (x + y)⃗ı + x2 − 2 ⃗ȷ, L : y = x2 + 4x + 3, y = 3x + 3.
Âàðèàíò  10.
(
)
1
M0 1; 2; √ .
6
(
)
(
)
2. F⃗ (x, y, z) = x2 (z − y)⃗ı + z 2 − 2xyz ⃗ȷ + x2 + 2xyz ⃗k.
x3 y 2
,
1. U (x, y, z) =
z
3. F⃗ (x, y, z) = (2x + 3yz)⃗ı + (2y + 3xz) ⃗ȷ + (2z + 3xy) ⃗k.
4. F⃗ (x, y) = (x + 4y)⃗ı + (2x − 5) ⃗ȷ, L : y = x2 + 3x − 2, y = 5x + 1.
Âàðèàíò  11.
(
)
1 1
1
1. U (x, y, z) = 2 , M0 2; ; √ .
x yz
3 6
(
)
(
)
2. F⃗ (x, y, z) = zx2 + 2y ⃗ı + zy 2 + 2x ⃗ȷ − z 2 (x + y) ⃗k.
71
3. F⃗ (x, y, z) = yz⃗ı + xz⃗ȷ + xy⃗k.
4. F⃗ (x, y) = (x − 4y)⃗ı + (5x − 2)⃗ȷ, L : y = 2x2 + 4x − 3,
Âàðèàíò  12.
(
2
1. U (x, y, z) =
x
,
y2z3
M0
√ √
2; 2;
y = −x + 4.
√ )
3
.
2
(
)
(
)
2. F⃗ (x, y, z) = zx2 − 2y 2 ⃗ı + zy 2 − 2x2 ⃗ȷ − z 2 (x + y) ⃗k.
(
)
(
)
(
)
3. F⃗ (x, y, z) = 2xy + z 2 ⃗ı + 2yz + x2 ⃗ȷ + 2xz + y 2 ⃗k.
4. F⃗ (x, y) = (2x + 5y)⃗ı + (3x + 2) ⃗ȷ, L : y = 3x2 + 4x + 1, y = x + 1.
Âàðèàíò  13.
)
(
1 1
1. U (x, y, z) = xyz, M0 1; ; √ .
3 6
(
)
2. F⃗ (x, y, z) = 2xyz ⃗ı + x2 + z 2 ⃗ȷ − xyz 2⃗k.
z4
y3
⃗
3. F (x, y, z) = ⃗ı +
⃗ȷ + xz 3⃗k.
4
3
4. F⃗ (x, y) = (2x − 5y)⃗ı + (1 − 3x)⃗ȷ, L : y = 2x2 + 5x + 2, y = 2x + 1.
Âàðèàíò  14.
(√
3
1. U (x, y, z) =
y
,
x2 z
M0
2
;
3
√
3 1
;
2 2
)
.
(
)
(
)
(
)
2. F⃗ (x, y, z) = x3 + y 3 ⃗ı + 3 x2 + y 2 ⃗ȷ − 3z x2 + 2y ⃗k.
3. F⃗ (x, y, z) = yz cos xy ⃗ı + xz cos xy ⃗ȷ + sin xy ⃗k.
4. F⃗ (x, y) = (x + 5y)⃗ı + (2 + 4x)⃗ȷ, L : y = x2 + 5x + 2,
Âàðèàíò  15.
(
)
1
2
1. U (x, y, z) = xy 2 z, M0 1; ; √ .
3 6
(
)
(
)
(
)
2. F⃗ (x, y, z) = x3 + y 3 ⃗ı + 3 x2 + y 2 ⃗ȷ − 3z x2 + 2y ⃗k.
72
y = 2x.
3. F⃗ (x, y, z) = 2xy 2 z 3⃗ı + 3x2 y 2 z 2⃗ȷ + 2x2 y 3 z ⃗k.
4. F⃗ (x, y) = (x − 5y)⃗ı + (1 − 4x) ⃗ȷ, L : y = 4x2 + 7x + 2,
Âàðèàíò  16.
y = 2x + 1.
(
)
x
1 1 1
1. U (x, y, z) = 2 , M0 √ ; √ ; √ .
yz
2 2 3
(
)
(
)
(
)
2. F⃗ (x, y, z) = x3 y + yz ⃗ı + 3 y 2 + xz ⃗ȷ − 3z x2 + 2y ⃗k.
3. F⃗ (x, y, z) = (2y + z)⃗ı + (2x − y) ⃗ȷ + (x − 2z) ⃗k.
4. F⃗ (x, y) = (5x + 2y)⃗ı + x2⃗ȷ,
L : y = −2x2 + 2x + 3,
Âàðèàíò  17.
y = 2x + 1.
(
2 3
√ √
2; 2;
√ )
3
.
2
y z
, M0
x2
(
)
(
)
2. F⃗ (x, y, z) = x z + y 2 ⃗ı − y (x + z) ⃗ȷ + z x − y 2 ⃗k.
1. U (x, y, z) =
3. F⃗ (x, y, z) = (x + 2z)⃗ı + (y + z) ⃗ȷ + (2x + y) ⃗k.
4. F⃗ (x, y) = (5x + y)⃗ı + (2x − 5) ⃗ȷ,
L : y = 2x2 + x − 3,
y = −2x + 2.
Âàðèàíò  18.
(
2 3
1. U (x, y, z) =
y z
,
x
M0
√ )
√
1
3
√ ; 2;
.
2
2
2. F⃗ (x, y, z) = (xy + xz)⃗ı − (xy + yz) ⃗ȷ + (xz − yz) ⃗k.
(
)
3. F⃗ (x, y, z) = 2xy ⃗ı + x2 − 2yz ⃗ȷ − y 2⃗k.
4. F⃗ (x, y) = y⃗ı + y (2x − 1) ⃗ȷ,
L : y = x2 + 3x − 3,
Âàðèàíò  19.
y
1. U (x, y, z) = 2 ,
xz
y = −x + 2.
(
M0
)
1 1 1
√ ;√ ;√ .
6 6 6
73
2. F⃗ (x, y, z) = x (y + z)⃗ı + y (x + z) ⃗ȷ − z (z + x + y) ⃗k.
3. F⃗ (x, y, z) = (2y + z)⃗ı + (y + 2x) ⃗ȷ + (x + 2z) ⃗k.
(
)
4. F⃗ (x, y) = (x + 2)⃗ı + x2 − y ⃗ȷ, L : y = 2x2 + x − 2, y = −x + 2.
Âàðèàíò  20.
yz 2
,
1. U (x, y, z) =
x
(
M0
)
1 1 1
√ ;√ ;√ .
2 2 3
2. F⃗ (x, y, z) = x (y − z)⃗ı + y (x − z) ⃗ȷ + z (z − x − y) ⃗k.
3. F⃗ (x, y, z) = −4z 2⃗ı + 2y ⃗ȷ − 8xz ⃗k.
4. F⃗ (x, y) = (2y − x)⃗ı + (3x − 1)⃗ȷ, L : y = x2 + 4x − 3,
Âàðèàíò  21.
(
2
1. U (x, y, z) =
z
,
x2 y 2
M0
y = −x + 3.
√ )
2
2
; 2;
.
3
3
(
)
(
)
2. F⃗ (x, y, z) = xy 2 + z ⃗ı − zy 2⃗ȷ + yz 2 − zy 2 ⃗k.
(
)
3. F⃗ (x, y, z) = z 2 + 2xy ⃗ı + x2⃗ȷ + 2xz⃗k.
4. F⃗ (x, y) = (3y − x)⃗ı + (2x + 5)⃗ȷ, L : y = 2x2 + 4x − 2,
Âàðèàíò  22.
(
2
1. U (x, y, z) =
x
,
y2z3
M0
√ √
2; 2;
y = x + 3.
√ )
3
.
2
(
)
(
)
(
)
2. F⃗ (x, y, z) = y 2 + yz 2 ⃗ı + z 2 + zx2 ⃗ȷ + x2 + y 2 x ⃗k.
3. F⃗ (x, y, z) = 2xy 2 z 2⃗ı + 2yx2 z 2⃗ȷ + 2y 2 zx2⃗k.
4. F⃗ (x, y) = (4y + x)⃗ı + (5x − 1) ⃗ȷ, L : y = 2x2 + 3x + 1,
Âàðèàíò  23.
1. U (x, y, z) = x2 yz 3 ,
(
M0
√ )
1
3
2; ;
.
3
2
74
y = 2x + 2.
(
)
(
)
2. F⃗ (x, y, z) = x z + 3z 2 ⃗ı − y (x + z) ⃗ȷ + z x − z 2 ⃗k.
3. F⃗ (x, y, z) = (ey + yex )⃗ı + (xey + ex ) ⃗ȷ − 2z ⃗k.
4. F⃗ (x, y) = (4y − x)⃗ı + (2x + 5)⃗ȷ, L : y = x2 + 3x + 2,
Âàðèàíò  24.
xy 2
1. U (x, y, z) = 3 ,
z
(
M0
1
; 2;
3
y = 4x + 2.
√ )
3
.
2
(
)
(
)
2. F⃗ (x, y, z) = x2 z + 3 ⃗ı + y 2 − 2yxz ⃗ȷ + (x − 2yz) ⃗k.
3. F⃗ (x, y, z) = (3x + y)⃗ı + (x − y) ⃗ȷ + (3x + 3) ⃗k.
4. F⃗ (x, y) = (x − y)⃗ı + x2⃗ȷ, L : y = x2 + 6x + 1,
Âàðèàíò  25.
1
1. U (x, y, z) = 2 ,
xy z
y = 3x + 5.
)
(
2 1
M0 1; ; √ .
3 6
2. F⃗ (x, y, z) = (x + 1) ey ⃗ı − (y + 1) ex⃗ȷ + z (ex − ey ) ⃗k.
3. F⃗ (x, y, z) = 3z ⃗ı + y ⃗ȷ + (3x − z) ⃗k.
4. F⃗ (x, y) = (2x + 2y)⃗ı + (x − 1) ⃗ȷ,
L : y = 2x2 + 4x + 2,
y = 3x + 5.
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ
Âàðèàíò 1.
Çàäàíèå 1.
Âû÷èñëèòå ãðàäèåíò ñêàëÿðíîãî ïîëÿ
x
U (x, y, z) = 2 3 â òî÷êå M0
y z
Ðåøåíèå.
(
√ )
3
1 √
√ , 2,
.
2
2
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå
∂U
1
∂U
2x
∂U
3x
= 2 3;
= − 3 3;
= − 2 4.
∂x
y z
∂y
y z
∂z
y z
75
Âû÷èñëèì çíà÷åíèÿ ÷àñòíûõ ïðîèçâîäíûõ â òî÷êå M0 :
∂U
∂x
M0
∂U
∂y
M0
∂U
∂z
M0
√
23
4 3
√ =
=
;
9
2·3 3
√
2·8
4 3
√
√ =−
= −√
;
9
2·2 2·3 3
√
4 2
3 · 16
=−
.
= −√
3
2·2·9
Òîãäà ñîãëàñíî ôîðìóëå (24)
(
grad U
√ )
√
√
√
√
1
3
4 3⃗ 4 3⃗ 4 2 ⃗
√ , 2,
=
i−
j−
k.
2
9
9
3
2
Çàäàíèå 2.
Ïðîâåðüòå, áóäåò ëè ñîëåíîèäàëüíûì âåêòîðíîå ïîëå
F⃗ (x, y, z) = y(x2 − z)⃗i + x(y 2 + z)⃗j + (y 2 − 4xyz)⃗k.
Ðåøåíèå.
Âû÷èñëèì äèâåðãåíöèþ âåêòîðíîãî ïîëÿ, ñì. ôîðìóëó (25):
( 2
)
( 2
)
( 2
)
∂
y(x
−
z)
∂
x(y
+
z)
∂
(y
−
4xyz)
div F⃗ (x, y, z) =
+
+
=
∂x
∂y
∂z
= 2xy + 2xy − 4xy ≡ 0.
Ñëåäîâàòåëüíî, âåêòîðíîå ïîëå ÿâëÿåòñÿ ñîëåíîèäàëüíûì â R3 .
Çàäàíèå 3.
Ïðîâåðüòå, áóäåò ëè ïîòåíöèàëüíûì âåêòîðíîå ïîëå
F⃗ (x, y, z) = (3 + 2xy)⃗i − xy⃗j + (x2 z − 2xy + 1)⃗k.
Ðåøåíèå.
Âû÷èñëèì ðîòîð âåêòîðíîãî ïîëÿ, ñì. ôîðìóëó (26):
rot F⃗ (x, y, z) =
⃗i
⃗j
⃗k
∂
∂x
∂
∂y
∂
∂z
=
(3 + 2xy) (−xy) (x z − 2xy + 1)
) (
)
(
∂(3 + 2xy) ∂(x2 z − 2xy + 1) ⃗
∂(x2 z − 2xy + 1) ∂(−xy) ⃗
−
i+
−
j+
=
∂y
∂z
∂z
∂x
2
76
(
)
∂(−xy) ∂(3 + 2xy) ⃗
+
−
k = −2x⃗i − (2xz − 2y)⃗j + (−y − 2x)⃗k.
∂x
∂y
⃗ (x, y, z) ̸≡ 0, ñëåäîâàòåëüíî, äàííîå ïîëå íå ÿâëÿåòñÿ ïîòåíÈòàê, rot F
öèàëüíûì.
Çàäàíèå 4.
Âû÷èñëèòå öèðêóëÿöèþ ïëîñêîãî âåêòîðíîãî ïîëÿ
F⃗ (x, y) = 2y⃗i + (5x − 1)⃗j
âäîëü çàìêíóòîãî êîíòóðà Γ : y = −x2 + 5x − 5, y = x − 2 (â ïîëîæèòåëüíîì íàïðàâëåíèè)
1) íåïîñðåäñòâåííî,
2) èñïîëüçóÿ ôîðìóëó Ãðèíà.
Ðåøåíèå.
I
Öèðêóëÿöèÿ âåêòîðíîãî ïîëÿ ýòî èíòåãðàë
2ydx + (5x − 1)dy.
Γ
Êîíòóð èíòåãðèðîâàíèÿ îáðàçîâàí äóãîé ïàðàáîëû è îòðåçêîì ïðÿìîé.
Äëÿ åãî ïîñòðîåíèÿ íåîáõîäèìî íàéòè òî÷êè ïåðåñå÷åíèÿ ïðÿìîé è ïàðàáîëû. Ðåøèì äëÿ ýòîãî ñîîòâåòñòâóþùóþ ñèñòåìó óðàâíåíèé:
{
y = x − 2,
⇔
y = −x2 + 5x − 5.
{
y = x − 2,
⇔
−x2 + 5x − 5 = x − 2.
{
y = x − 2,
x2 − 4x + 3 = 0.
Êîðíÿìè êâàäðàòíîãî óðàâíåíèÿ ÿâëÿþòñÿ ÷èñëà x1 = 1, x2 = 3. Òàêèì
îáðàçîì, òî÷êè ïåðåñå÷åíèÿ áóäóò A(1; −1) è B(3; 1).
Êîíòóð èíòåãðèðîâàíèÿ èçîáðàæåí íà ðèñ. 17.
1) Íåïîñðåäñòâåííîå èíòåãðèðîâàíèå.
Èíòåãðàë ïî êîíòóðó Γ ñëåäóåò ðàçáèòü íà ñóììó èíòåãðàëîâ ïî åãî ÷àñòÿì
îòðåçêó AB(Γ1 ) è äóãå ïàðàáîëû BCA(Γ2 ).
Ïðè âû÷èñëåíèè èíòåãðàëà ïî îòðåçêó Γ1 ñâåäåì åãî âû÷èñëåíèå ê âû÷èñëåíèþ îïðåäåëåííîãî èíòåãðàëà ïî ïåðåìåííîé x :
y = x − 2, dy = dx, x ∈ [1; 3].
∫
∫3
2ydx + (5x − 1)dy =
Γ1
∫3
(2(x − 2) + 5x − 1) dx =
1
(7x − 5)dx =
1
77
Ðèñóíîê 17 Êîíòóð èíòåãðèðîâàíèÿ ê çàäà÷å 4 âàðèàíòà 1
(
)3
7x2
7
63
=
− 5x
− 15 − + 5 = 18.
=
2
2
2
1
Àíàëîãè÷íî âû÷èñëÿåòñÿ èíòåãðàë ïî Γ2 :
y = −x2 + 5x − 5, dy = (−2x + 5)dx,
ïðè÷åì x èçìåíÿåòñÿ îò 3 äî 1.
∫
∫1
2ydx + (5x − 1)dy =
3
)
2(−x2 + 5x − 5) + (5x − 1)(−2x + 5) dx =
3
Γ2
∫1
(
)1
(
37
=
(−12x2 + 37x − 15)dx = −4x3 + x2 − 15x
2
3
37
333
− 15 + 108 −
+ 45 = −14.
2
2
Îêîí÷àòåëüíî íàõîäèì
I
2ydx + (5x − 1)dy =
= −4 +
Γ
78
∫
∫
2ydx + (5x − 1)dy +
Γ1
2ydx + (5x − 1)dy =
Γ2
= 18 − 14 = 4.
2) Âû÷èñëåíèå ïî ôîðìóëå Ãðèíà.
 íàøåì ñëó÷àå P (x, y) = 2y, Q(x, y) = 5x − 1.
∂P
∂Q
∂Q ∂P
= 2,
=5⇒
−
= 3.
∂y
∂x
∂x
∂y
Òîãäà
Ïî ôîðìóëå Ãðèíà (ñì.ôîðìóëó (31)) èìååì
∫∫
I
2ydx + (5x − 1)dy = 3
Γ
∫3
dxdy = 3
−x2∫+5x−5
dx
dy =
1
D
∫3
x−2
3
(−x + 4x − 3)dx = (−x + 6x − 9x) =
2
=3
3
2
1
1
= −27 + 54 − 27 + 1 − 6 + 9 = 4.
Âàðèàíò 2.
Çàäàíèå 1.
Âû÷èñëèòå ãðàäèåíò ñêàëÿðíîãî ïîëÿ
2 3
U (x, y, z) =
Ðåøåíèå.
y z
x2
(
√ )
√ √
3
.
â òî÷êå M0
2, 2,
2
Âû÷èñëèì ÷àñòíûå ïðîèçâîäíûå
−2y 2 z 3 ∂U
2yz 3 ∂U
3y 2 z 2
∂U
=
;
= 2 ;
=
.
∂x
x3
∂y
x
∂z
x2
Âû÷èñëèì çíà÷åíèÿ ÷àñòíûõ ïðîèçâîäíûõ â òî÷êå M0 :
∂U
∂x
M0
∂U
∂y
M0
√
√
3 6
2·2·3 3
=−
;
=− √
8
2 2 · 23
√
√
√
2 2·3 3 3 6
=
;
=
2·8
8
79
∂U
∂z
=
M0
3·2·3 9
= .
2·4
4
Òîãäà ñîãëàñíî ôîðìóëå (24)
(
grad U
√ √
2, 2,
√ )
√
√
3
3 6⃗ 3 6⃗ 9 ⃗
i+
j + k.
=−
2
8
8
4
Çàäàíèå 2.
Ïðîâåðüòå, áóäåò ëè ñîëåíîèäàëüíûì âåêòîðíîå ïîëå
F⃗ (x, y, z) = (5 + 3xy)⃗i + xy 2⃗j + (4 − 3yz − 3xyz)⃗k.
Ðåøåíèå.
Âû÷èñëèì äèâåðãåíöèþ âåêòîðíîãî ïîëÿ, ñì. ôîðìóëó (25):
div F⃗ (x, y, z) =
∂(5 + 3xy) ∂(xy 2 ) ∂(4 − 3yz − 3xyz)
+
+
=
∂x
∂y
∂z
= 3y + 2xy − 3y − 3xy = −xy ̸≡ 0.
Ñëåäîâàòåëüíî, âåêòîðíîå ïîëå íå ÿâëÿåòñÿ ñîëåíîèäàëüíûì â R3 .
Çàäàíèå 3.
Ïðîâåðüòå, áóäåò ëè ïîòåíöèàëüíûì âåêòîðíîå ïîëå
F⃗ (x, y, z) = (3x − 5yz)⃗i + (3y − 5xz)⃗j + (3z − 5xy)⃗k.
Ðåøåíèå.
Âû÷èñëèì ðîòîð âåêòîðíîãî ïîëÿ, ñì. ôîðìóëó (26):
rot F⃗ (x, y, z) =
⃗i
⃗j
⃗k
∂
∂x
∂
∂y
∂
∂z
=
(3x − 5yz) (3y − 5xz) (3z − 5xy)
)
(
)
∂(3x − 5yz) ∂(3z − 5xy) ⃗
∂(3z − 5xy) ∂(3y − 5xz) ⃗
−
−
i+
j+
=
∂y
∂z
∂z
∂x
)
(
∂(3y − 5xz) ∂(3x − 5yz) ⃗
−
k = (−5x + 5x)⃗i + (−5y + 5y)⃗j+
+
∂x
∂y
(
+(−5z + 5z)⃗k ≡ 0.
⃗ (x, y, z) ≡ 0, ñëåäîâàòåëüíî, äàííîå ïîëå ïîòåíöèàëüíî.
Èòàê, rot F
80
Çàäàíèå 4.
Âû÷èñëèòå öèðêóëÿöèþ ïëîñêîãî âåêòîðíîãî ïîëÿ
F⃗ (x, y) = (y + 2xy)⃗i + (x2 + 3x + y 2 )⃗j
âäîëü çàìêíóòîãî êîíòóðà Γ : y = x2 − 1, x + y = 1 (â ïîëîæèòåëüíîì
íàïðàâëåíèè)
1) íåïîñðåäñòâåííî,
2) èñïîëüçóÿ ôîðìóëó Ãðèíà.
Ðåøåíèå.
I
Öèðêóëÿöèÿ âåêòîðíîãî ïîëÿ ýòî èíòåãðàë
(y + 2xy)dx + (x2 + 3x + y 2 )dy.
Γ
Êîíòóð èíòåãðèðîâàíèÿ îáðàçîâàí äóãîé ïàðàáîëû è îòðåçêîì ïðÿìîé.
Äëÿ åãî ïîñòðîåíèÿ íåîáõîäèìî íàéòè òî÷êè ïåðåñå÷åíèÿ ïðÿìîé è ïàðàáîëû. Ðåøèì äëÿ ýòîãî ñîîòâåòñòâóþùóþ ñèñòåìó óðàâíåíèé:
{
x + y = 1,
⇔
y = x2 − 1.
{
y = 1 − x,
⇔
1 − x = x2 − 1.
{
y = 1 − x,
x2 + x − 2 = 0.
Êîðíÿìè êâàäðàòíîãî óðàâíåíèÿ ÿâëÿþòñÿ ÷èñëà x1 = −2, x2 = 1. Òàêèì
îáðàçîì, òî÷êè ïåðåñå÷åíèÿ áóäóò A(−2; 3) è B(1; 0).
Êîíòóð èíòåãðèðîâàíèÿ èçîáðàæåí íà ðèñ. 18
1) íåïîñðåäñòâåííîå èíòåãðèðîâàíèå.
Èíòåãðàë ïî êîíòóðó Γ ñëåäóåò ðàçáèòü íà ñóììó èíòåãðàëîâ ïî åãî ÷àñòÿì
îòðåçêó BA(Γ1 ) è äóãå ïàðàáîëû ACB(Γ2 ).
Ïðè âû÷èñëåíèè èíòåãðàëà ïî îòðåçêó Γ1 ñâåäåì åãî âû÷èñëåíèå ê âû÷èñëåíèþ îïðåäåëåííîãî èíòåãðàëà ïî ïåðåìåííîé x :
y = 1 − x, dy = −dx,
ïðè÷åì x èçìåíÿåòñÿ îò 1 äî −2.
∫
(y + 2xy)dx + (x2 + 3x + y 2 )dy =
Γ1
∫−2
=
(
)
(1 − x + 2x(1 − x)) dx + 3x + x2 + (1 − x)2 (−dx) =
1
81
Ðèñóíîê 18 Êîíòóð èíòåãðèðîâàíèÿ â çàäà÷å 4 âàðèàíòà 2
∫1
=−
(1 − x + 2x − 2x2 − 3x − x2 − 1 + 2x − x2 )dx =
−2
∫1
4x3
4x dx =
3
1
2
=
−2
−2
4
= (1 − (−8)) = 12.
3
Àíàëîãè÷íî âû÷èñëÿåòñÿ èíòåãðàë ïî Γ2 :
y = x2 − 1, dy = 2xdx, x ∈ [−2; 1].
∫
(y + 2xy)dx + (x2 + 3x + y 2 )dy =
Γ2
∫1
=
(
)
(
)
x2 − 1 + 2x(x2 − 1) dx + 3x + x2 + (x2 − 1)2 2xdx =
−2
∫1
(x2 − 1 + 2x3 − 2x + 6x2 + 2x3 + 2x5 − 4x3 + 2x)dx =
=
−2
82
(
∫1
(2x + 7x − 1)dx =
5
=
2
−2
)
x6 7x3
+
−x
3
3
(
1
=
−2
)
1 7
+ −1 −
3 3
(
)
64 56
−
−
+ 2 = −3.
3
3
Îêîí÷àòåëüíî íàõîäèì
I
(y + 2xy)dx + (x2 + 3x + y 2 )dy =
Γ
∫
∫
2
=
2
(y + 2xy)dx + (x2 + 3x + y 2 )dy =
(y + 2xy)dx + (x + 3x + y )dy +
Γ1
Γ2
= 12 − 3 = 9.
2) âû÷èñëåíèå ïî ôîðìóëå Ãðèíà (ñì. ôîðìóëó (31)).
 íàøåì ñëó÷àå P (x, y) = y + 2xy, Q(x, y) = x2 + 3x + y 2 .
Òîãäà
∂P
∂Q
= 1 + 2x,
= 2x + 3.
∂y
∂x
∂Q ∂P
−
= (2x + 3) − (1 + 2x) = 2.
∂x
∂y
Îáëàñòüþ èíòåãðèðîâàíèÿ D ÿâëÿåòñÿ îáëàñòü, îãðàíè÷åííàÿ êîíòóðîì
Γ (ñì. ðèñ. 18). Äâîéíîé èíòåãðàë, èìåþùèé â äàííîì ñëó÷àå ñìûñë óäâîåííîé ïëîùàäè îáëàñòè D, ëåãêî âû÷èñëÿåòñÿ íåïîñðåäñòâåííî:
Ñëåäîâàòåëüíî
I
∫∫
(y + 2xy)dx + (x2 + 3x + y 2 )dy = 2
Γ
∫1
dxdy = 2
∫1
)
(
x2 x3
−
(2 − x − x )dx = 2 2x −
2
3
1
2
=2
−2
=
−2
((
) (
))
1 1
8
=2
2− −
− −4 − 2 +
= 9.
2 3
3
83
dx
−2
D
∫1−x
x2 −1
dy =
Ëèòåðàòóðà
1 Øèïà÷åâ Â.Ñ. Âûñøàÿ ìàòåìàòèêà: Ó÷åá. äëÿ âóçîâ 5-å èçä., ñòåðåîòèï
/ Â. Ñ. Øèïà÷åâ. Èçä-âî ¾Âûñøàÿ øêîëà¿. Ì.: 2002. 479 ñ.
2 Êóäðÿâöåâ Ë. Ä. Êóðñ ìàòåìàòè÷åñêîãî àíàëèçà. Òîì 1 / Ë. Ä. Êóäðÿâöåâ. Èçä-âî ¾Äðîôà¿ Ì., 2003. 704 ñ.
3 Èëüèí, Â. À. Îñíîâû ìàòåìàòè÷åñêîãî àíàëèçà. ×àñòü 1 / Â. À. Èëüèí,
Ý. Ã. Ïîçíÿê. Èçä-âî ¾Ôèçìàòëèò¿. Ì., 2005. 648 ñ.
4 Áåðìàí, Ã. Í. Ñáîðíèê çàäà÷ ïî ìàòåìàòè÷åñêîìó àíàëèçó / Ã. Í. Áåðìàí. Èçä-âî ¾Ëàíü¿. ÑÏá., 2008. 608 ñ.
5 Äàíêî Ï. Å. Âûñøàÿ ìàòåìàòèêà â óïðàæíåíèÿõ è çàäà÷àõ / Ï. Å. Äàíêî, À. Ã. Ïîïîâ, Ò.ß. Êîæåâíèêîâà, Ñ.Ï. Äàíêî. Èçä-âà: Îíèêñ, Ìèð è
Îáðàçîâàíèå. Ì., 2008. 815 ñ.
6 Ëóíãó Ê. Í. Âûñøàÿ ìàòåìàòèêà: Ðóêîâîäñòâî ê ðåøåíèþ çàäà÷: Ó÷åáíîå ïîñîáèå / Ê. Í. Ëóíãó, Å.Â. Ìàêàðîâ. Èçä-âî Ôèçìàòëèò. Ì.,
2009. 381 ñ.
7 Âäîâèí, À.Þ. Âûñøàÿ ìàòåìàòèêà. Ñòàíäàðòíûå çàäà÷è ñ îñíîâàìè
òåîðèè / À.Þ. Âäîâèí, Ë.Â. Ìèõàë¼âà, Â. Ì. Ìóõèíà è äð. Èçä-âî
¾Ëàíü¿. ÑÏá., 2008. 256 ñ.
8 Êóçíåöîâ Ë. À. Ñáîðíèê çàäàíèé ïî âûñøåé ìàòåìàòèêå / Ë. À. Êóçíåöîâ. Èçä-âî ¾Ëàíü¿. ÑÏá., 2008. 240 ñ.
9 Áàðàíîâà Å. Ñ. Ïðàêòè÷åñêîå ïîñîáèå ïî âûñøåé ìàòåìàòèêå. Òèïîâûå
ðàñ÷åòû: Ó÷åáíîå ïîñîáèå / Å. Ñ. Áàðàíîâà, Í. Â. Âàñèëüåâà. Èçä-âî
¾Ïèòåð¿. ÑÏá., 2009. 320 ñ.
10 Çàïîðîæåö, Ã. È. Ðóêîâîäñòâî ê ðåøåíèþ çàäà÷ ïî ìàòåìàòè÷åñêîìó
àíàëèçó / Ã.È. Çàïîðîæåö. ÑÏá.: ¾Ëàíü¿, 2010. 464 ñ.
11 Áóòóçîâ, Â.Ô. Ìàòåìàòè÷åñêèé àíàëèç â âîïðîñàõ è çàäà÷àõ / Â.Ô. Áóòóçîâ, Í.×. Êðóòèöêàÿ, Ã.Í. Ìåäâåäåâ, À.À. Øèøêèí. Ì.: ÔÈÇÌÀÒËÈÒ, 2002. 480 ñ.
84
Ñîäåðæàíèå
Ââåäåíèå . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Êîíòðîëüíàÿ ðàáîòà  5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  5 . . . . . . . . . . . . . . . . . . . . . . . . .
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  5 . . . . . . . . . . . . . . . . . . . . . . .
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ . . . . . . . . . . . . . . . . . .
Êîíòðîëüíàÿ ðàáîòà  6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  6 . . . . . . . . . . . . . . . . . . . . . . . . .
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  6 . . . . . . . . . . . . . . . . . . . . . . .
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ . . . . . . . . . . . . . . . . . .
Êîíòðîëüíàÿ ðàáîòà  7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  7 . . . . . . . . . . . . . . . . . . . . . . . . .
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  7 . . . . . . . . . . . . . . . . . . . . . . .
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ . . . . . . . . . . . . . . . . . .
Êîíòðîëüíàÿ ðàáîòà  8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ñîäåðæàíèå êîíòðîëüíîé ðàáîòû  8 . . . . . . . . . . . . . . . . . . . . . . . . .
Îñíîâíûå ïîíÿòèÿ è ôîðìóëû . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Óñëîâèÿ çàäà÷ êîíòðîëüíîé ðàáîòû  8 . . . . . . . . . . . . . . . . . . . . . . .
Ïðèìåðû ðåøåíèÿ çàäà÷ òèïîâûõ âàðèàíòîâ . . . . . . . . . . . . . . . . . .
Ëèòåðàòóðà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3
5
5
5
9
15
24
24
24
26
34
44
44
44
49
57
66
66
66
69
75
84
Êàôåäðà ìàòåìàòèêè
Ìàòåìàòèêà (âòîðîé ñåìåñòð)
ó÷åáíîå ïîñîáèå äëÿ ñòóäåíòîâ çàî÷íîé ôîðìû
îáó÷åíèÿ
Òàòüÿíà Âàñèëüåâíà Ñëîáîäèíñêàÿ
Àëåêñåé Àíäðååâè÷ Ãðóçäêîâ
Òàòüÿíà Âèêòîðîâíà Âèííèê
Îòïå÷àòàíî ñ îðèãèíàë-ìàêåòà. Ôîðìàò 60 × 901/16
Ïå÷. ë. 5,25. Òèðàæ 50 ýêç.
Ñàíêò-Ïåòåðáóðãñêèé ãîñóäàðñòâåííûé òåõíîëîãè÷åñêèé èíñòèòóò
(Òåõíè÷åñêèé óíèâåðñèòåò)
190013, Ñàíêò-Ïåòåðáóðã, Ìîñêîâñêèé ïð., 26
Òèïîãðàôèÿ èçä. ÑÏáÃÒÈ(ÒÓ), òåë.: 4949365