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3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n dx (fg = nc n k f (k g (n k (1.3 k=0 g(x g(x d dx d 2 dx (fg = (f + 2f d 2 dx + f d2 g (1.4 dx2 f + 2f d + f d2 g(x dx dx 2

4 1 1.2 f(a+ x f(a x f (a x f(a + x f(a + f (a x (1.5 a (f(a, f (a a + x x 0 x dx f = f(a + x f(x x 0 df = f(a + dx f(a df = f (adx (1.6 df (1.6 df dx = f (a f(x = x 2 df = (a + dx 2 a 2 = 2adx + dx 2 (1.7 f (a = 2a 2adx dx 2 (1.6 dx x dx 2 dx = lim x 2 x 0 x = 0 (1.8 dx (1.6 a x df = f(x + dx f(x = f (xdx = df dx (1.9 dx

1.2. 5 df df dx dx y = f(u, u = g(x dy dx = dy du du dx (1.10 x (1.5 f(x [a, b] (a, b f(b f(a = f (c (1.11 b a c (a, b A(a, f(a, B(b, f(b AB b = a + x (1.11 f(a + x = f(a + f (c x (a < c < a + x (1.12 (1.5 f(x AB F (x = f(x f(a f(b f(a (x a (1.13 b a F (a = F (b = 0 F (x [a, b] F (x x x = c (a < c < b F (c = 0 f (c f(b f(a b a = 0 f(b f(a b a = f (c (1.14 F (a = F (b = 0 x = c (a < c < b F (x F (c = 0

6 1 x = 0 f(x = 2x + 3 (1.5 f(x f(0 x = 2 = f (0 f(x = f(0 + f (0x. (1.15 (1.5 a = 0, x = x f(x x (1.15 f(x = x 2 + 2x + 3 1 = f (0, 2 = f (0, 3 = f(0 (1.16 2 f(x = f(0 + f (0x + f (0 x 2 (1.17 2 n dn x n = n! dx n f(x = f(0 + f (0x + f (0 2! x 2 +... + f (n (0 x n. (1.18 n! x = 0 n f(0, f (0,..., f (n (0 (1.18 x (1.12 a 0, x x f(x f (2 (x f (2 (x = f (2 (0 + f (3 (cx (0 < c < x (1.19 f (1 (x = f (1 (0 + f (2 (0x + f (3 (c x 2 (1.20 2! f (1 (0 f(x = f0 + f (1 (0x + f (2 (0 2! x 2 + f (3 (c x 3 (1.21 3!

1.2. 7 f(x f0 + f (1 (0x + f (2 (0 x 2 2 x 3 f (3 (c x 3 3! f (n (x f(x = f0+f (1 (0x+ f (2 (0 2! x 2 +...+ f (n (0 x n + f (n+1 (c n! (n + 1! xn+1 (0 < c < x (1.22 f (n+1 (c x n+1 (Lagrange (n+1! n lim n f (n+1 (c (n + 1! xn+1 = 0 (1.23 1.18 n f(x = f(0 + f (1 (0x + f (2 (0 2! f(x = f(a+f (1 (a(x a+ f (2 (a 2! x 2 +... + f (n (0 x n +... (1.24 n! (x a 2 +...+ f (n (a (x a n +... n! (1.25 (Taylor series expansion x = a f(x = e x f (n (0 = 1 e x = 1 + x + 1 2! x2 + 1 3! x3 +... = n=0 1 n! xn (1.26 f(x = sin x f (2n (0 = 0, f (2n 1 (0 = ( 1 n 1 (n : sin x = x 1 3! x3 + 1 5! x5 +... = n=0 ( 1 n (2n + 1! x2n+1. (1.27

8 1 cos x = 1 1 2! x2 + 1 4! x4 +... = n=0 ( 1 n (2n! x2n. (1.28 (1.23 x e x lim n e c (n + 1! xn+1 = 0 (1.29 x x (lim n n = 0 n! x sin x, cos x 1 1 x = 1 + x + x2 +... (1.30 x < 1 x x > 1 1 x (lim n n x > 1 1 x 1.3 e x sin x, cos x e ix = cos x + i sin x (x :. (1.31 f(x = e ix, g(x = cos x+i sin x f (x = if(x, g (x = ig(x f(0 = g(0 = 1 (1.26 (1.27 (1.28 e ix (ix n = n! n=0 = n=0 ( 1 n (2n! x2n + i n=0 ( 1 n (2n + 1! x2n+1 = cos x + i sin x (1.32

1.4. 9 1.31 cos x = eix + e ix, sin x = eix e ix 2 2i (1.33 cos(ix = ex + e x 2 = cosh x, sin(ix = e x e x 2i = i sinh x (1.34 e i(α+β = e iα e iβ (1.35 cos(α + β + i sin(α + β = (cos α + i sin α(cos β + i sin β (1.36 (1.36 cos(α + β = cos α cos β sin α sin β sin(α + β = sin α cos β + cos α sin β. (1.37 1.4 x y = f(x d 2 y dx 2 = x2 + 2x + 3 (1.38 (1.38 2 2

10 1 y = f(x (1.38 x dy dx = x3 3 + x2 + 3x + c 1, y = x4 12 + x3 3 + 3 2 x2 + c 1 x + c 2 (1.39 2 c 1, c 2 n n (1.39 2 (1.39 c 1 = 0 c 2 x = 0 dy y dx c 1, c 2 (1.38 x y = f(x dy dx = αy (df(x dx y = αf(x (1.40 1 dy y dx = α d log y = α (1.41 dx log y = αx + c (c : y = e c e αx = c e αx (c = e c (1.42 c

1.4. 11 (1.40 dy y = αdx (1.43 dy y = αdx + c log y = αx + c (1.44 (1.42 dy = X(xY (y (1.45 dx dy Y (y = X(xdx (1.46 x y (1.46 1 Y (y dy = X(x dx + c (1.47 dy dx = (x + 1y2 (1.48 dy y = (x + 1 dx + c 1 2 y = 1 1 2 x2 + x + c y = 1 2 x2 + x + c. (1.49 t d 2 y dx 2 y = 0 (1.50

12 1 e x y = e λx (1.51 (1.50 (λ 2 1e λx = 0 λ 2 1 = 0 λ = ±1. (1.52 λ 2 1 = 0 f 1 (x = e x, f 2 (x = e x. (1.53 (1.50 y ( d2 dx 2 1(f 1 + f 2 = ( d2 dx 2 1f 1 + ( d2 dx 2 1f 2 = 0 + 0 = 0. (1.54 f 1, f 2 y = c 1 f 1 (x + c 2 f 2 (x = c 1 e x + c 2 e x (1.55 (1.55 c 1,2 d 2 y dx 2 + y = 0 (1.56 y = e λx λ 2 + 1 = 0 λ = ±i y = c 1 e ix + c 2 e ix (1.57 y = c 1 cos x + c 2 sin x (c 1 = c 1 + c 2, c 1 = i(c 1 c 2 (1.58 y = A sin(ωt + φ 0

1.5. 13 1.5 x φ(t, r ( r = (x, y, z 4 f(x, y, z f(x + x, y, z f(x, y, z lim x 0 x (1.59 f x f(x, y, z x (1.60 y, z f(x, y, z = x 2 y + 2z 3 f(x, y, z f(x + x, y, z f(x, y, z = lim x x 0 x [(x + x 2 y + 2z 3 ] [x 2 y + 2z 3 ] (2x x + x 2 y = lim = lim x 0 x x 0 x = 2xy (1.61 f(x, y, z = x 2 y + 2z 3 y, z x x, y, z dx, dy, dz f df f(x, y, z = xy 2 z 3 df = f(x + dx, y + dy, z + dz f(x, y, z = (x + dx(y + dy 2 (z + dz 3 xy 2 z 3 = (x + dx(y 2 + 2ydy(z 3 + 3z 2 dz xy 2 z 3 = y 2 z 3 dx + 2xyz 3 dy + 3xy 2 z 2 dz (1.62

14 1 dx, dy, dz f x = y2 z 3, (1.62 df = f x f y = 2xyz3, dx + f y f z = 3xy2 z 2 (1.63 f dy + dz (1.64 z (1.64 x y x, y f(x, y r, θ x, y x = r cos θ y = r sin θ (1.65 f(x, y r, θ r, θ df = f f dx + dy. (1.66 x y x, y (1.66 (1.67 dx = x x dr + r θ dθ. dy = y y dr + dθ. (1.67 r θ df = f x ( x x f dr + dθ + r θ y ( y r = ( f x x r + f y dr + ( f y r x x θ + f y dr + y θ dθ y dθ (1.68 θ

1.6. 15 f r = f x x r + f y y r f θ = f x x θ + f y y θ (1.69 f(x, y = x 2 + y 2 f x = x f = cos θ, x2 + y2 y = sin θ (1.70 (1.69 x y = cos θ, r r = sin θ x y = r sin θ, = r cos θ (1.71 θ θ f r = cos2 θ + sin 2 θ = 1, f = r cos θ sin θ + r sin θ cos θ = 0 (1.72 θ f(x, y = x 2 + y 2 = r (1.72 (1.69 ( f r f θ = ( x r x θ y r y θ ( f x f y. (1.73 1.6

16 1 x, y f(x, y (x, y f(x, y x, y dx, dy f(x, ydxdy f(x, y dxdy (1.74 D D (x, y dxdy (x, y ds = dxdy f(x, y ds (1.75 D f(x, y = x + 3y 2 D 0 x 1, 0 y 1, y x (1.76 x y x y (1.76 0 y x 1 x f(x, y dxdy = ( (x + 3y 2 dydx = D 1 0 = 7 12 [xy + y 3 ] x 0dx = 0 1 0 0 (x 2 + x 3 dx x, y 1 1 f(x, y dxdy = ( (x + 3y 2 dxdy = D 1 0 = 7 12 [ 1 2 x2 + 3y 2 x] 1 ydy = 0 y 1 (1.77 0 ( 1 2 + 5 2 y2 3y 3 dy (1.77 (1.78

1.7. 17 1.7 2 (x, y (u, v u = v = (u, v, (u + du, v, (u + du, v + dv, (u, v + dv A, B, C, D ABCD v u du dx, dy dv = 0 dx = x u du, y dy = du (1.79 u AB AB = ( x, y du AD = ( x, y u u v v dv AB AD sin α = AB 2 AD 2 ( AB AD 2 = ( x u 2 ( y v 2 + ( y u 2 ( x v 2 2 x x y y u v u v dudv = x y u v y x dudv (1.80 u v α AB AD ( x u x v J y u y v J = x y u v y x u v (1.81 (1.82 J (1.80 J dudv u u + du, v v + dv dudv J dudv

18 1 D x = ξ(u, v, y = η(u, v (1.83 D f(x, ydxdy = D f(ξ(u, v, η(u, v J dudv (1.84 Jdu = dx du = dx du (u, v = (r, θ ( ( x y cos θ sin θ r x θ r y θ = r sin θ r cos θ (1.85 J = r (1.86 f(x, ydxdy = f(r cos θ, r sin θrdrdθ (1.87 D D rdrdθ f(x, y = e (x2 +y 2 x 2 +y 2 = r 2 D 2π 0 { 0 e r2 rdr}dθ = 2π 0 [ 1 2 e r2 ] 0 dθ = 1 2 2π e (x2 +y 2 dxdy = ( (1.88 (1.89 e x2 dx( 0 dθ = π (1.88 e y2 dy (1.89 e x2 dx = π (1.90

1.8. 19 1.8 b f(xdx x a P Q (contour C x k x k f(x k δx ( x 0 C n k (x k, y k F (x k, y k s k n F (x, y lim n n F (x k, y k s k k=1 C F (x, yds. (1.91 C F (x, yds (1.92 C C R F (x, y = (x 2 + y 2 1 2 t x = ξ(t, y = η(t, ξ(t 2 + η(t 2 = R 2. (1.93 t t t + dt (x, y (x + dx, y + dy 2 ds ds = dx 2 + dy 2 = ( dx dt 2 + ( dy dt 2 dt = (ξ (t 2 + (η (t 2 dt (1.94

20 1 dt ds F (x, yds = F (ξ(t, η(t (ξ (t 2 + (η (t 2 dt (1.95 C C t t t ξ(t = R cos t, η(t = R sin t. (1.96 F (ξ(t, η(t = (ξ(t 2 +η(t 2 1 2 = 1 (ξ R (t 2 + (η (t 2 = R F (ξ(t, η(t 2π (ξ (t 2 + (η (t 2 dt = dt = 2π (1.97 C F (ξ(t, η(t = 1 R 2πR 1 2πR = 2π R S (surface S n k S k F (x, y, z F k n F k S k F (x, y, zds ( (1.98 lim n k=1 S ds V F (x, y, z dv F (x, y, zdv (1.99 V 0 1.9 - div rot E(x, y, z t

1.9. - div rot 21 V (x, y, z P Q (contourc t V t V tds (1.100 C t P Q P Q tds d s V d s (1.101 C P Q F (x, y, z F d s ( F d s (1.102 C S n V nds = V ds (1.103 S ds nds 2 S S E nds (1.104 S S

22 1 E E S (x, y, z V x, y, z dx, dy, dz x 2 A, B A, B (x, y, z, (x + dx, y, z x B V n x V (x + dx, y.z A, B V x (x + dx, y, zdydz V x (x, y, zdydz = V x(x, y, z dv x (dv = dxdydz (1.105 y z 2 V y(x,y,z dv V z(x,y,z dv y y z A dx, dy (x, y, z V y (x,y,z y div V dv (1.106 div V V x x + V y y + V z z. (1.107 div (divergence, S V V V divv dv (1.108 V

1.9. - div rot 23 S V V S divv dv = V ds (1.109 V rotv ds = V d s (1.110 S C S ds n (rotation rotv rotv = V z y V x z V y x C S V y z Vz x Vx y. (1.111 S (x, y, z z x, y dx, dy x A, B A, B (x, y, z, (x, y + dy, z x z n = (0, 0, 1 A V x V (x, y.z A, B V x (x, y, zdx V x (x, y + dy, zdx = V x y dxdy = V x ds (ds = dxdy y (1.112

24 1 ds y 2 Vy ds x ( V y x V x y ds = rot V n ds (1.113 n z z rotv n ds = rotv ds (1.114 S S C (1.110 rotv (1.110 ( O P O P C 1, C 2 ( F d s = ( F d s. (1.115 C 1 C 2 S

1.9. - div rot 25 F d s F d s = C 2 C 1 C F d s = 0 (1.116 C O C 1 P C 2 O t (1.116 rotf ds = 0 (1.117 S S C C S (1.117 rot F = 0 (1.118 F

27 2 2.1 3 A = (A 1, A 2, A 3, B = (B1, B 2, B 3 A B = A 1 B 1 + A 2 B 2 + A 3 B 3 (2.1 δ ij δ ii = 1, δ ij = 0 (for i j (2.2 δ ij (i, j (2.1 δ ij A i B j (2.3 i,j=1,2,3 A B A B A B A B A B 180 A B A B 2 = A B sin α

28 2 α A B 180 B A = A B. (2.4 A A = 0 A B V x, y, z i, j, k V = x i + y j + z k (x, y, z V e i (i = 1, 2, 3 e 1 = i A A = A i e i (2.5 i=1,2,3 e i e i e 1 e 1 = e 2 e 2 = e 3 e 3 = 0, e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2. (2.6 e i e j = ɛ ijk e k (2.7 k=1,2,3 ɛ ijk Levi-Civita i, j, k ɛ jik = ɛ ijk ɛ iik = 0 ɛ ijk i, j, k ɛ 123 = 1 (2.8 1,2,3 ɛ 213 = 1, ɛ 231 = 1 (2.9

2.2. nabula 29 A, B A B = ( A i e i ( B j e j = i=1,2,3 j=1,2,3 i,j=1,2,3 A i B j e i e j = i,j,k=1,2,3 ɛ ijk A i B j e k (2.10 e k A B k ( A B k ( A B k = ɛ ijk A i B j (2.11 i,j=1,2,3 k = 3 ( A B 3 = i,j=1,2,3 ɛ ij3a i B j = ɛ 123 A 1 B 2 + ɛ 213 A 2 B 1 = A 1 B 2 A 2 B 1 A B A B A 2 B 3 A 3 B 2 = A 3 B 1 A 1 B 3 (2.12 A 1 B 2 A 2 B 1 2.2 nabula (nabula = x y z. (2.13 f grad V div rot gradf = f, (2.14 div V = V, (2.15 rot V = V. (2.16 2.3 (

30 2 ( A ( B C (2.17 A = (A 1, A 2, A 3, B = (B1, B 2, B 3, C = (C 1, C 2, C 3 A ( B C A 1 A 2 A 3 = B 1 B 2 B 3 (2.18 C 1 C 2 C 3 Levi-Civita A ( B C = A k ( B C k = A k ɛ ijk B i C j = i,j,k=1,2,3 = i,j,k=1,2,3 k=1,2,3 ɛ ijk A k B i C j = i,j,k=1,2,3 k=1,2,3 ɛ jki A i B j C k i,j,=1,2,3 ɛ ijk A i B j C k (2.19 (i, j, k (j, k, i (2.18 A, B, C B C B, C B, C A B C θ A ( B C = A B C cos θ (2.20 A cos θ B, C (2.20 B, C A, B A ( B C = B ( C A = C ( A B. (2.21

2.3. 31 Levi-Civita ɛ ijk A i B j C k = ɛ ijk B i C j A k = ɛ ijk C i A j B k (2.22 i,j,k=1,2,3 i,j,k=1,2,3 i,j,k=1,2,3 i,j,k=1,2,3 ɛ ijka i B j C k i, j, k k, i, j i,j,k=1,2,3 ɛ kija k B i C j = i,j,k=1,2,3 ɛ ijkb i C j A k ( A ( B C (2.23 A B, C A B, C B C (2.23 (2.23 A B C A ( B C = ( A C B ( A B C (2.24 B C B C Levi-Civita (2.24 k ( A ( B C k = ɛ ijk A i ( B C j = i,j=1,2,3 = ɛ ijk A i ( i,j,l,m=1,2,3 i,j=1,2,3 l,m=1,2,3 ɛ lmj B l C m = i,j,l,m=1,2,3 ɛ ijk ɛ lmj A i B l C m ɛ jik ɛ jlm A i B l C m (2.25 j=1,2,3 ɛ jik ɛ jlm = δ il δ km δ im δ kl (2.26

32 2 (i, k (l, m (2.25 ( A ( B C k = (δ il δ km δ im δ kl A i B l C m i,l,m=1,2,3 = ( A i B i C k + ( A i C i B k i=1,2,3 i=1,2,3 = ( A CB k ( A BC k (2.27 (2.24 k (2.24

33 3 3.1 V ( 2 V f f : V V = f( V (3.1 y = x f( V 1 + V 2 = f( V 1 + f( V 2, (3.2 f(c V = cf( V (c :. (3.3 y = x V = x e 1 + y e 2, V = x e 1 + y e 2 (3.4

34 3 x, y ( e i (i = 1, 2 (3.2 (3.3 V = f(x e 1 + y e 2 = xf( e 1 + yf( e 2 (3.5 (3.5 ( x ( ( x = y f( e 1 f( e 2 (3.6 y ( f( e 1 f( e 2 (3.7 f( e i (i = 1, 2 2 2 M (3.6 ( ( x y = M x y (3.8 ( y = x ( 0 f( e 1 = e 2 = 1 f( e 2 = e 1 = ( 1 0 (3.9 ( 0 1 1 0 (3.8 ( ( ( x 0 1 x = y 1 0 y ( y = x (3.10 (3.11

3.2. 35 (x, y (y, x θ ( cos θ f( e 1 = sin θ f( e 2 = ( sin θ cos θ (3.12 ( cos θ O(θ = sin θ sin θ cos θ (3.13 3.2 i -1-1 180 i 90 i 90 i ( 0 1 1 0. (3.14 ( 0 1 1 0 2 = ( 1 0 0 1 (3.15

36 3-1 z = a + bi ( ( 1 0 0 1 a + bi a + b 0 1 1 0 ( a b = = ( cos θ sinθ a 2 + b 2 (tan θ = b b a cos θ sin θ a (3.16 z = a 2 + b 2 θ 3.3 (3.2 (3.3 f(x, g(x d df (f(x + g(x = dx dx + dg dx d df (cf(x = c dx dx (3.17 sin x, cos x a sin x + b cos x (3.18 sin x, cos x e 1 e 2 ( a (3.19 b

3.4. 37 d (a sin x + b cos x = b sin x + a cos x (3.20 dx ( b a ( 0 1 1 0 (3.21 (3.22 M = M M M (M t i i d f(x (3.23 dx ( 0 i i 0 (3.24 σ 2 i 3.4

38 3 2 x y ax 2 + 2bxy + ay 2 = 1 (a, b : (3.25 x, y ( 2 (3.25 M ( ( ( x a b x y M = 1 (M = (3.26 y b a y = x (x, y (3.25 x y 45 x y x y (x, y P x y (x, y P x, y e 1,2 ( x y = x e 1 y e 2 (3.27 e 1 = ( 2 1 1 2, e 2 = ( 1 2 1 2 (3.28 ( ( ( x 2 1 2 1 = x + y = y 1 2 x = x y 2 y = x +y 2 1 2 ( 1 2 1 2 1 2 1 2 O( π 4 = ( e 1 e 2 ( x y = O( π4 ( x y (3.29 (3.30

3.4. 39 45 x y x y 45-45 (3.29 M e 1 = (a + b e 1, M e 2 = (a b e 2 (3.31 (3.31 e 1,2 M a + b, a b M M ( e 1 t M e 2 = ( e 2 t M e 1 = 0. (3.32 t (transpose 1 ( e i t e j = δ ij (3.33 (3.30 O( π 4 (3.31 (3.32 (3.33 O( π 4 t MO( π ( t ( 4 = e 1 e 2 M e 1 e 2 ( t ( ( e = 1 t ( (a + b 0 ( e 2 t (a + b e 1 (a b e 2 = (3.34 0 (a b O( π 4 t MO( π 4 M O( π 4 t O( π 4 = I

40 3 (3.29 (3.26 (3.34 ( ( ( x x y M = (x y O( π y 4 t MO( π 4 x y = ( x y ( (a + b 0 0 (a b ( = (a + bx 2 + (a by 2 = 1 (3.35 x y (2bxy x y 3.5 M M V λ M V = λ V (3.36 (M λi V = 0 (I : (3.37 V = 0 M λi (3.37 V = 0 V M λi = 0 (3.38 (3.38 λ λ (3.37 V

3.5. 41 ( 2 1 M = 1 2 (3.39 2 λ 1 1 2 λ = 0 λ2 4λ + 3 = 0 λ = 3, 1 (3.40 λ = 3, 1 λ = 3 ( 1 1 V = 0 (3.41 1 1 1 ( 1 2 1 2 (3.42 e ±ix

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