1. 7.1 ux 1, x 2,, x n n 2 u, u,, u F x 1, x 2,, x n ; u, u, u,, u = 0 7.1 2. F u, u, u,, u a 0 x 1, x 2,, x n u + 3. fx 1, x 2,, x n 0 a i x 1, x 2,, x n u i = fx 1, x 2,, x n 7.2 a 0 x 1, x 2,, x n u + a i x 1, x 2,, x n u i = 0 7.3 4. 5. j=1 b j x 1,, x n j = Zx 1, x 2,, x n ; z 1
2 a i, f, b j, Z 6. {x 1, x 2,, x n }Du = ϕx 1, x 2,, x n 7.1 F x 1, x 2,, x n ; ϕ,,,, 0 u = ϕx 1, x 2,, x n 7.1D 7.1n = 2 F x, y, z,, = 0 7.4 z = ϕx, y, x, y Dx, y, zz = ϕx, y 7.47.1u = ϕx 1, x 2,, x n n + 1{x 1, x 2,, x n, u}7.1 7.2.1 7.2 7.1 n dy 1 = f 1x, y 1, y 2,, y n, dy 2 = f 2x, y 1, y 2,, y n, 7.5 dy n = f nx, y 1, y 2,, y n. ϕx, y 1, y 2,, y n 7.5 G dϕx, y 1 x, y 2 x,, y n x = 0 ϕx, y 1 x, y 2 x,, y n x = c 7.6 7.5ϕ7.5
7.2 3 7.5nϕ j x, y 1, y 2,, y n = c j j = 1, 2,, n Jacobi 1, ϕ 2,, ϕ n 1, y 2,, y n = G0Jacobi 1 2 1 1 2 2 1 1 2 1 1 2 2 1 n 1 n 2 2 1 n 2 n n 1 n 2 2 1 n 2 n nϕ j j = 1, 2,, n n n n n 7.7 7.2.2 n = 1 y = ψx, c dy = fx, y 7.8 ϕx, y = c 7.87.8yx ϕx, yx = c dϕx, yx = + dy = 0 + fx, y = 0 u = ϕx, y u + fx, y u = 0 7.9 7.8yx7.9u = ux, yx du = u + u dy = u + fx, y u = 0
4 ux, yx = ux, y = c7.8 7.1 ψx, y 1, y 2,, y n = c7.5g ψ + f ψ ψ ψ 1 + f 2 + + f n = 0 7.10 1 2 n 7.2 Atn nftna t ba t bt 0 nη x = Atx + ft 7.11 φta t b φt 0 = η 7.2x 0, y 0 1,, y 0 n G7.5 y j x 0 = y 0 j, j = 1, 2,, n y j = ϕ j xj = 1, 2,, n ψx, y 1,, y n = c x = x 0 ψx 0, y 0 1,, y 0 n + ψx, ϕ 1 x,, ϕ n x = d ψx, ϕ 1x,, ϕ n x = 0 f i x 0 ; y 0 1,, y 0 n i ψx 0, y 0 1,, y 0 n = 0 x 0, y 0 1,, y 0 n G7.10G 7.10G7.5 d ψ ψx, ϕ 1x,, ϕ n x = + f ψ ψ 1 + + f n = 0 1 n yj =ϕ j x, ψx, ϕ 1 x,, ϕ n x = ψx, y 1,, y n = c7.5 j=1,2,,n
7.2 5 7.2.3 7.2 7.5nϕ j x, y 1,, y n = c j j = 1, 2,, n7.5 7.5n ϕ j x, y 1, y 2,, y n = c j j = 1, 2,, n ϕ 1 x, y 1, y 2,, y n = c 1 ϕ 2 x, y 1, y 2,, y n = c 2 ϕ n x, y 1, y 2,, y n = c n 7.12 y j 7.5 7.127.57.5 :7.5 g 0 = dy 1 g 1 = dy 2 g 2 = = dy n g n g j = g 0 f j j = 1, 2,, nn + 1µ 0, µ 1,, µ n 1 µ 0 g 0 + µ 1 g 1 + + µ n g n = 0 2 µ 0 + µ 1 dy 1 + + µ n dy n ϕϕ = c 1 2 x = dy y = xz = dy yz = dz xy dz z + x 2 + y 2 + z 2 7.2.4 7.3 X i x 1, x 2,, x n u i = 0 7.13 X i x 1, x 2,, x n x 0 1, x 0 2, x 0 n D 1 X 1 = 2 X 2 = = n X n 7.14
6 7.147.13 7.14n 1n 1 ϕ i x 1, x 2,, x n = c i i = 1, 2,, n 1c i n 17.13 7.3 ϕ i x 1, x 2,, x n = c i i = 1, 2,, n 1 7.14 7.13 Ψ u = Ψϕ 1, ϕ 2,, ϕ n 1 7.15 ϕ i x 1, x 2,, x n = c i i = 1, 2,, n 17.14 Ψ ϕ 1 x 1, x 2,, x n, ϕ 2 x 1, x 2,, x n,, ϕ n 1 x 1, x 2,, x n = c X i i = 1, 2,, ndx n 0x n 7.14x j = ψ j x n j = 1, 2,, n 1ϕ i = c i i = 1, 2,, n 17.14ϕ i ψ 1 x n, ψ 2 x n,, ψ n 1 x n, x n = c i, i = 1, 2,, n 1c i Ψ Ψ ϕ 1 x 1,, x n,, ϕ n 1 x 1,, x n xj =ψ j xn j=1,2,,n 1 = ϕ 1 x 1,, x n,, ϕ n 1 x 1,, x n = c7.14 7.1u = Ψ ϕ 1 x 1,, x n,, ϕ n 1 x 1,, x n 7.13 7.137.13 u = Ψ ϕ 1 x 1,, x n,, ϕ n 1 x 1,, x n ϕ i x 1, x 2,, x n = c i i = 1, 2,, n 17.14 7.1u = ϕ i i = 1, 2,, n 17.13n 1u = ϕx 1, x 2,, x n 7.137.13n 7.13n X i x 1, x 2,, x n x 1, x 2,, x n = 0 i X i x 1, x 2,, x n jx 1, x 2,, x n i = 0, j = 1, 2,, n 1 1 1 1 n 1 n 1 n 1 X 1 x 1, x 2,, x n X 2 x 1, x 2,, x n = 0 X n x 1, x 2,, x n 7.16
7.2 7 X i i = 1, 2,, nd n, ϕ 1,, ϕ n 1, x 2,, x n = 1 1 1 n 1 n 1 n 1 0 ϕx 1, x 2,, x n, ϕ 1 x 1, x 2,, x n,, ϕ n 1 x 1, x 2,, x n ϕ i i = 1,, n ϕ i i = 1,, n 1 Ψ 1 ϕx 1, x 2,, x n = Ψ ϕ 1 x 1, x 2,, x n,, ϕ n 1 x 1, x 2,, x n Ψ7.157.13 7.37.13 7.13 P x, y u + Qx, y u = 0, x 0 < x <, < y < u x=x0 = ϕy, < y < 7.17 7.18 P x, y = dy Qx, y 7.18 ψx, y = c 7.19 7.17 u = Φψx, y 7.20 Φ Φψx 0, y = ϕy 7.21 ψ = ψx 0, y 7.22
8 7.22 y = ω ψ 7.23 7.227.217.237.21 Φ ψ = ϕω ψ 7.24 Φ7.20 7.17 3 x = 0, u = y 2 4 x 1 0 u = Φψx, y = ϕωψx, y 7.25 x u y u = 0 x 1 u + x 2 u + + x n u = 0 7.2.5 : n = 2 b i x 1,, x n, z i = Zx 1,, x n, z 7.26 ax, y, z + bx, y, z = Zx, y, z 7.27 ax, y, zbx, y, zzx, y, zx, y, z D R 3 a, b 7.27z = ux, y F x, y, z = 0 = F / F, / F F = 7.27 ax, y, z F + bx, y, z F + Zx, y, z F = 0 7.28
7.2 9 F x, y, z = 07.27F = F x, y, z7.28 7.287.27 ϕx, y, z = c 1, ψx, y, z = c 2 7.28 a = dy b = dz Z 7.37.28 F = Φ ϕx, y, z, ψx, y, z 7.29 Φ Φ 0 Φ ϕx, y, z, ψx, y, z = 0 z = zx, y7.27 / / Φ Φ = Φ Φ, = ax, y, z Φ + bx, y, z Φ + Zx, y, z Φ = 0 a x, y, zx, y + b x, y, zx, y = Z x, y, zx, y z = zx, y7.27 7.27z = ζx, yψ [ ] Ψ ϕ x, y, ζx, y, ψ x, y, ζx, y 0 ϕ, ψ7.297.3 ϕ x, y, ζx, y ψ x, y, ζx, y γx, y = ϕ x, y, ζx, y, κx, y = ψ x, y, ζx, y γ = + ζ, γ = + ζ a γ + b γ = a + b + a ζ + b ζ = a + b + Z 0
10 a κ + b κ 0 { a γ + b γ = 0 a κ + b κ = 0 a, b Jacobi γ,κ,y Ψ γ κ γ κ 0 = 0γx, yκx, y [ ] Ψ ϕ x, y, ζx, y, ψ x, y, ζx, y = Ψ[γx, y, κx, y] 0 7.27: 7.4 ϕ i x 1,, x n ; z = c i i = 1, 2,, n 1 b 1 = 2 b 2 = = n b n = dz Z 7.30 nφn Φϕ 1, ϕ 2,, ϕ n = 0 7.31 z = zx 1, x 2,, x n 7.317.26 5 7.307.26 { x + y + x2 = z zx, y x=2 = y 4 6 7 y = z ωx 1 0 x 1 + x 2 + + x n = ωz
7.3 CAUCHY 11 7.3 Cauchy 7.3.1 v = P x, y, z, Qx, y, z, Rx, y, z Dx, y, zl x, y, zτ =, dy, dzv P x, y, z, Qx, y, z, Rx, y, z lτ v P = dy Q = dz R 7.32 l7.32 n nv z = zx, yn = ux, y, z = 0n = n v = 0 7.33,, 1 7.33 P + Q = R 7.34 u, u, u 7.33 P u + Q u + R u = 0 7.35 n7.347.35 π 7.347.35 7.32 γ πp 0 x 0, y 0, z 0 γ 0 γ 0 π π D7.32 πp 0 x 0, y 0, z 0 γ 0 πp 0 x 0, y 0, z 0 γ 0 π
12 7.32 ψ 1 x, y, z = c 1, ψ 2 x, y, z = c 2 7.36 γ 0 ψ 1 x, y, z = c 0 1, ψ 2 x, y, z = c 0 2 7.37 c 0 1 = ψ 1 x 0, y 0, z 0, c 0 2 = ψ 2 x 0, y 0, z 0 7.38 7.4π Φ ψ 1 x, y, z, ψ 2 x, y, z = 0 7.39 Φp 0 π Φ ψ 1 x 0, y 0, z 0, ψ 2 x 0, y 0, z 0 = 0 7.38 Φc 0 1, c 0 2 = 0 7.37γ 0 Φ ψ 1 x, y, z, ψ 2 x, y, z = 0 γ 0 π 7.3.2 Cauchy Γ : x = ασ, y = βσ, z = ζσ, σ I σ7.35π : z = fx, y Γ ζσ = f[ασ, βσ] α σ, β σ, ζ σα 2 σ + β 2 σ 0 7.5 7.35Cauchy 1 α σ β σ P ασ, βσ, ζσ Q ασ, βσ, ζσ Cauchy
7.3 CAUCHY 13 2 Γ α σ = P ασ, βσ, ζσ β σ = Q ασ, βσ, ζσ ζ σ R ασ, βσ, ζσ Cauchy 3 Γ α σ P ασ, βσ, ζσ β σ Q ασ, βσ, ζσ Cauchy 1 x y = z 2 Γ : x = t, y = 3t, z = 1 + t 2, t > 0 Γ : z = x, x 2 + y 2 = 1 y x = 0