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Âû÷èñëèòü öèðêóëÿöèþ ïëîñêîãî âåêòîðíîãî ïîëÿIP (x, y)dx + Q(x, y)dyLäâóìÿ ñïîñîáàìè: íåïîñðåäñòâåííî è ïî ôîðìóëå Ãðèíà.12N123456789101112131415161718192021222324252627282930L∆ABCA (2, 1) B (2, 3) C (4, 3)x2 /9 + y 2 /4 = 1x2 + y 2 = 4yx2 + y 2 = 16x2 + y 2 /16 = 1y = sin x, y = 0, 0 6 x 6 πx2 /9 + y 2 = 4y = 4x2 , y = 4y = 5x2 , y = 10x∆ABCA (0, 1) B (2, 5) C (0, 5)x = 4y 2 , x = 16x2 + y 2 = 25y = x2 , y = 8xx = 9y 2 , x = 3yx2 + y 2 = 16x2 /25 + y 2 /4 = 14y = x2 + 4, y = 3x − 4∆ABCA (0, 0) B (2, 3) C (0, 3)|x| + |y| = 4x2 + y 2 = 36x2 /16 + y 2 /9 = 1y = x2 , y = 4x − 3∆ABCA (3, 3) B (5, 5) C (3, 5)√y = x, 4y = x + 3∆ABCA (2, 1) B (1, 4) C (2, 4)√y = 4 x, y = 4xx2 + y 2 = 4∆ABCA (3, 4) B (5, 6) C (3, 6)x2 + y 2 = 9x2 /4 + y 2 /25 = 1P (x, y)Q (x, y)x2 + y 22 (x + y)2xy + x + yxy + 3−xy 22x + 2yy2x2 y 2xy 2(x + y)2xy + 2x − 2yxy − 2x + 3y2x2 y−2x + 2yxyx2 + 4x−y(x − y)23xy 2x3 + 4x2y 2 + xyy 2 + x24xyxy2x + 3y 25y + x2x + 2yx2 + xyx2 + y 35x22x2 + 3y 23x − 2y 2−3xx − 2y4xyx2 + y 22 (x + y)2x + y2x+yx2 + 4xy−2 (x − y)2x3 /3x2 − y 24xy + y 2x2 + y 2(x + y)2x2 − 5yy2 + 2y2xyx + 2yx2 + 5y2y 2 − xy24x2 yy − 2x−x2 − y 2(x + y)2x + y2y + x2x − y2−x13pÇÀÄÀ×À 7.
Íèæå r = xi+yj+zk, |r| = x2 + y 2 + z 2 , c -ïîñòîÿííûéâåêòîð.1. Íàéòè rot(cf (|r|)).2. Íàéòè rot[c, rf (|r|)].3. Äîêàçàòü, ÷òî div[a, b] = b rot a − a rot b.4. Íàéòè div(u grad u).5. Íàéòè óãîë ϕ ìåæäó ãðàäèåíòàìè ïîëÿ u = x/(x2 + y 2 + z 2 ) âòî÷êàõ A(1, −2, 2) è B(3, 1, 0).6. Äîêàçàòü, ÷òî rot(ua) = u rot a − [a, grad u].7.
Íàéòè div (b(r, a)).8. Íàéòè grad u, ãäå u =µ|[c, r]|.¶ωyωx9. Íàéòè rot a, ãäå a = −,,z .2π x2 + y 2 2π x2 + y 210. Íàéòè div rot a.11. Íàéòè rot grad u.12. Íàéòè óãîë ϕ ìåæäó ãðàäèåíòàìè ïîëÿ u = y/(x2 + y 2 + z 2 ) âòî÷êàõ A(1, 2, 2) è B(3, 2, 0).13. Äîêàçàòü, ÷òî div(ua) = (a, grad u) + u div a.14. Íàéòè rot a, ãäå a = [grad u, b], u = y 2 − 2xz + z 2 , b = i + 2j − 3k.15. Íàéòè ïðîèçâîäíóþ ïîëÿ u = x2 + y 2 − 3x + 2y â òî÷êå M0 (0, 1, 2)ïî íàïðàâëåíèþ îò òî÷êè M0 ê òî÷êå M (3, 1, 6).16.
Íàéòè rot(f (|r|)r).17. Íàéòè div[b, r], ãäå b = x2 i + y 2 j.18. Íàéòè rot a, ãäå a = (yi + zj + xk)/|r|.19. Íàéòè óãîë ϕ ìåæäó ãðàäèåíòàìè ïîëÿ u = z/(x2 + y 2 + z 2 ) âòî÷êàõ A(2, 1, 1) è B(−3, −2, 1).20. Íàéòè rot a, ãäå a = [grad u, b], u = x2 − 2yz + y 3 , b = 2i − 3j + 6k.21. Íàéòè div[b, r], ãäå b = y 2 i − x2 k.22.
Íàéòè div(f (|r|)r).23. Íàéòè ïðîèçâîäíóþ ïîëÿ u = xy+yz−2y+4z â òî÷êå M0 (−1, 2, −3)ïî íàïðàâëåíèþ îò òî÷êè M0 ê òî÷êå M (−4, 2, 1).24. Íàéòè rot a, ãäå a = (zi + xj + yk)/|r|.25. Íàéòè ïðîèçâîäíóþ ïîëÿ u = y 2 z − 2xyz + z 2 â òî÷êå M0 (3, 1, 1)ïî íàïðàâëåíèþ âåêòîðà a, åñëè a îáðàçóåò ñ êîîðäèíàòíûìè îñÿìèîñòðûå óãëû α, β, γ , α = π/3, β = π/4.26. Íàéòè rot a, ãäå a = [grad u, b], u = xyz − 2y + z 3 , b = 2i − 3j − 4k.1427.
Íàéòè div[b, r], ãäå b = xyi − yzj + x2 k.28. Íàéòè óãîë ϕ ìåæäó ãðàäèåíòàìè ïîëÿ u = (z − x)/(x2 + y 2 + z 2 )â òî÷êàõ A(−2, 1, 3) è B(3, 4, −2).ÇÀÄÀ×À 8. Âû÷èñëèòü ïëîùàäü ÷àñòè ïîâåðõíîñòè σ , çàêëþ÷åííóþ âíóòðè öèëèíäðè÷åñêîé ïîâåðõíîñòè Ö.NσÖ21 x=√2yzy + z2 = 42 y = 9 − x2 − z 2x2 + y 2 = 43 x=3−y−zy 2 + z 2 = 2z4 y 2 = x2 + z 2x2 + z 2 = 4x5 y 2 + z 2 = 1, z > 0x2 + y 2 = 16 x2 + y 2 + z 2 = 4, z 6 0x2 + y 2 = 2x7 x2 + y 2 + z 2 = 16, x > 0 y 2 + z 2 = 98 x2 = y 2 + z 2 , x 6 0y2 + z2 = 19 2z = xyx2 + y 2 = 410 2z = x2 + y 2x2 + y 2 = 211 y 2 = 2xz0 6 x 6 2, 0 6 z 6 22212 z = 9p− x − yx2 + y 2 = 513 x = p y 2 + z 2y 2 + z 2 = 4z14 z = y 2 − x2x2 + y 2 = 815 2x = y 2 − z 2y2 + z2 = 1¡ 2¢216 2y = x2 + z 2x + z 2 = 2xz17 8 − z = (x2 + y 2 )3/2x2 + y 2 = 4¡¢218 y = x2 + z 24 x2 + z 2 = x2 − z 2¡ 2¢219 x2 + y 2 = z 2x + y 2 = 9xy¡ 2¢220 x2 + y 2 + z 2 = 1y + z 2 = 2yzp¢2¡ 221 z = x¡2 − y 2 ¢x + y 2 = x2 − y 222 z 2 = 4 x2 + y 2x2 + y 2 = 4y¡ 2¢223 4z = x2 + y 2x + y 2 = 8xy24 x2 + y 2 + z 2 = 4y 2 + z 2 = 2y25 x2 + y 2 + z 2 = 36, z 6 0 x2 + y 2 = 1626 x2 = y 2 − z 2y 2 + z 2 = 2z27 4z = x2 + y 2 , z 6 1y 2 = 3x2¡ 2¢228 z = 6 − 2x + 3yx + y 2 = 25xy29 y 2 + z 2 = 3, z > 0x + y = 0, x − y = 02230 2z = x − yx2 + y 2 = 115ÇÀÄÀ×À 9.
Íàéòè ïîòîê âåêòîðíîãî ïîëÿ a ÷åðåç çàìêíóòóþ ïîâåðõíîñòü σ äâóìÿ ñïîñîáàìè: 1) íåïîñðåäñòâåííî, âû÷èñëÿÿ ïîòîêè÷åðåç âñå ãëàäêèå êóñêè ïîâåðõíîñòè σ ; 2) ïî òåîðåìå Îñòðîãðàäñêîãî-Ãàóññà.N123456789101112131415161718192021222324252627282930axi + y 2 j − 2zkxi − yj + z 2 kxzi − 2xyj + k(1 − y)xi + yzj + zkxyi + xyj − xzk3xi + 2yj + z 2 k2i − 3y 2 j + zkx2 i − z 2 j + y 2 kyz(i − j) + 2xki + 3j + 2z 2 kxi + 2yj + 3zkxzi + 3yzj + xzkzi + y 2 j + xzk2xi − 3j + yzkx2 j − z 2 ki − yj + x(3 + z)kyi − zj + xyzkx2 i − 2yj + z 2 kyzi + xyj + zk3xyi + zyj + xzkxi + y 2 j + z 2 kxyzi + 2xyj − z 2 k−xi + yj − 2zkxi + 3y 2 j + 3z 2 kxzi + y 2 j + yzkzi − 3yj + xyzkxi + yj + 3xzkxi − 2yj + 8zkx2 j + zkx2 i + yj + zkσ2z =− y2, z = 0z 2 = x2 + y 2 , z = 4x2 + y 2 + z 2 = 1, x > 0(2 − z)2 = x2 + y 2 , z = 03z = 9 − x2 − y 2 , z = 0x2 + y 2 + z 2 = 4, y > 04z = x2 + y 2 , z = 9x2 + y 2 + z 2 = 9, x > 0y = 1 − x2 − z 2 , y = 05 − z = x2 + y 2 , z = −4y 2 = 4(x2 + z 2 ), y = 6x2 + y 2 + z 2 = 16, z > 0x2 = y 2 + z 2 , x = 79z = x2 + y 2 , z = 13z = 4 − x2 − y 2 , z = 1(2 − x)2 = y 2 + z 2 , x = 5x2 + y 2 + z 2 = 4, x 6 0y = x2 + z 2 , y = 8y 2 = x2 + z 2 , y = −2z = 9(x2 + y 2 ), z = 364z = 16 − x2 − y 2 , z = 3x2 + y 2 + z 2 = 9, y 6 0x2 = y 2 + z 2 , x = −43y − 2 = x2 + z 2 , y = 6z 2 = 4(x2 + y 2 ), z = 4y = 1 − x2 − z 2 , y = −3x2 + y 2 + z 2 = 16, z 6 0z = 25 − x2 − y 2 , z = 92z = 2 − x2 − y 2 , z = 0z = x2 + y 2 , z = 49 − x216ÇÀÄÀ×À 10.
Íàéòè öèðêóëÿöèþ âåêòîðíîãî ïîëÿ a ïî êîíòóðó Γäâóìÿ ñïîñîáàìè: 1) íåïîñðåäñòâåííî, âû÷èñëÿÿ ëèíåéíûé èíòåãðàë âåêòîðíîãî ïîëÿ ïî êîíòóðó Γ; 2) ïî òåîðåìå Ñòîêñà.N123456789101112131415161718192021222324252627282930azi − yj + y 2 k3zi + y 2 j − 2ykyzi − x2 jyi + xyj − zkyzi + xj + xzkxy(i − j) − zkzi − xyj + x2 kzi + x2 j − yky 2 i + zj − xkz 2 i + x2 j − ykzyi + 2j + xkzi − 2xj + x2 ky 2 i − z 2 j + zkzi + 2xj − x2 k2zi − 3yj − x2 k3zi + x2 j + 2xkxz(i + j + k)yi − xj + xkyzi + 2xj − ykxzi − zj + 2ykyi − zj + xk2zi + yzj − xk4yi − z 2 j + xk3xi + 2xzj − yk2xyi − 3xj − y 2 k−zi + yj + 2xkyj + 2k2zi − x2 j + 3xkyi − x2 j + xkxy(i + j + k)Γ= 9 − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 = 4, x + y + z = 2z 2 = 2 − x − y, x = 0, y = 0, z = 0 (1 îêòàíò)x + y + z = 2, x = 0, y = 0, z = 0x2 + y 2 = 1, y = zx2 + y 2 = 1 − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + z 2 = 1, x = y + 1x + y + 2z = 4, x = 0, y = 0, z = 0x2 + z 2 = 9, y = z + 12x + 3y + z = 6, x = 0, y = 0, z = 0x2 = 1 − y − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 = 4, x + y + z = 3x + 2y + z = 3, x = 0, y = 0, z = 0y 2 = 2 − x − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 = 9, x + y + z = 12x2 + y 2 = 1, z = y − 12x + y + 3z = 6, x = 0, y = 0, z = 0x2 + z 2 = 4 − y, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 = 4, z = x + 2y 2 + z 2 = 16, x + y + z = 4x2 + y 2 + z 2 = 9, x = 0, y = 0, z = 0 (1 îêòàíò)x = y2 + z2, x = 9x2 + z 2 = 1, x = yx + 2y + z = 4, x = 0, y = 0, z = 0x2 + y 2 = 4 − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 + z 2 = 1, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 + z 2 = 16, z = yx2 + y 2 = 9 − z, x = 0, y = 0, z = 0 (1 îêòàíò)x2 + y 2 = 4, z = y + 2x + 2y + z = 4, x = 0, y = 0, z = 0x2+ y217ÊÀËÅÍÄÀÐÍÛÉ ÏËÀÍ ÓÏÐÀÆÍÅÍÈÉ13.
Íåîïðåäåëåííûé èíòåãðàë. Âû÷èñëåíèå èíòåãðàëîâ ñëåäóþùèõòèïîâ:ZZZAx + BPn (x)dx, α = 1/2, 1;dx;R(x, xm/n , xp/q )dx;2α(ax + bx + c)Rn (x)ZZ sin ax ln axPn (x) cos ax dx;Pn (x) arctg ax dx; axe...½¾ZZZsinbxR(sin x, cos x)dx;sinp x, cosq xdx;sin axdx.cos bx4. Îïðåäåëåííûé èíòåãðàë.5. Ïðèëîæåíèÿ îïðåäåëåííîãî èíòåãðàëà.6. Êîíòðîëüíàÿ ðàáîòà.7. Ðàçáîð îøèáîê êîíòðîëüíîé ðàáîòû. Íåñîáñòâåííûå èíòåãðàëû.8,9.
Äâîéíîé èíòåãðàë.10. Òðîéíîé èíòåãðàë (ñôåðè÷åñêèå êîîðäèíàòû ïî óñìîòðåíèþ ïðåïîäàâàòåëÿ).11. Ñêàëÿðíûå è âåêòîðíûå ïîëÿ.12. Êðèâîëèíåéíûé èíòåãðàë. Öèðêóëÿöèÿ.13. Ïîòîê âåêòîðíîãî ïîëÿ.14. Òåîðåìû Îñòðîãðàäñêîãî-Ãàóññà è Ñòîêñà.15,16. Ïðèåì òèïîâîãî ðàñ÷åòà.ÒÅÎÐÅÒÈ×ÅÑÊÈÅ ÂÎÏÐÎÑÛ Ê ÝÊÇÀÌÅÍÓ (ÇÀ×ÅÒÓ)1. Îïðåäåëåíèå ïåðâîîáðàçíîé, òåîðåìà î ìíîæåñòâå âñåõ ïåðâîîáðàçíûõ. Íåîïðåäåëåííûé èíòåãðàë.
Ñâîéñòâî ëèíåéíîñòè.2. Íåîïðåäåëåííûé èíòåãðàë. Òåîðåìà î çàìåíå ïåðåìåííîé. Ôîðìóëà èíòåãðèðîâàíèÿ ïî ÷àñòÿì.3. Îáùàÿ ñõåìà èíòåãðèðîâàíèÿ ðàöèîíàëüíûõ ôóíêöèé.4. Èíòåãðèðîâàíèå ïðîñòåéøèõ äðîáåé.5. Èíòåãðèðîâàíèå òðèãîíîìåòðè÷åñêèõ ôóíêöèé.6. Èíòåãðèðîâàíèå äðîáíî-ëèíåéíûõ èððàöèîíàëüíîñòåé.187. Èíòåãðèðîâàíèå êâàäðàòè÷íûõ èððàöèîíàëüíîñòåé. Òðèãîíîìåòðè÷åñêèå ïîäñòàíîâêè.8. Îïðåäåëåííûé èíòåãðàë: îïðåäåëåíèå, ãåîìåòðè÷åñêèé è ìåõàíè÷åñêèé ñìûñë.
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