2 .RichardFeynmann2 . .Stan Ulam . -- . , . , Mathematica. Mathematica - . Mathematica - -, .. Mathe-matica . . -, -ii . - . -. - 17o, -. 1637, 1666, -. . , . . , , . ,- . -, . TychoBrache . Kepler (). - Newton.Newton,-, . .,-,-. -, ; . - . .iii Mathematica.. . . , - -. 202008iv 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . 21.3 . . . . 41.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 R3. . . . . . . . . . . . . . . . . . . . . 161.5.1 R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.2 R3. . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Mathematica . . . . . . . . . . . . . . . . . . . . . 292.1 . . . . . . . . . . . . . . . 292.1.1 Plot3D . . . . . . . . . . . . . . . . . . . . . . . . 302.2 . . . . . . . . . . . . . . . . . . 332.2.1 ContourPlot . . . . . . . . . . . . . . . . . . . . 332.3 . . . 362.4 . . . . . . . . . . . . . . . . . 382.4.1 . . 382.4.2 . 412.5 GraphicsContourPlot3D. . . . . . . . . . . . . . . . . 422.6 GraphicsParametricPlot3D. . . . . . . . . . . . . . 442.7 GraphicsShapes . . . . . . . . . . . . . . . . . . . . . . . 462.8 . . . . . . . . . . . . . . . . . . . . . . 472.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.8.2 . . . . . . . . . . . . . . 482.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.9 . . . . . . . . . . . . . . . . . . . . . . . . . 49ii 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 . . . . . . . . . . . . . . . . . . . 533.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.7 Mathematica . . . . . . . . . . . . . . 843.7.1 . . . . . . . . . . . 843.7.2 . . . . . . . . . . . . 863.7.3 . . . . . . . . . . . . . . . . . . . . . 873.7.4 . . . . . . . . . . . . . 883.7.5 . . . . . . . . . . . . 893.7.6 . . . . . . . . . . . 903.7.7 Mathematica. . . . . . . . . . 923.7.8 . . . . . . . . . . . . . . . . . . 984 . . . . . . . . 994.1 . . . . . . . . 994.2 . . . . . . . . . 1044.3 . . . . . . . . . . 1054.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.5 . . . . . . . . . . 1124.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.8 Mathematica . . . . . . . . . . . . 1244.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.8.3 . . . . . . . . . 1274.8.4 . . . . . . . . . . . . 1294.8.5 Mathematica. . . . . . . . . . 1304.8.6 . . . . . . . . . . . . . . . . . . 134iii5 . . . . . . . . . . . . . . . .1355.1 . . . . . . . . . . . . . . . . . 1355.2 Euler . . . . . . . . 1415.3 . . . . . . . . . . . . . . . . . . . . 1445.4 Taylor . . . . . . . . . . . 1475.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.5.1 . . . . . . . . . . . . . . . . 1505.5.2 . . . . . . . . . . . . . . . . . . . 1525.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.7 . . . . . . . . 1555.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.12 Mathematica1805.12.1 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1805.12.2 Mathematica. . . . . . . . . . 1825.12.3 . . . . . . . . . . . . . . . . . . 1886 . . . . . . . . . . . . . . . . . . . . . . . . . . .1896.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.2 , . . . . . . . . . . . . . . . . 1906.3 . . . . . . . . . . . . . . . . . . . . . . . 1926.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.4.1 . . . . . . . . . . . . . . 1936.4.2 . . . . . . . . . . . 1966.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.5.1 R3. . . . . . . . . . . . . . . . . . . . . . 2016.5.2 - R3. . . . . . . . . . . . . . . . . . . . . . . . . . 2036.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2146.8 Mathematica . . . . . . . 2156.8.1 . . . . . . . . . . . . 2156.8.2 . . . . . . . . . . . . . . . . 2166.8.3 . 2176.8.4 Laplace. . . . . . . . . . . . . . . . . . . . . . . . . . 218iv 6.8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.8.6 . . . . . . . . . . . . . . . . . . 2247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2257.1 . . . . . . . . . . . . . . . . . . . 2257.2 . . . . . . . . . . . . . . . . . . 2277.3 . . . . . . . . . . . . 2327.4 . . . . . . . . . . . . . . . . . . . 2337.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2347.5.1 Langrange. . . . . . . . . . . . . . . . . 2357.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.8 Mathematica . . . . . . . . . . . . . . . . 2608 . . . . . . . . . . . . . . . . . . . .2718.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729 A-. . . . . . . . . . . . . .29510 B-Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . .29910.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29910.1.1 . . . . . . . . . . . . . . . . . . . . 29910.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 29910.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . 30110.1.4 Mathematica . . . 30310.1.5 Mathematica. . . . . . . . . . . . . . . . . . . . . 30510.2 . . . . . . . . . . . . . 30810.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 30810.2.2 . . . . . . . . 30810.2.3 . . . . . . . . . . . . . . . . . . . . . . 30910.2.4 . . . 31010.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31110.2.6 . . . . . . . . . . . . . . . . . . 31210.3 . . . . . . . . . . . . . . . . . . 31310.3.1 . . . . . . . . . 31310.3.2 . . . . . . . . . . . . 31610.3.3 . . . . . . . 31710.3.4 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321v10.4 . . . . . . . . . . . . . . . . . . . 32210.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 32210.4.2 . . . . . . . . . . . . . . . . . . . 32310.4.3 . . . . . . . . . . . . . . . . . . 32410.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 32510.4.5 . . . . . . . . . . . . . . . . . . . . 32710.4.6 . . . 33010.4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.4.8 . . . . . . . . . . . . 33010.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33110.5.1 . . . . . . . . . . . . 33110.5.2 . . . . . . . . . . . . . . . . 33510.5.3 . . . . . . . . . . . . . . . . . . 33710.6. . . . . . . . . . . . . . . . . . . . . . . . . 33810.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34210.7.1 . . . . . . . . . . . . . . . 34310.7.2 . . . . . 34310.7.3 . . . . . . . . . . . . . . . . . 34410.7.4 . . . . . . . . . . . . . . . . . . . . . . . . 34610.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34710.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . 34810.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34910.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . 35110.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35310.9.1 . . . . . . . . . . . . . . . . . . . . . . . . 35310.10 . . . . . . . . . . . . . 35610.10.1Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 35710.10.2 . . . . . . . . . . . . . . . . 36010.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36110.10.4 Evaluate. . . . . . . . . . . . . . . . . . . . . . . . . 36110.11 . . . . . . . . . . . 36210.12 . 36410.12.1ParametricPlot . . . . . . . . . . 36410.13GraphicsFilledPlot . . . . . . . . . . . . . . . . . . . . 36610.13.1 FilledPlot . . . . . . . . . . . . . . 36610.13.2 FilledListPlot . . . . . . . . . . . 36810.14GraphicsImplicitPlot . . . . . . . . . . . . . . . . . . . 369vi 10.15 . . . . . . . . . . . . . . . . . . . . . . 37110.16 . . . . . . . . . . . . . . . . . . . . . 37510.17GraphicsPolyedra . . . . . . . . . . . . . . . . . . . . . 379. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .381 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3831. , AlbertEinstein1.1 - . , ,-, .-4-2024-4-2024-1-0.500.51-4-20241.1. f(x, y) = sin(_x2+y2). - 2 , . - - . , -(. 1.1).,,- . .1.2 , f(x, y) , , Mj(x, y) D R2 - (. 1.2).,,z= f(x, y) = (1 x2y2)1/2 ,D, x2+y2= 1. , x, y z . x, y z.xf(x,y)yD(,)f(x,y)f(,) 1.2. f R2 .1.2 3 R2, f(x, y).1.1:F(x, y) = (x2+y21)1/2+ log(4 x2y2).: F(x, y) x2+y21 0 4 x2y2> 0. {(x, y) : x2+y2 1, x2+y2< 4} x2+ y2 1 x2+ y2< 4 1.3.yx0 1 21.3. , x2+ y2= 1 F(x, y). 1.2: G(x, y) = (x2y2)1/2+ (x2+y21)1/2: G {(x, y) :x2 y2, x2+y2 1} x2+y2= 1 x = y, x = y( .1.4) :1.1:f: D E, , f -4 1.4. A, A - G. D Rn E R1 z= f(x1, x2, x3, ...xn).n- x1, ..., xn z.1.3 - - . (x, y) P(x, y),x, y . P(x, y)P(x, y),d(PP) =_(x x)2+ (y y)2. (1.1) , .1.3.1 , .- , ( n ) 1.3 5 , . (x0, y0). 1.2: M0(x0, y0) R2 , M0(M0, ) = {(x, y) :_(x x0)2+ (y y0)2 }.(M0, ) M0 (. 1.5)(x,y)0 01.5. (M0, ). |xx0| |y y0|( . 1.6).xy022(x,y)0 01.6. (M0, ).6 , (x0, y0). . , , - . . 1.3: D P() D.(.1.7)r1.7. . P r , P.1.3: ,,;: , P , r (r 0) . x22 y22=z > 0 (.1.16). O ,-OyO, Ox. 7 .-1-0.500.51-1-0.500.51-1-0.500.51-0.500.51-1-0.500.51.17. .1.5 R319_x_2+_y_2=_z_2,(.1.17), x22+y22 z22= 1(.1.18).-2-1012-2-1012-2-1012-2-1012-2-10121.18. x22+y22= 1 y= x2 (.1.19).z0xyz0xy1.19. . Mathematica ( 2)20 - .1.7: f(x, y, z) = 9x216y2+ 144z= 0: f(x, y, z) x y z, f(x, y, z) =0 . z z=y29x216 .1.6.,, -. ,. . (xy) f(x, y)-k, k.1.7:f(x, y) f(x, y) = z z= k, k (.1.20.) f(x, y). k=nh, n=0, 1, 2, 3, .. h. f-., 1.6 21xyz0z=4k=1k=4k=2k=5k=3k=6f(x,y)=k1.20. f(x, y) = k k.. , - -. f(x, y) = ax+by +c ax+by= kc, f(x, y) = x2+y2+a2 ( 1.6). f(x, y) = x2y2 (.1.21).-10-50510-10-50510-100-50050100-10-505-10 -5 0 5 10-10-505101.21. f= x2y2 (0, 0).22 - - . w=f(x, y, z) -f(x, y, z)= k. f(x, y, z) = x2+y2+z2 x2+y2+z2= k . 1.6: f(x, y) =_9 x2y2k = 0, 1, 2, 3.: f(x, y) =kx2+ y2=9 k2. k= 0, 1, 2 (0, 0)9 k2(.1.22).-202-20210152025-202 -3 -2 -1 0 1 2 3-3-2-101231.22. f= x2+y2 (0, 0).1.71.1::i) f1(x, y) = ln(1 x2y2), ii) f2(x, y) =11 x2y2,iii) f3(x, y) = ln(y x).:1.7 23(i) , x = r cos , y= r sin ,f1(x, y, z) f1(r) =ln(1 r2) 0 r0y >x. f3 y= x.1.2:f(x, y) =1x y+1x +y+ 2.yx 0y=xy=-x1.23. (x y)1/2+ (x + y)1/2+ 2.: f(x, y) x y> 0 x +y> 0, x > y x >y. .1.23 ( x = y x = y).1.3f(x, y, z) =1 z23 _4 x2y2.: 1z2 0 4 x2y2 0. , 24 z21 1 z1 r24. , x2+y2= 4 2.1.4 f(x, y, z) =arcsin x + arcsin y + arcsin z.: 1 x 1, 1 y 1 1 z 1. (x, y, z) f x= 1,y= 1, z= 1.1.5:f(x, y) =_|x| +|y| 2.f;;:|x| +|y| 2 0, (2,0), (-2,0), (0,2), (0,-2). f(x, y) 0 (x, y) .1.6:z=_cos(x2+y2),.: , cos(x2+y2)0, 2k /2x2+y22k+ /2 k=1, 2,... z _/2 [(4k 1)/2]1/2 [(4k+ 1)/2]1/2, k= 0, 1, 2,....(.1.24)1.7: U(x, y)(x, y)(xy), -,- . U=Ax2+By22x2y2, A, B . (0, 0).: 1.7 251.24. z=_cos(x2+ y2).Ax2+By22x2y2= c (1.4)c.(0, 0),x2y2Ax2 By2,, (1.4) (Ax2+ By2) 2c , , (2c/A)1/2 (2c/B)1/2.-0.4-0.200.20.4-0.4-0.200.20.400.10.20.3-0.4-0.200.20.4-0.4-0.20 0.20.4-0.4-0.200.20.41.25. 1.8a=2, b = 3, c = 1.1.8: x2a2+y2b2= 2cz, a, b, c.: z . x =0 y =0 , 26 z=p (x2/(2a2cp)) + (y2/(2b2cp))=1, p. 1.25.1.81.1: :i) f(x, y) = 3y29x + 5 ln x2,ii) f(x, y, z) = z(x2+y21)1/2+ ln[z2(4 x2+y2)],iii) f(x, y) = x(1 y)/(y22y + 1),iv) f(x, y, z) = (x2+y2z)1/2+ ln(z2+x2+y2).1.2: :i) f(x, y) = x/(y24x),ii) f(x, y) = [6 (2x + 3y)]1/2,iii) f(x, y) = [x2+y24] ln (16 x2y2),iv) f(x, y) = (x2y2)1/2+ (x2+y21)1/2.1.3: :f(x, y) = xln(y 1) + (1 x2)1/2.1.4: f(x, y) = y/(x2+y2).1.5: :i) f(x, y) = ln(x +y),ii) f(x, y) = arccos(xy),iii) f(x, y) = ln(x2+y),iv) f(x, y) = arcsin(x) + (xy)1/2.1.6::i) f(x, y) = ln(a x2+y2) + (x2+y2b),ii) f(x, y) = (x2y2)1/2+ (x2+y2)1/2.1.8 271.7::i) f(x, y, z) = ln(xyz),ii) f(x, y, z) = (1 x2y2z2)1/2.1.8: u=xy v= x2y2.1.9: :i) V= x +y +z,ii) U= ln(1 x2y2z2).28 2. MATHEMATICA , , .PeterD.Lax(1985)2.1 . -, . Mathematica . .f(x, y)Mathematica Plot3D.Plot3D[f [x, y], {x, xmin, xmax}, {y, ymin, ymax}]xmin, xmax, ymin, ymax x, y-., f(x, y). 2.1 sin xsin y 10 x 10 10 y10, Clear[f]30 MATHEMATICAf[x_, y_] :=Sin[x] Sin[y];Plot3D[f[x, y], {x, 10, 10}, {y, 10, 10}];-10-50510-10-50510-1-0.500.51-10-5052.1. f(x, y) = sin xsin y - 2.1.2.1.1Plot3D Mathematica . :2.1. Plot3D. Boxed TrueLighting TrueMesh TrueShading TrueViewPoint {1.3, 2.4, 2}GrayLevel[s]Hue[s]Boxed False: .2.1 31Lighting False: .Mesh False: .Shading False: .ViewPoint {x, y, z} : .{1.3, 2.4, 2} {0, 2, 0} {0, 2, 2} {0, 2, 2} {2, 2, 0} {2, 2, 0} {0, 0, 2} .2.2Clear[f]f[x_, y_] :=Sin[x] Sin[y];Clear[pl1, pl2, pl3, pl4]pl1=Plot3D[f[x, y], {x, 10, 10}, {y, 10, 10},DisplayFunction Identity];pl2=Plot3D[f[x, y], {x, 10, 10}, {y, 10, 10},Boxed False, DisplayFunction Identity];pl3=Plot3D[f[x, y], {x, 10, 10}, {y, 10, 10},Lighting False, DisplayFunction Identity];pl4=Plot3D[f[x, y], {x, 10, 10}, {y, 10, 10},Mesh False, DisplayFunction Identity];Show[GraphicsArray[{{pl1, pl2}, {pl3, pl4}}],DisplayFunction $DisplayFunction];-2.3.32 MATHEMATICA-10-50510-10-50510-1-0.500.51-10-50510-10-50510-10-50510-1-0.500.51-10-50510-10-50510-10-50510-1-0.500.51-10-50510-10-50510-10-50510-1-0.500.51-10-505102.2. f(x, y)=sin xsin y Plot3D.2.3Clear[plt1, plt2, plt3]plt1 =Plot3D[Sin[x + y^2], {x, 1, 1}, {y, 1, 1},Boxed False, DisplayFunction Identity];plt2 =Show[%, ViewPoint {0, 2, 2},DisplayFunction Identity];plt3 =Show[%, ViewPoint {0, 2, 0},DisplayFunction Identity];Show[GraphicsArray[{{plt1, plt2, plt3}}]];2.3. f(x, y) = sin xsin y- . .2.2 332.4Clear[col1, col2]col1=Plot3D[{Sin[x y], GrayLevel[x/3]}, {x, 0, 3}, {y, 0, 3},DisplayFunction Identity];col2=Plot3D[{Sin[x y], Hue[x/3]}, {x, 0, 3}, {y, 0, 3},DisplayFunction Identity];Show[GraphicsArray[{{col1, col2}}]];01230123-1-0.500.51012301230123-1-0.500.510123 2.4. f(x, y) = sin xsin y - .2.2 ContourPlot f(x, y) xmin x xmax, ymin y ymax.ContourPlot[f [x, y], {x, xmin, xmax}, {y, ymin, ymax}]2.2.1ContourPlot ??ContourPlot - ContourPlot . :ColorFunction Hue: -.34 MATHEMATICA2.2. ContourPlot. ColorFunction AutomaticContours 10PlotRange AutomaticContourShading TrueContourLines TrueContourStyle AutomaticPlotPoints 15Contours : , .PlotRange All {zmin, zmax}: {zmin, zmax} All.ContourShading False: .ContourLines False: -.ContourStyle: .PlotPoints : Plot.2.5Clear[f, c1, c2, c3, c4, c5, c6]f[x_, y_] :=Sin[x] Sin[y];c1=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity];c2=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity, ContourShading False];2.3 35c3=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity, ColorFunction Hue];c4=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity, ContourLines False];c5=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity,ContourStyle {RGBColor[1, 0, 0]}];c6=ContourPlot[f[x, y], {x, 2, 2}, {y, 2, 2},DisplayFunction Identity, Frame False, Axes False,ContourShading False, PlotPoints 50];Show[GraphicsArray[{{c1, c2, c3}, {c4, c5, c6}}]];Contours f(x, y) = c1, c2, . . ..2.6Clear[f, c1, c2, c3]f[x_, y_] :=x2 2 x + y2 2 y 2;c1=ContourPlot[f[x, y], {x, 10, 10}, {y, 10, 10},ContourShading False, DisplayFunction Identity,PlotPoints 60];c2=ContourPlot[f[x, y], {x, 10, 10}, {y, 10, 10},ContourShading False, DisplayFunction Identity,Contours 20, PlotPoints 60];c3=ContourPlot[f[x, y], {x, 10, 10}, {y, 10, 10},ContourShading False, DisplayFunction Identity,Contours {4, 24, 40}, PlotPoints 60];Show[GraphicsArray[{{c1, c2, c3}}]];2.3 - .ContourPlot Plot3D -.36 MATHEMATICA-2-1 0 1 2-2-1012-2-1 0 1 2-2-1012-2-1 0 1 2-2-1012-2-1 0 1 2-2-1012-2-1 0 1 2-2-1012 2.5. f(x, y) = sin x sin y ContourPlot.-10-5 0 5 10-10-50510-10-5 0 5 10-10-50510-10-5 0 5 10-10-50510 2.6. f(x, y) = x22x+y22y 2 Contours.Mathematica .2.7Clear[ap]ap =Plot3D[Sin[x + y], {x, 0, 3}, {y, 0, 3}]; Plot3D .Show[ContourGraphics[ap]];Show[GraphicsArray[{%, %%}]];2.4 3701230123-1-0.500.510122.7. f(x, y) = sin(x + y).0 0.5 1 1.5 2 2.5 300.511.522.532.8. f(x, y)=sin(x + y) .0 0.51 1.5 2 2.5 300.511.522.5301230123-1-0.500.5101232.9. 2.7- .2.4 ParametricPlot3D . 38 MATHEMATICA2.3. .Show[ContourGraphics[g]] .Show[SurfaceGraphics[g]] .Show[Graphics[g]] .Options[ParametricPlot3D].2.4.1ParametricPlot3D[{x[t], y[t], z[t]}, {t, tmin, tmax}] x=x(t), y=y(t), z=z(t), a t b.ParamatricPlot3D ParametricPlot, .2.8Clear[pp1, pp2]pp1=ParametricPlot3D[{Cos[2 t], Sin[2 t], t/5},{t, 0, 8 Pi}, PlotPoints 120, Ticks None,DisplayFunction Identity];pp2=ParametricPlot3D[{t Cos[2 t], t Sin[2 t], t/5},{t, 0, 8 Pi}, PlotPoints 120, Ticks None,DisplayFunction Identity];Show[GraphicsArray[{{pp1, pp2}}]]; ParametricPlot3D :2.9- g(x, y) = cos(x +siny) x = 5.Clear[pp1, pp2, g, p]g[x_, y_] =Cos[x + Sin[y]];2.4 392.10. 2.8.g[x, y] [0, 4] [0, 4] 2.11.p =Plot3D[g[x, y], {x, 0, 4 Pi}, {y, 0, 4 Pi},DisplayFunction Identity]; x = 5xy x = 5, x=5, y =t, z =0. 2.11.pp1=Show[ParametricPlot3D[{5, t, 0}, {t, 0, 4 Pi},DisplayFunction Identity]];Show[GraphicsArray[{{p, pp1}}]];05100510-1-0.500.51051002.557.5100510-1-0.500.5102.557.5102.11. g(x, y)=cos(x + siny)x = 5x=5g[x, y] gx=540 MATHEMATICA x=5, y=t, z=g[5, t] 2.12.pp2=ParametricPlot3D[{5, t, g[5, t]}, {t, 0, 4 Pi}];02.557.5100510-0.500.5102.557.5102.12. g(x, y) = cos(x + siny) x = 5 2.9.2.10- f[x, y] = x3sin[4y]+y2cos[3x] x2+y2= 1. (x,y) f .Clear[f, pp1, pp2, p]f[x_, y_] =x3 Sin[4 y] + y2 Cos[3 x];p =Plot3D[f[x, y], {x, 1.5, 1.5}, {y, 1.5, 1.5},DisplayFunction Identity]; :x=cos[t], y=sin[t], z=0 [0, 2].pp1=Show[ParametricPlot3D[{Cos[t], Sin[t], 0}, {t, 0, 2 Pi},DisplayFunction Identity]];Show[GraphicsArray[{{p, pp1}}]]; x2+y2= 1 f[x, y] - f - x=cos[t], y=sin[t]t, z =f[cos[t], sin[t]]. 2.14.pp2=ParametricPlot3D[{Cos[t], Sin[t], f[Cos[t], Sin[t]]},{t, 0, 2 Pi}];2.4 41-101-101-202-101-1-0.500.51-1-0.500.51-1-0.500.51-1-0.500.51-1-0.500.512.13. f[x, y] =x3sin[4y] + y2cos[3x] x2+ y2= 1.-1-0.500.51-1-0.500.51-0.500.51-1-0.500.51-1-0.500.512.14. f[x, y] = x3Sin[4y] +y2Cos[3x] x2+y2= 1 2.10.2.4.2ParametricPlot3D[{x[t, u], y[t, u], z[t, u]},{t, tmin, tmax}, {u, umin, umax}]x=x(t,u), y=y(t,u),z=z(t,u), tmin t tmax.2.11Clear[pp1, pp2]pp1=Show[ParametricPlot3D[{t, u, Sin[t u]}, {t, 0, 3}, {u, 0, 3},DisplayFunction Identity]];pp2=Show[ParametricPlot3D[{t, u2, Sin[t u]}, {t, 0, 3}, {u, 0, 3},DisplayFunction Identity]];42 MATHEMATICA01230123-1-0.500.5101230123012302468-1-0.500.510123024682.15. - 2.11, p1(t, u, sin(tu) p2= (t, u2, sin(tu).Show[GraphicsArray[{{pp1, pp2}}]];ParametricPlot3D , , ...2.12Clear[pp1, pp2, pp3]pp1=Show[ParametricPlot3D[{Sin[t], Cos[t], u}, {t, 0, 2 Pi},{u, 0, 4}, DisplayFunction Identity]];pp2=Show[ParametricPlot3D[{Cos[t] (3 + Cos[u]),Sin[t] (3 + Cos[u]), Sin[u]}, {t, 0, 2 Pi}, {u, 0, 2 Pi},DisplayFunction Identity]];pp3=Show[ParametricPlot3D[{Cos[t] Cos[u],Sin[t] Cos[u], Sin[u]}, {t, 0, 2Pi}, {u, Pi/2, Pi/2},DisplayFunction Identity]];Show[GraphicsArray[{{pp1, pp2, pp3}}]];2.5GraphicsContourPlot3D ContourPlot.ContourPlot3D Con-tourPlot.ContourPlotf x y. Contour-Plot3D fx,y,z. 2.5GraphicsContourPlot3D43-1-0.500.51-1-0.500.5101234-1-0.500.51-4-2024-4-2024-1-0.500.51-4-2024-1-0.500.51-1-0.500.51-1-0.500.51-1-0.500.51-1-0.500.51 2.16. , ParametricPlot3D.ContourPlot3D[f [x, y, z], {x, xmin, xmax},{y, ymin, ymax}, {z, zmin, zmax}] f(x, y, z) = 0.ContourPlot3D Mathe-maticaGraphicsContourPlot3D.ContourPlot3D Options[ContourPlot3D] = f(x, y, z) =_x2+y2+z2 0 > 03.5 67 (M0, ) |f f(M0)| < f(M0) 0. (x, y) - , . , .|2x3y3| 2|x|3+|y|3= 2|x|x2+|y|y2,, |x| _x2+y2 |y| _x2+y2,68 -1-0.500.51-1-0.500.51-2-1012-1-0.500.53.6.,(2x3 y3)/(x2+ y2) .|2x3y3| (x2+y2)12(2x2+y2) 3(x2+y2)32. 2x3y3x2+y2 3__x2+y2_3 33. =33 > 0 =3_/3 |f 0| . 0 0() f3(x, y) =x2yx4+y2() f4(x, y) =x+y+x2+y2xy() f5(x, y) =x6+y6x2+y2() f6(x, y) =x4+y4x3+y3() f7(x, y) = |x||1y|()f8(x, y) =x2x2+y2+|yx|3.5 71:() y= xlim(x,y)(0,0)f1(x, y) = 1, y= xlim(x,y)(0,0)f1= 1/5,.() y =ea|x|, a 1 ,f2 f2= |r|1+a| sin |2+a. a1 . y= m|x|af(x, y) =y|x|2+a_x2+y2=m|x|_1 +m2|x|2a=m_1 +m2|x|a f m x 0..3.13 ( ) :() F1(x, y, z) =x2y2+2y3zx2+y2+z2.() F2(x, y) =(x1)2+y21x2y2(1, 0).() F3(x, y) =x22xy2+y4x2+y4.() F4(x, y) =x22xy+y2x2yy3(1, 1).:() limx0[ limz0[ limy0F1]] = 176 limy0[ limz0[ limx0F1]] = 1.() -.() y2=mx, m.() .F4(x, y)=(xy)2x(x2y2)=xyx(x+y) (x, y) (1, 1).3.14():() f1(x, y) =x+yxyx (0, 1).() f2(x, y) =xy3x2+y6 .() f3(x, y) =x2y2+11x2+y2 .: () (0, 1) (M0, ).() y3=mx, m=, , m.() . - _x2y2+ 1 + 1(x, y)(r, ). r.3.15f(x, y) =sin(y x) + sin(y x) cos2x 2 cos xsin(y x)x4y x5.3.5 77: f(x, y) =(1 cos x)2sin(x y)x4(y x).,lim(x,y)(0,0)f(x, y) =limx0(1 cos x)2x4lim(y,x)0sin(y x)y xlimx0(1 cos x)2x4=14 limx0sin xx 1 cos x = 2 sin2(x/2) lHopital, 1,14.3.16:() f1(x, y) =___(3+x5) sin(x5y5)x5y5+xysinx siny (x, y) = (0, 0)2 (x, y) = (0, 0)() f2(x, y) =___x3yxy3x2+y2 (x, y) = (0, 0)0 (x, y) = (0, 0): () f1(x, y) 4 - (0, 0).() f(x, y)=r2f1() .3.17 DR2 .() f1(x, y) = exysin(x +y)() f2(x, y) = xln(xy)() f3(x, y) =___2x2y22x2+y2 (x, y) = (0, 0)0 (x, y) = (0, 0)78 () f4(x, y) =x6+x3y3+y3x3+y3:() (x, y).() (xy) xy >0, -.() (xy) (0, 0).() R2{x = y}, (xy)y= x.3.18f(x, y, z) =xsin x +y siny +z sin zx2+y2+z2 (x, y, z) = (0, 0, 0). f (0, 0, 0) R3.: f(x, y, z) =xsin x +y siny +z sin zx2+y2+z2 sinxx+_sin xxsin xx_x2x2+y2+z2+_sinyysin xx_y2x2+y2+z2+_sin zzsin xx_z2x2+y2+z2., limw0sin ww= 10 x2x2+y2+z2 1, 0 y2x2+y2+z2 1,3.5 790 z2x2+y2+z2 1 . , ,,, 2. f(0, 0, 0) = 1.3.19:f(x, y) =___(1+y2) sinx cos x2x x = 0 x = 0(0, 0).: flim(x,y)(0,0)f(x, y) =limx012sin 2x2x(1 +y2)=12limx0sin 2x2xlimy0(1 +y2) =12 1 =12. f =12.(.3.9)-2-1012-2-1012012-2-1013.9. 3.20 = 1/2.3.20:f(x, y) =___xy(x2y2)x2+y2 x = 0, y = 00 x = y= 080 .: |x2y2| |x2+y2| |xy|x2y2x2+y2 < |xy| (x2+y2)lim(x,y)(0,0)|f(x, y)| lim(x,y)(0,0)(x2+y2) = 0 .-1-0.500.51-1-0.500.51-0.200.2-1-0.500.53.10. 3.22.3.21f(x, y) =sinx sinytan x tan y, D : [0, /4] [0, /4] {P(x, y)/x = y}. f x = y,D;: f(x, y) f(x, y) =(sin x sin y) cos xcos ysin xcos y sin y cos x. :f(x, y) =2 cosx+y2sinxy2cos xcos ysin(x y). y x, f(x, y) cos3x. , f(x, y) 3.5 81f(x, y) =___sinxsinytan xtan y, x = ycos3x, x = y [0, /4] [0, /4].3.22:()f1(x, y) =___x2y32x2+y2 (x, y) = (0, 0)0 (x, y) = (0, 0)()f2(x, y) =___xyx2+xy+y2 (x, y) = (0, 0)0 (x, y) = (0, 0)()f3(x, y) =___2x2y22x2+y2 (x, y) = (0, 0)0 (x, y) = (0, 0)()f4(x, y) =___x2y2x2+y4 (x, y) = (0, 0)0 (x, y) = (0, 0):() , lim(x,y)(0,0)f(x, y) =limr0(r3)g()) = 0 g . f R2.() y=mx.f R2{0, 0}.() ().() y= x , . , , 82 x2< 2x2+y4.3.23lim(x,y)(,)ex+yx2+y2_1 + sin_3x +y__x+y.:lim(x,y)(,)ex+yx2+y2_1 + sin_3x +y__x+y=_lim(x,y)(,)ex+yx2+y2__lim(x,y)(,)_1 + sin_3x +y__x+y_.() ,x = r cos , y= r sin , limrecos +sinr= e0= 1.() x +y= tlimt_1 + sin_3t__t= 1 .R(t) =_1 + sin_3t__tln R(t) = t ln_1 + sin_3t__ =ln_1 + sin_3t__1t=f1(t)f2(t)limtln R(t) =limtf1(t)limtf2(t)=00.,.lH opital limtf1(t)f2(t)= 3 limtln R(t) = 3 limtR(t) = e3. 1 e3= e3.3.6 833.63.1: f(x, y) =x2y4(x2+y4)3(x, y) (0, 0). () y= mx() y= kx2.3.2: f(x, y) =xyx2+y2, (x, y) (0, 0).3.3 : (0, 0).i) f(x, y) =xx+y, ii) f(x, y) = x yx2+y2,iii) f(x, y) = x2+yx2+y2, iv)f(x, y) =xy2(x2+y4)3.3.4:x4+y4x2+y2 0f(x, y) = 0, x = y= 0. f xy=0, y x=0 .3.10:f(x, y)=e1xyx =y. f(x, y) R2;84 3.7Mathematica3.7.1.Mathematica, - f(x). f(x) =1x2+ 1. (3, 3). - 0 1.Clear[f]f[x_] :=1/(1 + x2);Plot[f[x], {x, 3, 3}]-3 -2 -1 1 2 30.20.40.60.813.11. 1/(1 + x2). 0, x0..-0.10.1,-0.010.01...Random .3.7 Mathematica 85pinak =Table[Random[Real, {10n, 10n}], {n, 1, 8}]{0.0532542, 0.00803469, 0.000720242, 0.0000919018,8.63277 106, 5.15603 107, 6.66202 108,6.76187 109} .fpinak =Map[f, pinak]{0.997172, 0.999935, 0.999999, 1., 1., 1., 1., 1.}x f.times =Table[{pinak[[i]], fpinak[[i]]}, {i, 1, 8}];TableForm[times]0.0532542 0.9971720.00803469 0.9999350.000720242 0.9999990.0000919018 1.8.63277 1061.5.15603 1071.6.66202 1081.6.76187 1091.,f 0 1, (x, y) (0,0) >0, 1.3.7.2Mathematica .-:Limit[f [x], x > a].86 f(x)xa. .- /. . , cos(x2)0 .Cos[x2]/.x 01 , sin(x)/x x = 0.t =Sin[x]/xSin[x]x x0,0/0 .t/.x 0Power :: infy : Infinite expression10encountered.Infinity :: indet : Indeterminate expression 0 ComplexInfinityencountered.Indeterminate Limit, .Limit[t, x 0]1, . , sin(1/x) 0.,0 -11. Limit - Interval, .3.7 Mathematica 87Interval[{xmin, xmax}]sin(1/x)0,Mathematica .Limit[Sin[1/x], x 0]Interval [{1, 1}]Limit .Limit[x f[x], x 0]Limit[x f [x], x 0]3.7.3 , . Lim-it Direction.Limit[expr, x x0, Direction 1] x x0 ,limxx0(expr).Limit[expr, x x0, Direction 1] , x x0 , limxx+0(expr). 1/x0, . .Limit[1/x, x 0, Direction 1] .Limit[1/x, x 0, Direction 1]88 -1 -0.5 0.5 1-100-75-50-252550753.12. 1/x.3.7.4 Limit - . , . NLimit, .NLimit[expr, x x0] NLimit Mathematica,"Numer-icalMath". .0f[x_, y_] :=1/; x =y =0f[x_, y_] :=If[x2 + y2 >0, (1 + x y)/(x2 + y2), 1] g f f.g[x_, y_] := (1 + x y)/(x2 + y2) , .96 Limit[Limit[g[x, y], x 0], y 0] ==Limit[Limit[g[x, y], y 0], x 0] == Truef (0, 0) , .f[0, 0] =1 = True5. () f(x, y) =x4+y4(x +y)4+x3y5 .Clear[f]f[x_, y_] := (x4 + y4)/((x + y)4 + x3 y5)f[x, y]/.y m xx4+ m4x4m5x8+ (x+ mx)4FullSimplify[%]1+ m4(1+ m)4+ m5x4%/.x 01+ m4(1+ m)46. lim(x,y,z)(0,0,0)xy2z2(x2+y2+z2)2Clear[f, x, y, z, r, phi, thita, h]3.7 Mathematica 97-0.4-0.200.20.4-0.4-0.200.20.4-1001020-0.4-0.200.20.4-0.4-0.20 0.20.4-0.4-0.200.20.43.19. f(x, y) =x4+y4(x+y)4+x3y5.f[x_, y_, z_] :=x y2 z2/(x2 + y2 + z2)2 0 > 0 ,|x x0| < |f(x) f0| < . f: D E D RnE Rm, f=(f1, f2, ..., fm)x0 f1, f2, ..., fm x0. f(t) = (f1, f2, ..., fm)dfdt=m

i=1dfidt ei. f(x), xfxi=_f1xi, f2xi, f3xi_.df= (df1, df2, df3) =_3

i=1f1xidxi,3

i=1f2xidxi,3

i=1f3xidxi._...6.1 xu(x, y, z)=(x + y + z) ex + (x2+ y2+z2) ey + (x3+y3+z3) ez:ux=(x +y +z)x ex +(x2+y2+z2)x ey +(x3+y3+z3)x ey= ex + 2x ey + 3x2 ez 6.2 f(x, y) =(x2+y2) ex +xy ey.2fxy.:192 fx= 2y ex +x ey2fxy= ey. 6.3 f(x, y) = (x2+y2) ex +xy ey.: P(x, y)=x2+ y2,Q(x, y)=xy.df= dP ex +dQ ez=_Pxdx +Qy dy_ ex +_Pxdx +Qy dy_ ey= (2xdx + 2ydy) ex + (ydx +xdy) ey6.3 f(x) n0 (.6.3)yzx0 0 0Q(x,y,z)Q(x,y,0)P(x,y,z)0 0P(x,y,0)C6.3. .Dn0f(x) =limh0f(x +n0h) f(x)h(6.1) Dn0f fx0n0.6.4 193Dn0f=fx cos +fycos +fzcos (6.2)n0=(cos ,cos ,cos ), cos , cos ,cos 1.3.: (6.1) Dn0(x0) =_ddh [f(x0 +hcos , y0 +hcos , z0 +hcos )]_h0=__fx_0_(x0 +hcos )h_+_fy_0_(y0 +hcos )h_..._=_fx_0cos +_fy_0cos +_fz_0cos , n0= exDexf(x0) =_fx_0, .6.4 6.4.1f(x, y, z)- f(x) =lim|h|0=f(x +h) f(x)|h|f(x) =fx ex +fy ey +fz ez. (6.3)Dn0f(x0) = (f(x0)) n0. (6.4) (6.2) . n = |n|n0194 Dnf(x0) = |n|Dn0f(x0) (6.5)(.) (6.5) , -(f) n0, Dn0f(x0).: (6.4) Dn0f(x0) ., f(x, y, z) =0 P(x, y, z) f= 0, (f)P P. r= x ex+y ey+z ez M(x, y, z) . dr =dx ex+dy ey+dz ez M. df=fxdx +fydy +fzdz= 0_fx ex +fy ey +fz ez_(dx ex +dy ey +dz ez) = f dr= 0 fdr . 6.4 ().yzx0 0 0F(x,y,z)CSPr(r)06.4. f M06.4 1956.4:f(x, y) =x2+y2.: f(x+h)f(x) =f(x) h +O(h) O(h) h.f(x +h) f(x) = f(x +h1, y +h2) f(x, y)= [(x +h1)2+ (y +h2)2] (x2+y2)= [2 h1 + 2yh2] + [h12+h22]= [2x ex + 2y ey] h +|h|2f(x) = f(x, y) = 2x ex + 2y ey. 6.5 : -f(x, y)=(x 1)2 y2, .: n0= (n1, n2) M0(x0, y0) f.Dn0f= (f) n0f=fx ex +fy ey, n0= n1 ex +n2 ey,f n0= 2(x 1)n12yn2Dn0f|M0= 2(x01)n12y0n2. (6.6)6.6: f(x, y) = x33x2y +3xy2+ 1R2.fM(3, 1) ,N(6, 5).: f M0196 (f)M0=_fx_M0 ex +_fy_M0 ey= 12 ex9 ey n0=_661,561_. (Dn0f)M0= (f)M0 n0= (12 ex9 ey)_661 ex +561 ey_=2761.6.4.2f =f1 ex+f2 ey +f3 ez f=f1x1+f2x2+f3x3.. f -, U(x, y, z) =U1(x, y, z) ex+ U2(x, y, z) ey+U3(x, y, z) ez. V =xyz (. 6.5) . 0. -. 6.5. V .6.4 197xz U2y0U2xz. 0_U2(x, y, z) +U2yy_xz, U2(x, y +y, z) = U2(x, y, z) +U2yy.y - 0_U2(x, y, z) +U2yy_xz 0U2(x, y, z)xyz= 0U2yV 0U1xV 0U3zV. 0_U1x+U2y+U3z_V= 0 UV. U=0- . - U> 0 U< 0 . . .- f , U(x, y, z) = U1(x, y, z) ex+U2(x, y, z) ey+U3(x, y, z) ez. dVdt=dxdtyz +dydtxz +dzdtyx.198 x dxdt= [U1(x +x, y, z) U1(x, y, z)]. Taylor dxdt= [U1(x, y, z) +U1xx U1(x, y, z)] =U1xx. x y d(V )dt=_U1(x, y, z)x+U2(x, y, z)y+U3(x, y, z)z_V= ( U)V. U= 0V . - . , , R(x(t), y(t), z(t), t)dRdt=Rt+Rx dxdt+Ry dydt+Rz dzdt=Rt+U R. .6.4.3, . :f= ex ey ezx1x2x3f1f2f3==_f3x2f2x3_ ex +_f1x3f3x1_ ey +_f2x1f1x2_ ez..- 6.4 199= ez,= (.6.6).6.6. z . P(x, y, z) r =x(t) ex+y(t) ey+z ezx(t) =d cos(t), y(t) =d sin(t). v(t) = x(t) ex+ y(t) ey=d sin(t) ex+d cos(t) ey= y ex + x ey. v= 2 ez. . (F= 0) -. , , -( - ). B J. (6.7),.200 6.7 (fA) = A f+f A: (fA) =_x ex +y ey +zez_ (fA)=(fAx)x+(fAy)y+(fAz)z= f_Axx+Ayy+Azz_+Axfx+fy+fz= f A+A f6.8: , .: F F= f(r)er., ( )(f er) = f er +f er., er= 0f(r) =dfdrr =dfdrrr.F(r) =dfdrrr er= 0 F. , .6.9:, M.: MF= GMr3r, r M G . , F= 0. 6.5 201 (fr) = r f+f r F= r _GMr3_+_GMr3_ r=3GMr33GMr3= 0. q1, q2F= q1q2r3r. . -.1. (f) = 02. (A) = 03. (AB) = A( B) B( A) + (B )A(A )B4. (A) = ( A) 2A- . 2=2x21+2x22+2x23Laplace.-Laplace.6.56.5.1R3. , f(x, y, z) = 0,202 (f)M0 M0(x0, y0, z0). , (r r0) f= 0,r , r0 M0(6.7) .yzx0f|M=MCf(x,y,z)=00rM00rRR-r0r-r06.7. . (x x0)_fx_0+ (y y0)_fy_0+ (z z0)_fz_0= 0. (6.8) , z =f(x, y), (6.8) z=_fx_0x +_fy_0y, x = x x0, y= y y0 z= z z0.6.5 203,- z =f(x, y) M0(x0, y0, z0). f(x, y, z) = 0 (r r0) f= 0, (6.9) (r r0) f. (6.9) (x x0)_f/x_0=(y y0)_f/y_0=(z z0)_f/z_0. (6.10)(6.10) f(x, y, z) = 0 (x0, y0, z0).6.5.2R3f(t), g(t), h(t) CR3. - C M0(x0, y0, z0).yzx0rM00rRTuCPMK6.8. C T204 R = f(t) ex +g(t) ey +h(t) ez , -T0=dRdtM0 M0.rr0 M(x, y, z) M0(x0, y0, z0),(. 6.8) , r=r r0 T0.(r r0) T0= 0, M0. x x0f(t)=y y0g(t)=z z0h(t)., M0(x0, y0, z0)(r r0) T0= 0(x x0)f(t) + (y y0)g(t) + (z z0)h(t) = 0., M0C2- F(x, y, z)= 0 G(x, y, z)= 0.- C2 M0 M0, _Fx_M0(x x0) +_Fy_M0(y y0) +_Fz_M0(z z0)_Gx_M0(x x0) +_Gy_M0(y y0) +_Gz_M0(z z0). ,6.5 205G(x,y,z)=0ABCF(x,y,z)=0M0T6.9.(x x0)_D(F,G)D(y,z)_M0=(y y0)_D(F,G)D(x,z)_M0=(z z0)_D(F,G)D(x,y)_M0 (x x0)_D(F, G)D(y, z)_M0+ (y y0)_D(F, G)D(x, z)_M0+ (z z0)_D(F, G)D(x, y)_M0= 0. (6.11)6.10:--x2+ y2+ z2=42 x2+ y2=2xM0(, , 2).: F= x2+y2+z242= 0 G = x2+y22x = 0 C1. _Fx_M0=_Fy_M0= 2,_Fz_M0= 22,_Gx_M0=_Gz_M0= 0,_Gy_M0= 2.206 _D(F, G)D(y, z)_M0= 422,_D(F, G)D(z, x)_M0= 0,_D(F, G)D(x, y)_M0= 42, y= , x +z2 = 3, z= x2.6.11: x2+ y3=2(2 + yz)x2+ 1 z2= 2y2(1 2) M0(1, 1, 2).: F= x2+y32(2+yz) = 0 G = x2+1z2 2y2(1 2) = 0 M0, F G (F)M0(G)M0= 0.F= 2x ex + 3(y22z) ey2y ezG = 2x ex2(2 4)y ey2z ez.(F)M0= 2 ex ey + 2 ez(G)M0= 2 ex + 2(2 4) ey 4 ez, (F)M0 (G)M0= 0.6.12:, x2+ 3y2+ 2z2=9 x2+y2+z28x8y 6z +24 = 0 M0(2, 1, 1).: F= x2+3y2+2z29 = 0 G = x2+y2+z28x8y6z+24 = 0 M0, (F)M0= (G)M0. FG M0(F)M0= 4 ex + 6 ey + 4 ez6.6 207(G)M0= 4 ex6 ey4 ez .6.66.1: () F G , F G -.() f(x, y, z) g(x, y, z) , f g= 0.:(), [F G] = 0 F=G = 0. , (F G) = (F)G(G)F, F= (f1, f2, f3) G = (g1, g2, g3).F G = (f2g3f3g2) ex + (f3g1f1g3) ey + (f1g2f2g1) ez (F G) =x(f2g3f3g2) +y(f3g1f1g3)+z(f1g2f2g1) =f2x g3f3x g2 +f3yg1f1yg3 +f1zg2f2zg1 +g3x f2g2x f3 +g1yf3g3yf1 +g2zf1g1zf2=_f3yf2z_g1 +_f1zf3x_g2 +_f2x f1y_g3_g3yg2z_f1_g1zg3x_f2 +_g2x g1y_f3=(F) G(G) F(), (f) = 0 ().6.2:f(x) = |x + y +z|R3. x0=(x0, y0, z0)x0 + y0 +z0= 0,fx0.208 :n0=(cos a1, cos a2, cos a3)-. Dn0f= limt0f(x0 +tn0) f(x0)t= limt0|x0 +t cos a1 +y0 +t cos a2 +z0 +t cosa3|t= limt0|t|t | cos a1 + cos a2 + cos a3|, (6.12) x0 +y0 +z0= 0.(6.16) cos a1 + cos a2 + cos a3= 0. (6.13)(6.13) , .6.3: () f(x, y)=xey,fP(2, 0) P Q(5, 4).() f ; ;:() f(x, y) =_fx, fy_ = (ey, xey), f(2, 0) = (1, 2).PQ=(3, 4) u=(3/5, 4/5), fPQ,PDuf(2, 0) = f(2, 0) u = (1, 2) _35, 45_ =35+85=115.() f f(2, 0) = (1, 2). |f(2, 0)| = |(1, 2)| =5.6.4 : -T(x, y) = excos y +eycos x.()(0,0);6.6 209() (0,0);:T(x, y) =Tx ex +Ty ey= (excos y eysin x) ex+ (eycos x exsin y) ey.() (0,0) T(0, 0) = ex + ey.() T(0, 0) = ex ey.6.5: f(x, y) =(x3+ y3)/3tr(t) = a cos t ex +b sint ey.:ft-d[f(r(t))]dt. d[f(r(t))]dt=_fxdxdt+fydydt+fzdzdt_ = f (dr(t)dt).f= x2 ex +y2 ey. x(t) = a cos t y(t) = b sin t,f(r(t)) = a2cos2t ex +b2sin2t ey.dr(t)dt= a sint ex +b cos t ey,d[f(r(t))]dt= sin t cos t(b3sin t a3cos t).210 6.6:ff(x, y) =_y y2x1/2+ 2x_ ex +_x2y1/2 x + 1_ ey.:fx=y y2x1/2+ 2x,fy=x2y1/2 x + 1. f/xxf(x, y) = xy xy +x2+(y) (y)x. yfy=x2y1/2 x +d(y)dy. f/y d(y)dy= 1 (y) = y +c.f(x, y) = xy1/2x1/2y +x2+y +c. f(x, y) = xy1/2x1/2y +x2+ y + c .6.7: , f(x, y, z)=(x2/a2) + (y2/b2) +(z2/c2) , n=MO, , 2f(M)/r r.:f M(x0, y0, z0)f= (2x/a2) ex + (2y/b2) ey + (2z/c2) ezDnf= (f) n0|n| =1r_2x02a22y02b22z02c2_= 2f(M0)r.6.6 2116.8:() (AB) = B (A) A (B)() 2( A) = (2A):() (AB) =x(AyBzAzBy) +y(AzBxAxBz)+z(AxByAyBx) = Bx_AyzAzy_+By_AzxAxz_+Bz_AxyAyx_Ax_ByzBzy_Ay_BzxBxz_Az_BxyByx_ = B (A) A (B).()(A) = ( A) 2A [(A)] = [2A+( A]0 = (2A) +2( A) (2A) = 2( A)6.9:F= (x + 2y +z) ex + (x 3y z) ey + (4x +y + 2z) ezf= 2x2yz3.() , , , .()F f, (F )f, (F) f, (F )f()g F= g;,.212 :()FF= 0. = 4, = 2, = 1.() F f= (F )f= (x + 2y + 4z)4xyz3+ (2x 3y z)2x2z3+ (4x y + 2z)6x2yz2.(F) f= 0.(F )f= [(2x 3y z)4xyz3(4x y + 27)2x2z3]i+ [(4x y + 27)4xyz3(x + 2y + 4z)6x2yz2]j+ [(x + 2y + 4z)2x2z3(2x 3y z)4xyz3]k(), F= 0 = g.F= (x + 2y + 4z) ex + (2x 3y z) ey + (4x y + 2z) ey=gx ex +gy ey +gz ezg(x, y, z) =_xa(t + 2y + 4z)dt +_yb(2a 3t z)dt+_zc(4a b + 2t)dt=x223y22+z2+ 2xy yz + 4zx +C.6.10: x2+ y2+ z2=a2M0(1, 1, 1): F=x2+y2+z2a2 M0 Fx= Fy= Fz= 2. 2[(x 1) + (y 1) + (z 1)] = 0 x +y +z= 3. x 1 = y 1 = z 1.6.6 2136.11 : f(x, y, z) = x2+y2+z29 = 0, g(x, y, z) = x2+y2z 3 = 0M0(2, 1, 2).: f =0g= 0 M(x, y, z) (f)M (g)M= |f|M|g|M cos . (f)M0= 4 ex2 ey + 4 ez, (g)M0= 4 ex2 ey ez,cos =(4 ex2 ey + 4 ez) (4 ex2 ey ey)42+ 22+ 4242+ 22+ 12=16621.6.12- z= x2+y2 M(1, 1, 2).: F= x2+y2z= 0_Fx_M0= 2,_Fy_M0= 2,_Fz_M0= 1.z 2 = 2(x 1) + 2(y 1) = 2(x +y 2)z 2 = x 12= y 12.6.13-x = a cos , y= a sin , z= b, = 2.:dxd= a sin = y,dyd= a cos = x,dzd= b., M0(x0, y0, z0)x x0y0=y y0x0=z z0b y0(x x0) + x0(y y0) + b(z z0)=0=2. x=a, by= az 2ab ay +bz 2b2= 0.214 6.14 xyz= 3,.: f= xyz a3= 0.M0(x0, y0) (x x0)_fx_M0+ (y y0)_fy_M0+ (z z0)_fz_M0= 0(x x0)y0z0 + (y y0)x0z0 + (z z0)x0y0= 0.z z1=x0y0z0=3. x1=3, y1= 3., V=16x1y1z1=169.6.76.1: f(x, y, z) =xcosysinz 2 ex ey+4 ez M0(1, ,4).6.2: :i)f1= 3x +y +z2, ii)f2= 2x +y3+z4iii)f3= xy +yz + 3xz, iv)f4= x3+y3+z33xz6.3: f(x) =_

ni=1xi2,x1, x2, . . . , xn x ei, i = 1, 2, . . . , n Rn.6.4: f(x, y)=x3y4(6, 1) v= 2 ex + 5 ey.6.5: , ( v) =0, v .6.8 Mathematica 2156.6:,,() = 0.6.7: f, f(x, y) = y2 ex + (2xy 1) ey.6.8 : x2y+xy2(1,4) 2x3y4 = 0 . , x2y +xy2(1,4).6.9: -U(r, ) =(Pcos)/r2, P, DmaxU=P(sin2+4 cos2)12/r3 (r, ).6.8Mathematica6.8.1 f(x, y, z) = f1(x, y, z) ex+f2(x, y, z) ey+f3(x, y, z) ez= {f1(x, y, z), f2(x, y, z), f3(x, y, z)}Mathematica :1. :Clear[f, x, y, z]f[x_, y_, z_] = {x2 + y, y2 + z, z2 + x}{x2+ y, y2+ z, z2+ x}2. :Clear[f, x, y, z, i, j, k]^ex= {1, 0, 0};^ey= {0, 1, 0};^ez= {0, 0, 1};f[x_, y_, z_] = (x2 + y)i + (y2 + z)j + (z2 + x)k{x2+ y, y2+ z, z2+ x}216 6.8.2 f(x, y, z)f(x, y, z) =fx ex +fy ey +fz ez Grad, Mathematica, CalculusV ectorAnalysis. ,- .0)&&(fxx[0, 0] >0)Truef[3, 3]9Clear[g, x, y]g[x_, y_] = Normal[Series[f [x, y], {x, 3, 3}, {y, 3, 3}]]9+ (3+ x)2+ 2(3+ x)(3+ y) + 2(3+ y)2()Clear[f , fx, fy, fxx, fyy, fxy, x, y, simia, d]f[x_, y_] :=x4 + y4fx[x_, y]:=D[f[x, y], x]fy[x_, y]:=D[f[x, y], y] simia = Solve[{fx[x, y] == 0, fy[x, y] == 0}, {x, y}]{{x 0, y 0}, {x 0, y 0}, {x 0, y 0},{x 0, y 0}, {x 0, y 0}, {x 0, y 0},{x 0, y 0}, {x 0, y 0}, {x 0, y 0}}262 fxx[x_, y_] :=D[f[x, y], x, 2]fyy[x_, y_] :=D[f[x, y], y, 2]fxy[x_, y_] :=D[f[x, y], x, y]d[x_, y_] = fxx[x, y] fyy[x, y] fxy[x, y]2144x2y2d[x, y]/.simia{0, 0, 0, 0, 0, 0, 0, 0, 0}fxx[x, y]/.simia{0, 0, 0, 0, 0, 0, 0, 0, 0}()Clear[f , fx, fy, fxx, fyy, fxy, x, y, simia, d]f[x_, y_] :=y4x4fx[x_, y]:=D[f[x, y], x]fy[x_, y]:=D[f[x, y], y] simia = Solve[{fx[x, y] == 0, fy[x, y] == 0}, {x, y}]{{x 0, y 0}, {x 0, y 0}, {x 0, y 0},{x 0, y 0}, {x 0, y 0}, {x 0, y 0},{x 0, y 0}, {x 0, y 0}, {x 0, y 0}}fxx[x_, y_] :=D[f[x, y], x, 2]fyy[x_, y_] :=D[f[x, y], y, 2]fxy[x_, y_] :=D[f[x, y], x, y]d[x_, y_] = fxx[x, y] fyy[x, y] fxy[x, y]2144x2y27.8 Mathematica 263d[x, y]/.simia{0, 0, 0, 0, 0, 0, 0, 0, 0}fxx[x, y]/.simia{0, 0, 0, 0, 0, 0, 0, 0, 0}()Clear[f , fx, fy, fxx, fyy, fxy, x, y, simia, d]f[x_, y_] :=3 x y x3 y3fx[x_, y]:=D[f[x, y], x]fy[x_, y]:=D[f[x, y], y]Needs[MiscellaneousRealOnly] .simia = Solve[{fx[x, y] == 0, fy[x, y] == 0}, {x, y}]NonReal :: Warning : NonReal numberencountered.{{x 0, y 0}, {x 1, y 1}{x NonReal , y NonReal }, {x NonReal , y NonReal }}fxx[x_, y_] :=D[f[x, y], x, 2]fyy[x_, y_] :=D[f[x, y], y, 2]fxy[x_, y_] :=D[f[x, y], x, y]d[x_, y_] = fxx[x, y] fyy[x, y] fxy[x, y]29+ 36xyd[x, y]/.simiaNonReal :: Warning : NonReal numberencountered.{9, 27, NonReal , NonReal }fxx[x, y]/.simia264 NonReal :: Warning : NonReal numberencountered.{0, 6, NonReal , NonReal } (0, 0) f(x, y) .d[0, 0] 0)&&(fxx[1, 1] 0)&&(fxx[1/3, 1/3] >0)Truef[1/3, 1/3]16 :Clear[f1, f2, f3, t, s, g, h]f1[x_, y_] =f[x, 0]x2266 f[0, 0]0(0,0)f(x, y).f2[x_, y_] =f[0, y]y +5y22g[y_] := y +5y22t =Solve[{D[g[y], y] ==0}, {y}]{y 15}D[f2[x, y], {y, 2}] ==5 >0Truef[0, 1/5]110 (0, 1/5) .f3[x_, y_] =f[x, x + 1]1 x+ x2+ 2x(1+ x) +52 (1+ x)2h[x_] := 1 x +x2 +2x(1 +x) + (5/2) (1 +x)2s =Solve[{D[h[x], x] ==0}, {x}]{{x 611}}D[h[x], {x, 2}] ==11 >07.8 Mathematica 267Truef[6/11, 5/11]322 (6/11, 5/11) .3. f(x, y)=xy I (0, 1) (1, 0).Clear[f, x, y, g, t]f[x_, y_] :=x yg[x_] :=f[x, x 1]t =Solve[{D[g[x], x] ==0}, {x}]{{x 12}}D[g[x], {x, 2}] == 2 0)&&(fxx[1/3, 1/3] >0)Truef[1/3, 1/3]13ap =Sqrt[%]135. f(x, y, z) =xyz g= xy +yz +zx 1 = 0.Clear[f , x, y, z, g, f1, m, s, g1, fmax, fmin]f[x_, y_, z_] :=x y zg[x_, y_, z_] :=x y + y z + z x 17.8 Mathematica 269f1[x_, y_, z_] :=f[x, y, z] + m g[x, y, z]t = Solve[{D[f1[x, y, z], x] == 0, D[f1[x, y, z], y]== 0,D[f1[x, y, z], z] ==0}, {x, y, z}]{{x 0, y 0, z 0}, {x 2m, y 2m, z 2m}}g[x, y, z]/.t{1, 1+ 12m2}g1[m_] := 1 + 12m2s =Solve[{g1[m] ==0}, {m}]{{m 123}, {m 123}}t/.s{{x 0, y 0, z 0}, {x 13, y 13, z 13},{x 0, y 0, z 0}, {x 13, y 13, z 13}, }fmax=f[13,13,13]133fmin=f[13, 13, 13]133270 8. , . , . . .... . . . .P.A.M.Dirac,19778.1 -. . . , , ,.-,,,Tay-lor, , ,, -272 . . - . , -. .8.28.1: F- , F = V , V (x, y, z) ,. m (r(t)) mdvdt=F ( E= mv2/2 +V) .: md2rdt2= mdvdt= F (8.1) r= x(t) ex +y(t) ey +z(t) ez v(t) =vx(t) ex +vy(t) ey +vz(t) ez . . (8.1) F= V (x(t), y(t), z(t))mdvdt= V (x(t), y(t), z(t)). (8.2) (8.2) vmv dvdt= v V8.2 273v dvdt= vxdvxdt+vydvydt+vzdvzdt=d[(v2x +v2y +v2z)/2]dt=d(v2/2)dt. v V (x(t), y(t), z(t)) = vxVx+vyVy+vzVz=dxdtVx+dydtVy+dzdtVz=dVdt. (8.2) ddt_mv22+V_ = 0. .: E.8.2: m ( ) N(r(t), p(t), t) (dN/dt =0) LiouvilleNt+p xN+F pN= 0p F . x p x=x ex +y ey +z ezp=px ex +py ey +pz ez.:N(r, p, t)=N(x, y, z, px, py, pz, t)274 dNdt=Nt+Nx dxdt+Nydydt+Nzdzdt+Npxdpxdt+Npydpydt+Npzdpzdt= 0Nt+p xN+dpdt pN= 0.Newtondpdt=F Liouville.Liouville DNDt= F vNDDt=t+p x. -6.:Liouville . 8.3: T(x, y)=A + x2 y2, .r(t) = x(t) ex + y(t) ey(2, 1) .: T= 2x ex2y ey DnoT Tn. x(0) = 2 y(0) = 1 T, - T, x = 2x, y= 2y x(t) = c1e2t, y(t) = c2e2t8.2 275 r(t) = 2e2t ex +e2t ey.8.4: Mi(i, i), (i =1, 2, . . . , n)-. M(x, y) - mi -MMi . ;: f(x, y) =n

i=1mi[(x i)2+ (y i)2]. f(x, y)fx= 2n

i=1mi(x i) = 0,fy= 2n

i=1mi(y i) = 0. fx = 0, fy= 0 x =m11 +m22 +. . . +mnnm1 +m2 +. . . +mny =m11 +m22 +. . . +mnnm1 +m2 +. . . +mn. (8.3) A =2fx2=2fy2= = 2n

i=1mi, B=2fxy= 0. B2A< 0 A > 0 .: Mi mi,(8.3)- . 8.5: 12m2. .:, () x, y, z. V =xyz, 276 2xz +2yz +xy = 12. zz=12 xy2(x +y),V V= xy 12 xy2(x +y)=12xy x2y22(x +y).Vx=y2(12 2xy x2)2(x +y)2, Vy=x2(12 2xy y2)2(x +y)2. VVx=Vy=0, x=0y= 0 . 12 2xy x2= 0, 12 2xy y2= 0.x2=y2, x=y.x= y12 3x2=0,x=2, y= 2, z= 1. V, x = 2,y= 2,z= 1 4m3.8.6: P0(x0, y0), C:g(x, y) =0, P1(x1, y1). ( ) P0P1 . PC . P , , , , . 12 (. 8.1) . C.:xy P0PP18.2 277y(x,y)(x,y)g(x,y)=00 0 0(x,y)e121 1e1 8.1. P0 P P1-(Fermat ).f(x, y) =_(x x0)2+ (y y0)2+_(x x1)2+ (y y1)2 (x, y) C, g(x, y)=0. f e1 e2, f= e1 +e2,e1=(x x0, y y0)_(x x0)2+ (y y0)2, e2=(x x1, y y1)_(x x1)2+ (y y1)2. P0PP1f= g, e1+e2= g. -. e1 +e2C,.e1 + e2= ge1 + e2 g C: g(x, y) = 0.8.7:,(1)A v1, B v2 (2) (. 8.2), Snellsinv1=sinv2.278 n =v1v2.y1:V1x2:V28.2. .: x2+a2 =_y2+b2. t(x, y) =x2+a2v1+_y2+b2v2. x y D= x +y= l, t = Dxv1x2+a2=yv2_y2+b2.. 8.2 sin =x/x2+a2. sin =y/_y2+b2 Snellsinv1=sinv2.8.8:() z(x, t) 2zx2 1c22zt2= 0 (8.4)c . (: -(8.4) u=x + ct, v =8.2 279x ct, c).() z(r, t):2zr2+2rzr 1c22zr2= 0 (8.4) .:() z(u(x, t), v(x, t))x t,zx= zuux +zvvx= zu +zv, zt= zuut +zvvt= c(zuzv)zxx= zuu + 2zuv +zvv(8.5)1c2ztt= zuu2zuv +zvv(8.6) (8.6) (8.5) 2zuv= 0 (8.7)(8.7)zv= F(v),F. z=_F(v)dv +f(u) z= g(v) +f(u) (8.8)= f(x +ct) +g(x ct)()2r2(rz) = rzrr + 2zr,2t2(rz) = rztt _2r2 1c22t2_(rz) = 0. (8.9)280 (8.4) z=1r[f(r +ct) +g(r ct)].: .(8.4) .8.9: Maxwell , E=, E= Ht, H= 0, H= J+Et. J J+ /t=0. ;: H= J+Et (H) = J+( E)t. 0 = J+t , J= v.8.10: 2fx2+2fy2+2fz2 1u22ft2= 0 (8.10)u .() Maxwell c( ) (:Maxwell 8.2 281 E= 0, E= 1cBt, B= 0, B=1cEtE(x, t), B(x, t) -).: () Maxwell - A = (A) 2A(E) = 1ct(B) 2E=1c22Et2.(B) =1ct(E) 2B=1c22Bt2. E, B c ( ).8.11: B=4cJ E= 1cBt B= 0 (8.11) OhmJ= _E +v Bc_(8.12) c ( ), Bt= (v B) +2B. (8.13) ( ) .: Ohm Bt= c(E)= c_J v Bc_= (v B) cJ= (v B) c_c4B_= (v B) c24(B) (8.14)282 ( B)= ( B) 2B B= 0 (8.14) Bt= (v B) +2B. = (c2/(4)).: (8.14) v.8.12: J= E( Ohm), . Maxwell, ( = 0),(E)(H)-2 =1c22t2+c2t, c = ()12. /(c2).(:Maxwell(MKS) E=, E= Ht, H= 0, H= J+Et , .): Maxwell (H) = J+ t(E)2H= (E) 2Ht22H=1c22Ht2+c2Ht. .8.2 283:. - -( = 0). 8.13: m q v= (vx, vy, 0) (B= B0 ez.) F= v B/c. :1. .2. .(: , , d2dt2+2 = 0, - . =a sin(t + ), a .- .): - mdvdt= qv Bc. -mv dvdt= qv v Bc= 0. d(mv2/2) = 0. . mdvxdt= qvyB0c284 mdvydt= qvxB0c. d2vxdt= q2B20c2m2vx= 2vx.vy. - . , (x, y). 8.14: P, V, T, U -, , .VT-, P=P(V, T)U=U(V, T), P+_UV_TT_PT_V= 0 (8.15) (U/V )T U(V, T)/V , . V P, T =T(V, P) U= U(V, P).:TUV P, dT=_TV_FdV+_TP_VdPdU=_UV_FdV+_UP_VdPU P , dUdP. dP=dT _TV_PdV_TP_V8.2 285dU=_UP_VdT+(U,T)(V,P)dV_TP_V, P U T V, - dP=_PT_VdT+_PV_TdVdU=_UT_VdT+_UV_TdVdPdUdTdV ,_PT_V=1_TP_V,_UV_T=(U,T)(V,P)_TP_V (8.15), P_TP_V+(U, T)(V, P) T= 08.15: (F= g ez) dudt= P g ez,, , P(x, y, z) , u g.12u2+P+gz=.:uddt_12u2_ = u P guz= dPdtd(gz)dtddt_12u2+P+gz_ = 0286 12u2+ P+ gz ( , Bernoulli). 8.16: Q , R,I,Q=I2R. ,R1,R2R3(. 8.3), ;R1 1R 22R 338.3. .: I ,I= I1 +I2 +I3 Q(I1, I2, I3) = I21R1 +I22R2 +I23R3. - Q I1+I2+I3= I = Q = I21R1 +I22R2 + (I I1I2)2R3. QI1= 2I1R12(I I1I2)R3= 0QI2= 2I2R22(I I1I2)R3= 0. I1=IRR1, I2=IRR2, I3=IRR3 Q Qmin= I2R8.2 2871R=1R1+1R2+1R3. 8.17: . . Descartes, . - Carl B. Boxer ( Princeton University Press). - . i r. i r n ( Snell )n =sin isin r. (n=c/v). 1.331.342 . i, r (.8.4)r-rrri-rrii A/2/2C8.4. ABC (i r) +( r) +/2 = = 4r 2i.288 = 4r 2i. (i, r, n) = 4 arcsin_sinin_2i. . ( . 8.5). 1637 Descartes (/i 0). in8.5. .(i, r; n)i= 0 n.: i (i, r)(i, r)i= 4cos in_1 sin2in22 = 4cos i_n2sin2i2 = 0sini =_(4 n2)/38.2 289in= arcsin(_(4 n2)/3). 1.33()< n