010-104-2002-2-final - > ; o 2 eK 2002 12 14{ 1r 3r...

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Unformatted text preview: > ; o 2 eK 2002 12 14{ 1r 3r 4 Z 9 < : s2: j h ]_ \ s&` "r r(8& 120&). HH x h 1. (10&) /_ & (0, 0, 0)\"' (1, 1, 1)t curve N h B h f C = {(x, y, z) | x2 = y, y 2 = xz, 0 x 1} 9 : h \ f{ M & r xy dx + 3zx dy - 5x2 yz dz C \ . 2. 3. (10&) /S = {(x, y, z) | x2 + y 2 = 1, 0 z x + 1} _ surface G h B integral S z d\ . (15&) /\ # %% N h B Q i D = {(x, y, z) | 1 x 2, 0 xy 2, 0 z 1} / h \ @K & r (x2 y + 3xyz) dxdydz D \ . 4. 5. (20&) /_ S = {(x, y, z) | x2 + y 2 + z 2 = 4} 0\" vector field N h B Af F = x2 i + xzj + 3zk_ flux integral S F n d\ . N B /\ vector field F(x, y, z) = #4 e . QR (a) (10&) curl(F) = 0e` #. h (b) (10&) xy _ " C = {(x, y, z) | 3x2 + 2y 2 = 1, z = 0}\ h ' : r> ~ 3 [ M_ flow integral C F dr\ . t x2 -y x i+ 2 j 2 +y x + y2 6. N B i f p < < /_ %% D\" &_ y p0 f g #4 e a r x QR S\ outward orientation` D_ boundary . H (a) (10&) div(f g) = f div( g) + h (b) (10&) h #. S f D ge` #. f div( g) - gdiv( f ) dV e` (f g - g f ) n d = 7. 2032 12 xx{ :{ _ @: {{ M. 4 xa G Z 9 9 D / 99 x o 2002 ` /p&\" $&` e F L r h<f h Ga H H H j qo 9 / / p A s] `<_ _ |'` lu ? @ 5o\ 0 t x 4 : G K s. #\ :K \ F _ t r r x p 9< t_ x x\ (x 18), y\ 1(y 0)s ` M H q p : f (x, y) = e-((x-18) 2 +y 2 ) Q #. r 8 s a x b, {s c x 9 y d\ 7 " F \ tt |_ qs f G H d c a b f (x, y) dxdy Q #. (a) (10&) F _ 8tt h G x и &` sr. 0 18 f (x, y) dxdy\ . , > (b) (15&) %@ \ $K s xs 1s y r h i/ r3 : p q + X |s ȳ S r x + 82 , if 18 x 118, (x, y) = 200 0, otherwise. Q j F / # ` . z] G 8 @q H x p q > ' p+ Y (G\ 1t . , e-10000 0s K a. r Ҹ % ~ ...
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