1. Интегралы ФНП Диф_ур (853736), страница 26
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8.4. 8.5395 1) "#$ ) y = ex=2 + e;x=2 ( x1 = 0 x2 = 2):2) "#$ x = (t2 ; 2) sin t + 2t cos t y = (2 ; t2) cos t + 2t sin t ( = 0 = ):3) ( = a(1 + cos ') (a > 0):4) ppy = x ; x2 + arcsin x:31) e ; e;1 2) 3) 8a 4) 2:3||||| z = a z = b (a < b) ( .8.6). @ S (z ) * 6 z = const * <a b], (J ZbV = S (z )dz:(8:4);a. 8.6396A , 0 , y = y(x) y = 0x = a x = b ( . 8.7), * x = const S (x) = (y(x))2 :;. 8.72(J ZbV = (y(x))2 dx:a(8:5) 8.5. % (J , 6 - ( z = x4 + y2 * z = 1:< 9 z = 0 z = 1 ( . 8.8).22;.
8.8% *( z2 = const 0 < z 21 { (2 , 2yx 6 4 + 2 = z (2px z )2 + py 2 = 1: ( 2z)22, 6 xa2 + yb2 = 1 ab: )pp, S (z ) = 2pz 2z = 2 2z: 4 397S (z ) <0 1] (, z = 0 $ S (0) = 0). 6, (8.4):1 pS = 2 2zdz = 2z 2 = 2:00Z1pp> 22 %3 , # % x4 + y9 ; z2 = 1 #(# z = ;1 z = 2:. 36:||||| 8.6. % (J , -;; , ( y2 = 4x x = 1 0 (p . 8.9)..
8.9< A $ p , y = 2 x0 y = 00 x = 1 0 (p . 8.10).. 8.10398 (J (8.5).py(x) = 2 x0 a = 00 b = 1:V =Z10; 14xdx = 2x2 = 2:0 %3, / (/ ), x + e;xe y = 2 y = 0 x = 1 x = 2 /( # 0x: e4 ; e;4 ; e2 + e;2 + 4:8|||||399 9 ! . # D = f(x y)g { $ Oxy. @ $ (x y) 2 D z , $ D f (x y) x y: x y * . G$ D ( * f (x y):@ , ( ( $ , $ , * . , z = f (x y) f (x y ): K ( .; (x y) $ z * ( . 9.1).;. 9.1 9.1.
- ( vu22uxytz = 1; ; :a2 b2< % 2 x y,2 $ 1 ; xa2 ; yb2 , ..222222xyxyxy1 ; a2 ; b2 0 a2 + b2 1: 9 a2 + b2 = 1 - 6 , ( , 6 6 ( . . 9.2). >400;. 9.2 9.2. - ( z = ln(y2 ; 4x + 8):< 2( z = ln t { ft : t > 0g:6 ( { $ (x y) , y2 ; 4x +8 > 0: ;, y2 = 4x ; 8 { ( 3 x = 2 y = 0: 9 02 ; 4 0 + 8 > 0 (0,0)$ ( .
9 (3,0), (, ( , 02 ; 4 3+8 < 0. B y2 ; 4x + 8 $ 0, .. (, y2 ; 4x + 8 < 0 * ( > 0 * . ), ( ( , $ ( y2 = 4x ; 8 ( . . 9.3). >;. 9.3 9.3. - ( z = px + y + px ; y:< 2( * $ (x y) (x+y 0x ; y 0401 , x + y = 0 x ; y = 0 * ( . . 9.4). >;. 9.4 %# ():1) z = R2 ; x12 ; y2 2) z = px1+ y + px1; y ;;3) z = ln xy:1) "# #(#$ - #(* ( (5# x2 + y2 = R2 (#.
9.5).. 9.5x+y > 02) x ; y > 0 { / #$ - / (%- )), (#(#$ # x + y = 0 x ; y = 0 (#. #. 9.6).. 9.6402;3) ,%#$ xy > 0 -%5 #. 9.7.. 9.7||||| 9.4. /( 1) z = x2 + y2:;;< ( ( . 9.8). >. 9.8q2) z = x2 + y2: , < C( 222: zz =0x + y , .. ( . 9.9). >. 9.9403q3) z = 1 ; x2 ; y2:222z=1;x;y, z 0 ..< C ( 222 x + y + z = 1 ( . 9.10). >z0;;. 9.104) z = x2 ; y2:< K z = x2 ; y2 ( * ( ( () ( .9.11). >. 9.11 D = f(x y z )g { $ Oxyz: @ $ (x y z ) 2 D u, , $ D f (x y z ) x y z: x y z * .G$ D f (x y z ):% f (x y z ) * f (x y z ) = const $ * (, ( , ).404 9.5.
- ( qu = R2 ; x2 ; y2 ; z 2 + px2 + y21+ z 2 ; r2 R > r , x2 + y2 + z 2 = 2 * . , 3 x2 + y2 + z 2 = R2.< - ( 92222R ;x ;y ;z 0 =x2 + y2 + z 2 ; r2 > 0 9x2 + y2 + z 2 R2 = :x2 + y2 + z 2 > r2 ), ( * 3 r2 < x2 + y2 + z 2 R2 r R.2, x2 + y2 + z 2 = 2 = const qu = R2 ; 2 + p21; r2 * .9 (, x2 + y2 + z 2 = 2 r < R * u = u(x y z ).F *qu() = R2 ; 2 + p 1 (r R]:2 ; r2- 3 011 ;q @p 1Aq=;+R2 ; 2 (2 ; r2)3R2 ; 2 (2 ; r2)3 : 0 < r < R u0 () < 0 , u() (. 6 3 u() = R, .. x2 + y2 + z 2 = R2:>u0 () = ; p405 !2 M0(x0 y0) .. $ M (x y) (x ; x0)2 + (y ; y0)2 < 2; { M0(x0 y0) * ( ( (M0) ( . 9.12).. 9.12@ (M0) M0 { M0(x0 y0) * ( ( (M0 ):L A f (x y) M0(x0 y0) 8" > 0 9 (M0) : 8M 2 (M0) jf (x y) ; Aj < ": 6 3A = lim! f (x y ):;xy!y00x% ( .
9.13).* M0(x0 y0). 9.132,x ; x0 = cos y ; y0 = sin 406limf (x0 + sin y0 + cos ):! f (x y ) = lim!0xy(9:1)!y00x (9.1) x y * : () , (9.1) $ 6 ( , lim!! f (x y ) .) 9.6. % 22x+yp2 2lim:!x+y+1;1!xy00xyx0y00! x = cos 222x+yp2 2p2< lim= y = sin = lim= 0 =!!0x+y+1;1+1;1! p 22 + 1 ; 1 2=2 = = 2: > = lim!0 2 =2 ! 0xy00 9.7. % p22x y + 1 ; 1:limx2 + y2!!00xypx2y2 + 1 ; 1 x=cos=< lim =22!y=sinx+y!xy00 0 ! p4 cos2 sin2 + 1 ; 1+1;1=2= lim==2!00 ! 0 =22242sin2 = 0: >cossin= lim=lim!0!0228q 9.8.
% 221;cos(x+y):lim!(x2 + y2)!xy004070! x = cos 2221;cos1;cos(x+y)= 0 == y = sin = lim< lim2 + y2)2!!0(x!xy00 1 ; cos 2 =2 4 = lim 2 = 0:= ! 0 = lim!0 22 !0 2> "#$:sin(x3 + y3) 1) limx!0x2 + y22) lim(1 + x2y2);1=(x2+y2 ):x!0!!y 0y 01 7 / 5 #$-/$ / -$ lim(1 + z)1=z = e:z!02) 1:1) 0||||| !"#$% &'!() !#("*%(+ ,!! +4 f (x y) (x0 y0) lim! f (x y ) = f (x0 y0 ):xy!xy00 9.9. / * x = 0y=0f (x y) = x2x+y y2 2 2f (0 0) = 0: x = cos 2 2xyf (x y) = lim= y = sin =< lim2 + y2!!x!!xy00xy004 cos2 sin2 = lim 2 sin2 2 = 0:= lim!0!024/, lim! f (x y ) = f (0 0) = 0 ! (0 0): >xy00408 9.10.
/ x = 0 y = 0< ; 2y2xf (x y) = x4 + y4 x = cos y = sin f (0 0) = 0:4 cos2 sin2 = sin2 2 :) lim!044F , ( , 6 (0,0) .), (0 0): > 8##/$ /#$ x = 0 y = 0:1) f (x y) = x2 xy+ y2 f (0 0) = 03 32) f (x y) = x2x+y y2 f (0 0) = 0:1) 9-/2) /.|||||C $ ( . 9.11. - 22x :z = yy2 +; 2x< 4 , ( . 6 * { (y2 = 2x: >409 ( -/ ()1) z = x ;1 y 2) z = sin1x + sin1y :1) y = x:2) x = m y = n (m n | ) #).|||||410 10 - . $ z = f (x y) (x0 y0): @ (x0 y0) x y * f (x0 + 4x y0) ; f (x0 y0)fx0 (x0 y0) = 4limx!04xf (x0 y0 + 4y) ; f (x0 y0) :fy0 (x0 y0) = 4limy!04y% (x y) @z @z : A z = f (x y) (* $ @x@y * @z y ( , @x@z ), @yx:!@z = d z @x dx y=const @z = d z @y dy x=const : 10.1.
- $ - .1) z = x2y3 :< ) y , $ $ , .. (Cu)0 = Cu0 C = const @z = y3 2x = 2xy3@x222 2 x , @z@y = x 3y = 3x y : >2) z = 5xy4 + 3x3y2:< ) y , 6 , ..(u + v)0 = u0 + v0 411 x ,@z = 5y4 + 9x2y20@x@z = 20xy3 + 6x3y: >@y3) z = x2y3 sin x cos y:< $ , * * * * * , .. (uv)0 = u0 v + uv0 :@z = 2xy3 sin x cos y + x2y3 cos x cos y@x@z = 3x2y2 sin x cos y + x2y3 sin x (; sin y): >@y2xsin y :4) z = y3 ++ cosx (<u !0 u0 v ; uv0v =v2 @z = 2x(y3 + cos x) ; (x2 + sin y) (; sin x) =@x(y3 + cos x)23(x2 + sin y) sin x = 2x(y + cos(yx3)++cosx)2@z = cos y (y3 + cos x) ; (x2 + sin y) 3y2 =@y(y3 + cos x)23y ; 3(x2 + sin y)y2 : >= (y + cos x)(ycos3 + cos x)24125) z = xpy + pyx :34< ;3 * * (:z = xy1=3 + yx;1=4:- @z = y1=3 + y ; 1 ! x;5=4 = py ; py @x44 x5@z = x 1 y;2=3 + x;1=4 = px + p1 : >@y33 y2x33446) z = (5x3y ; y4 + 7)4:< A $ z = z (u(x y)) z (u) {3 , u(x y) { .
5 ,@z = @z @u @z = @z @u @x @u @x @y @u @y@z = 4(5x3y ; y4 + 7)3 5 3x2y = 60x2y(5x3 y ; y4 + 7)3@x@z = 4(5x3y ; y4 + 7)3 (5x3 ; 4y3): >@y7) z = sin(x2y3):< ; z (u) = sin u a u = x2y3 :@z = @z @u = cos(x2y3) 2xy3 = 2xy3 cos(x2y3 )@x @u @x@z = @z @u = cos(x2y3) x2 3y2 = 3x2y2 cos(x2y2): >@y @u @y4138) z = exy :2@z = exy x 2y = 2xyexy : >@y9) z = sin2(x3y4 ):< A $ z = z (u(v(x y))) z (u) = u2 u(v) = sin v v(x y) = x3y4 : - :@z = @z @u @v =@x @u @v @x= 2 sin(x3y4) cos(x3y4) 3x2y4 = 3x2y4 sin(2x3y4 )0@z = @z @u @v =@y @u @v @y@z = exy y2 = y2 exy < @x2222= 2 sin(x3y4) cos(x3y4) x3 4y3 = 4x3y3 sin(2x3y4 ): >10) z = xy :< x * *@z = yxy;1 @x y { *@z = xy ln x: >@y # -/:3 + y31) z = x3y ; y3x2) z = xx2 +y2 4) z = e;x=y :3) z = xpy + p3yx 414@z = 3x2y ; y31) @x@z = x4 + 3x2y2 ; 2xy3 2) @x(x2 + y2)2@z = py ; py 3) @x3 3 x4@z = ; 1 e;x=y 4) @xy@z = x3 ; 3y2x@y@z = y4 + 3x2y2 ; 2x3y @y(x2 + y2)2@z = x + p1 @y 2py 3 x@z = x e;x=y :@y y2|||||L u = u(x y z ) * :@u = d u @x dx y =constz =const@u = d u @y dy x=constz =const@u = d u @z dz x=consty =const: 10.2.
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