10 0 -10
50
0 -50 0
-50 50
z
∈ C
x = Re z
z = x+ x +iy z
y = Im z : :
z1 + z2 = (x1 + x2 ) + i(y1 + y2 ), z1 z2 = (x1 x2 y1 y2 ) + i(x1 y2 + y1 x2 ).
:
z
x, y
∈ R
−
→ ¯z,z , z¯ = x − iy, |z| = x2 + y2 = |z¯| = √ zz¯ ≥ 0, d(z1 , z2 ) = |z1 − z2 | = |z2 − z1 | ≥ 0.
: :
z1 + z2 = z = z 2 + z1 ,
z1 z2 = z = z 2 z1 ,
z1 + (z ( z2 + z3 ) = (z1 + z2 ) + z3 = z 1 + z2 + z3 ,
z1 (z2 z3 ) = (z1 z2 )z3 = z = z 1 z2 z3 ,
z1 (z2 + z3 ) = z 1 z2 + z1 z3 .
(z1 + z2 ) = z¯1 + z¯2 , i z=0
(z1 z2 ) = z¯1 z¯2 ,
i i2 =
z = z, z¯,
|z| = |z¯| = |z|.
i3 = i i4 = 1 x=y=0
−1
−
⇐⇒ z ∈ R.
|z1z2| = |z1||z2| |z1 + z2| ≤ |z1| + |z2|, |z1 − z2| ≥ | |z1| − |z2|| . ez
≡ ex+iy = ex (cos y + i sin y)
+ i sin θ ) z = reiθ z = r (cos θ + i z n = r n einθ
z i(θ1 +θ2 )
r1 r2 e
z n = r n (cos nθ + nθ + i sin nθ) nθ). iθ n
e
= einθ
(cos θ + i + i sin θ)n = cos nθ + nθ + i i sin nθ
z1 z2 =
5i = 1 + 2i, 2+i i(1
( 1 + i)7 =
−
√ √ √ − i 3)( 3 + i) = 2 + 2i 3,
−8(1 + i).
√
√
(1 + i 3)−10 = 2−11 ( 1 + i 3),
−
−π < arg z < π, |z| > 3 1 < |z − 2i| < 2 |2z +23| > 5 Im(z ) > 0 Im(z 2 ) ≥ 0 Re z1 ≤ 12 |z − 3| > |z| {a} |z − 1 − i| = 1, {b} |z − 1| = |z + i|, {c} |z − 1 + i| ≥ |z − 1 − i|, {d} |z + 4i| + |z − 4i| = 10. √ z = 6eiπ/ 3 e−3 3 |eiz | a)
b) z1 = 6 + 17i,
z2 = 6 + 8i
|z − 12| = 5 , |z − 4| = 1, |z − 8i| 3 |z − 8|
1 + i| |z2 − 2i| = 4, ||z + = 1. z − 1 − i| z1 = 1 − i, z2 = −1 + i
z 2 = i;
z2 = 3
z 7 + 1 = 0;
− 4i;
z 8 = 1 + i;
z3 =
−1; z6 = 64; z¯ = z 3 ; |z | − z = 1 + 2i;
z cos(z
− i) = 2.
(z − 1 + i)(z − 1) = z. 2
3
z = 2kπ + i(1
− ln(2 ± √ 3)), k ∈ Z
z1 = (1
(1 + cos α + i sin α)2n
− i)/2 =
n
1 + i tan α 1
− i tan α
=
z2 = (3 + i)/5
2cos α
2n
einα , 2 1 + i tan nα , 1 i tan nα
−
ez
z
|ez | = eRe z
ez+2πi = e z .
5i . 3
tan z = zk = i ln2 + π
1 2
+ k , k ∈
z1 z2
Z
z3 2
|z| |z | ||zz || 1 2 3
z z1 z2 z3
2 2 2
z¯ z¯1 z¯2 z¯3
= 0.
1 1 1 1
z1 = 1
z3 = 1 + i (x + 1)2 + y 2 = 5 z1 z2
−i
z2 = 2i
z3
−z z¯−z¯ −z = z¯−z¯ −z z¯ −z¯ −z z¯ −z¯
z z z3 z3
1
1
2
2
1
3
1
2
3
2
k=1
|z − z1| = k, |z − z2| z1
z2
k
|z − z1| + |z − z2| = a > 0 z1
z1
z2
a > z1
| − z2|
z2
a=0
a < 0 az 2 + bz + c = 0
a b
c z P (z) P (¯z ) = P (z).
z a0 z n + a1 z n−1 + e
··· + an−1z + an = 0
n
π
a0 , . . . , an eπ
a)
√ 3 + √ 2;
√ 4 3
b)
− 2i. (ξ , η , ζ )
z = x + iy ξ =
x , 1+ z 2
||
x =
η =
y , 1+ z 2
ξζ , ξ 2 + η 2
||
y =
ζ =
|z|2 , 1 + |z |2
ηζ . ξ 2 + η 2
e+π
(ξ , η , ζ )
P (z )
P ( z),
P (¯z ),
−
P
1 z
.
( ξ, η, ζ ) (ξ, η, ζ ) (ξ, η, 1
− −
− − ζ )
−
z1 z2
z3 z1 + z2 + z3 = 0 z1 z2
z3
z
z − 1 + z| + | z − 1 − z| = |z − 1| + |z + 1|.
|
2
2
z1 z2
z3
z12 + z22 + z32 = z 1 z2 + z2 z3 + z3 z1 . (z3
− z1)(z2 − z3) = (z1 − z2)2,
(z1
− z2)(z3 − z1) = (z2 − z3)2. z
ζ
|z + ζ |2 + |z − ζ |2 = 2|z|2 + 2|ζ |2 |zζ ¯ + 1|2 + |z − ζ |2 = (1 + |z|2)(1 + |ζ |2) |zζ ¯ − 1|2 − |z − ζ |2 = ( |z|2 − 1)(|ζ |2 − 1) |z + ζ | ≤ |z| + |ζ | |z − ζ | ≥ | |z| − |ζ | | ≥ |z| − |ζ | |z + ζ | ≥ | |z| − |ζ | | ≥ |z| − |ζ | |z − ζ | ≤ |z| + |ζ | u
(u1 , u2 , u3 ) θ
v
≡ (v1, v2, v3)
u v = u 1 v1 +u2 v2 +u3 v3
·
u v = u v cos θ
·
| || |
z1 z2 = Re(¯ z1 z2 ).
·
u
≡ (u1, u2, u3)
v
≡ (v1, v2, v3) w ≡ (w1 , w2 , w3 ) w1 = u 2 v3 − u3 v2 w2 = u 3 v1 − u1 v3 w3 = u 1 v2 − u2 v1 |w| = |u||v| sin θ θ z1
u
≡ (u1, u2, u3)
≡ (v1, v2, v3) z1 = 2 + i z2 = −1 − i v
× z2 = Im(¯z1z2).
= u z3 = 3
| × v|
≡
w = u
×v
z1 z2
z3
2
(z − z )|z | , 2
3
1
4iz1
π
π z = a + ib 2
×
2 z = a + ib
Z =
→
a b −b
a
,
a, b
∈R
|z| in a) un = , n
n2 in l´ım 3 = 0; n→∞ n + 1
a)
(1 + i)n b) un = . n
b)
l´ım
n
→∞
n n + 3i
−
in = 1 n+1
− i.
∞ 1 + i/3 + (i/3) + ··· = (i/3) − , 2
n 1
n=1
a a)
∞ n− e
a in
,
b)
n=1
∞ n− e
a iπ/n
a
2
n=1
iπ/n
−1
,
∞ (a + 1)(a + 2) ··· (a + n) i . n
n!
n=1
cos kθ
c)
n=1
d)
sin kθ
,
∞ n + 1− e
k
sin θ k = 1/n
cos θ k = 5
n
k k = 1/2 n
sin x + sin 2x +
··· + sin nx, cos x + cos 2x + ··· + cos nx, sin x + sin 3x + ··· + sin(2n − 1)x, cos x + cos 3x + ··· + cos(2n − 1)x, sin x − sin2x + ··· + (−1)n−1 sin nx. eix
(1
− i)4i sin z = 2
R R = [log 2, log 3]
f (z ) = e z
× [−π, π]
(ez + 1)3 + 8 = 0 (cos i)i t
eit + 1
∈ (−π, π) (1 i)1+i cos z = cosh z Q = z : 4 z f (z) = z
−
≤ | | ≤ 9, −π/2 ≤ Argz ≤ π/2}
√ {
(
−∞, 0] + 16i − 1)4 = 16 √ 2
(z
. (ai)i
1 + i tan2 1
n
=
− i tan2 { z ∈ C
: 8
1 + i tan(2n) 1 i tan(2n)
−
si n = 1, 2, 3 . . .
≤ |z| ≤ 27, − 3π4 ≤ arg z ≤ 0 }
√ z
(
−∞, 0]
3
sin(ez ) = z
∈ C
|z| = 1
a
1 , 2
z
∈ C.
∈C
z − a 1 − a¯z = 1. A=
D = M i
B = 7π/4 M i C = 7π/4 + M i M > 0
−
− √ −
2z 4 + 1 3i = 0 cosh z sinh z = 1 e(z−1) = 1 f (z) z f (1) = 1 f ( 1) f (2i)
√
−
−Miiz e
2
a
−
D=
C
− {z = −iy, y ≥ 0}
√ i i
d 1 (Re f (z)) = f (z). dz 2
|z| ≥ 1 0 ≤ arg z ≤ π/42 1/z
z3
√ − i = − 3. z 1+i = 4
cosh(z + w) = cosh z cosh w + sinh z sinh w,
dz C
C z + 2 = 1
|
z
∀z, w ∈ C.
.
{z ∈ C, | Re z| ≤ 1, | Im z| ≤ 1, }
|
sin z + cos z = 2.
√
( 2)
−
√ z
2
.
|z| ≤ 4 −π/4 ≤ arg(z) ≤ π/4 √ z ≡ √ re i(π+θ/2)
C
2z + 3i¯ z = 1/¯ z
√ −i i
1 w=
1/z 2
z
∈ C
z C cosh2 z = sinh2 z
∈
≤ | z| ≤ 2
π/2
≤ Argz ≤ 3π/4
cosh z = sinh z u
u =
v
−v z
∈ C cos z¯ = cos z.
f (z )
√ z 4
π T (z ) = (z
i 2π
C
− [0, ∞) e
− i)/(z + i)
− − i)/√ 2 A = {z ∈ C : |z | ≤ 1, Im z ≥ 0 }
√
f ( 1) = (1
z 4 + 8 + 8 3i = 0.
(1 +
√
3i)cos π
f (i)
f (z )
z0
l´ım f (z) = f (z0 ).
z
→z
0
z z0 = x 0 + iy0
z0 u
l´ım
z
f (z) = u(x, y) + iv(x, y) (x0 , y0 ) f (z )
v
→z
0
f (z) z
z
− f (z0) − z0
z0
f (z0 )
f
f (z ) z0
z0
z0
z
z0 f (z)
z0 w = f (z) = u(x, y) + iv(x, y)
R ∂u ∂v = , ∂x ∂y
∂u = ∂y
w
R
∂v − ∂x .
R f = u + iv
df ∂u(x, y) ∂v(x, y) 1 ∂u(x, y) ∂v(x, y) = +i = + . dz ∂x ∂x i ∂y ∂y z f (z)
z
P (z) = a 0 + a1 z + a2 z 2 + w = P (z)
··· + an−1zn−1 + an zn.
P (z) , Q(z)
Q(z) ez = e x+iy = e x (cos y + i sin y)
a az = e z ln a , ln a
a
eiz
sin z =
− e−iz , 2i
cos z =
eiz + e−iz , 2
sec z csc z tan z cot z ez
sinh z =
− e−z , cosh z = ez + e−z ,
2 tanh z coth z z = ew
2
w = ln z
w
z w = ln z = ln r + i(θ + 2kπ),
arcsin z =
z = rei(θ+2kπ ) .
k = 0, 1, 2, . . . ,
± ±
1 ln iz + i
1 − z , arc cos z = 1 ln z + z − 1 , 2
2
i
sin−1 z cos−1 z ln
arctan z
arcsinh z = ln z + z + 1 , arccosh z = ln z + z − 1 , 2
2
arctanh z zα α
f (z )
∈C
≡ eα ln z ,
g(z)
z f (z)g (z)
≡ eg(z) ln f (z) ,
w P 0 (z)wn + P 1 (z)wn−1 + P 0 (z) = 0 P 1 (z) . . . Pn (z)
··· + P n−1(z)w + P n(z) = 0, z
f (x, y)
R R
R
R2
∂ 2 f (x, y) ∂ 2 f (x, y) + = 0. ∂x 2 ∂y 2 u(x, y) u
v(x, y)
v
∂ 2 u ∂ 2 v = , ∂x 2 ∂x∂y
∂ 2 u = ∂y 2
2
∂ v , − ∂y∂x
∂ 2 u ∂ 2 u ∂ 2 v ∂ 2 v + = 0 = + . ∂x 2 ∂y 2 ∂x 2 ∂y 2
|z|2
|z|2 e2z
sin z
||
−
z=0
||
∈ C
|xy|
f (z) =
w = z 2 , Re z > 0; w = z¯2 , z < 1; 1 w = , z < 1; z 1 1 1 w = z + , z < 2; 2 z 2 1 1 w = z + , z < 1, 4 z
z
||
0 < arg z < π/2.
||
f (z)
z 2
∂ 2 + ∂x 2 ∂y 2
∂
|f (z)|2 = 4|f (z)|2.
f (z) = z 2
z 10 + 1 , z →i z 6 + 1
b) l´ım
1/2
e−1/2
a) l´ım 5/3
1/2
z
1
→0
− cos z ,
c) l´ım
z2
f 1 (z) = xy + iy;
z
1
− cos z ,
→0 sin z2
2
d) l´ım (cos z)1/z . z
→0
f 2 (z) = e y (cos x + i sin x). R2
f (z ) u(x, y) +1 u(x, y) = sin x cosh y + 2 cos x sinh y + x2 w = z 3 + 3z 2 + C w = sin z + z 2 = x 3
−
3xy 2
f (z) h(z) = f (¯ z) v
+ 3x2
−
3y 2
− y2 + 4xy 2 − i(2sin z + 2z ) f (z ) = u(x) + iv(y) g(z ) = f (¯z )
C
u f (z ) = (ax + b) + i(ay + c)
uv
eu cos v
a,b,c
sin z sin z = sin x cosh y + i cos x sinh y u(x, y) v(x, y) u = 2x(1 u = 2x
− y)
− x3 + 3xy2
f (z)
u = sinh x sin y u = y/(x2 + y2 ) u = cos x sin y f 1 (z) = iz 2 + 2z + C f 2 = 2z
− z3 + C f 3 = −i cosh z
f 4 = i/z + C f 5 u(x, y) = 2ex cos y
w = f (z) f (0) = 2 2ez u v
ui
vi U V
D
D a) b) c) d)
U = au bv, V = bu + av, a, b C. U = au 1 + bu2 , V = av 1 + bv2 , a, b U = u 1 u2 v1 v2 , V = u 1 v2 + v1 u2 . U = e u cos v, V = e u sin v.
e)
U = e u −v cos2uv, V = e u −v sin2uv. u 2 v 2 v 2 u2 U = e uv cos , V = e uv sin . 2 2
−
−
2
2
2
2
−
f)
a) u = φ(ax + by), a, b
∀ ∈ R,
2
∀ ∈ ∀ ∈ C.
2
a) u = φ(x + y ),
b) u = φ(x
C ln(x2 + y 2 ) + K
C (x2
−
b) u = φ(x +
2
x + y ), 2
2
c) u = φ(x2 + y).
2
2
− y ),
− y2) + K
c) u = φ(y/x),
d) u = φ
C arctan(y/x) + K
2
x + y x
.
Cx/ (x2 + y2 ) + K
w = u + iv
u
w w
|w|
w f (z ) = z/(1+ z )
||
C
2z + 1 z(z 2 + 1) z3 + i z 2 3z + 2
−
(z + 2) −1 (z 2 + 2z + 2) −1
z¯ f (z) = 0z
2
z = 0
− iv
si z = 0
si z = 0, z = 0
|w| < 1
f (z) = x 3
− i(y − 1)3 ≡ u + iv ∂u ∂v +i = 3x2 , ∂x ∂x
3x2
f (z ) f (z)
f (z ) 2x + xy 2 i z z¯ ex (cos y i sin y)
−
z0 = i
−
sin z sinh z
f 1 (z) = 3x + y + i(3y
− x);
f 3 (z) = e −y (cos x + i sin x); f 1 = (3
− i)z
f 2 = sin z f 3 = e iz f 4 = (z 2
f (z)
tan z
f 2 (z) = sin x cosh y + i cos x sinh y; f 4 (z) = (z 2
− 2)e−z z0
− 2)e−x(cos y − i sin y). u = Re f
v = Im f
f (z0 ) = u x (x0 , y0 ) + ivx (x0 , y0 ) f (z0 ) = v y (x0 , y0 )
− iuy (x0, y0) f (z0 ) = u x (x0 , y0 ) − iuy (x0 , y0 ) f (z0 ) = v y (x0 , y0 ) + ivx (x0 , y0 )
|f (z0)|2 = ux2 + uy 2 = ux2 + vx 2 = uy 2 + vy 2 = vx 2 + vy 2 u(x, y)
∂u/∂x ∂u/∂y u(x, y) v(x, y) ∂u/∂x ∂v/∂x
∂u/∂y ∂v/∂y u
v
R2
u(x, y) = C 1
v(x, y) = C 2 (x0 , y0 )
u(x, y)+ iv(x, y) x (z + z¯)/2 y g
→ (z − z¯)/2i
→
u = C 1 v = C 2 ∂u/∂x ∂v/∂y x y g(z, z¯) = u(x(z, z¯), y(z, z¯)) + iv(x(z, z¯), y(z, z¯)) z¯ u + iv
f (z) = u(x, y) + iv(x, y) v f (z) f (z) = u(z, 0) + iv(z, 0),
u f (z) = u(0, iz) + iv(0, iz).
−
f (z) = u(x, y) + iv(x, y) u f (z) = 2u(z/2, iz/2)+ f (z) = 2iv(z/2, iz/2)+
− −
−
u v
v
v f
u
f (z) = u(x, y) + iv(x, y)
u
v f (z)
f (z) = u(z/2, iz/2) + iv(z/2, iz/2).
−
−
f (z) = u(x, y) + iv(x, y) u(z/2, iz/2) = iv(z/2, iz/2).
−
−
f (z) a) u(x, y) = x 4 6x2 y 2 +y4 ,
−
c) v(x, y) = x 3 2y 2 ,
b) v(x, y) = sinh x cos y,
−
f (z) = u(x, y) + iv(x, y) f (z ) = u(az, i(1 a)z) + iv(az, i(1
v a
f
− − a)z)
− −
d) u(x, y) = y/(x2 +y2 ).
a = 0 a = 1
u a = 1/2
a f (z) = z 1/2 z = re iθ w =
r > 0, 0
√ z2 − 1 z = reiθ
f = u(r, θ) + iv(r, θ)
∂u 1 ∂v = , ∂r r ∂θ
||
f (z) f (z) = 1/(2f (z))
√ z2 + 1 w =
f = z 27 f = z z 2
≤ θ < 2π
∂v = ∂r
. − 1r ∂u ∂θ
f = r 2 + θ + 2ir 2 θ zi
u(r, θ)
i
z = u + iv z = π + i log (2 + 51/2 )
w = sin z
√ z √ −1 = −i
f (z) = [0, + )
∞
C
√
f (z) = 1/2 z
RT = 2 RM = 3
RJ = 10 T T = 1 T M = 2
v(x, y) = x 3 a
− 3axy2
u(x, y)
v(x, y) f (z) = u + iv u =
f (z)
f (z) = u + iv g(z )
u
T J = 12
v
v = u + v
u+v
v(r, θ)
u(x, y) = xy + (ey + e−y )cos x u u z
f (z)
Re f (z) = u(x, y) f (0) = 2 + i
u(x, y) = ax 3 + bx2 y + cxy 2 + dy 3 v(x, y)
.
u(x, y) + iv(x, y)
u(x, y) = x 2
− y2 + 5x + y − x2 +y y2 ,
u v f (z )
u f (z ) = u + iv
x
y
φ(xy)
f (z)
z0 = 0 f (z) + i2 f (z) = 0
f (0) = 0 f (0) = i
2
f (z )
2
u(x, y) = e x −y cos(2xy)
∇2u = 0 z
R2
f (z ) = u + iv
f (0) = 1
f = u + iv
u(x, y) = x f (z) f (z) = α, z →0 z m l´ım
α
∈ C,
m
∈ N. m
α f (z)
≡ f (x+iy) = u(x, y)+iv(x, y) f (0) + f (0) = i, u + 2v = 2x(1 − y + x) − 2y(y − 2). f (x + iy) = cos x(cosh y + a sinh y) + i sin x(cosh y + b sinh y) a b f (z ) f
z
− e−y sin x
f (z) : f (z)
C
→C
∀(x, y) ∈ R2
≡ f (x + iy) = f (x) − f (y) + 2xyi a,b,c
∈C
f (z) = az 2 + bz z¯ + c¯ z2 ,
z1 z2 z2 = z 1
C z2
(x2 ,y2 )
f (z)dz =
z1
(x2 ,y2 )
(u(x, y) + iv(x, y))(dx + idy) =
(x1 ,y1 )
(udx
− vdy) + i
t2
f (z)dz =
z1
C
t2
{u(x(t), y(t))x(t) ˙ − v(x(t), y(t))y(t) ˙ } dt+i
t1
t1
{v(x(t), y(t))x(t) ˙ + u(x(t), y(t))y(t) ˙ } dt.
y = y(x)
C z2
x2
f (z)dz =
z1
(vdx + udy)
(x1 ,y1 )
{x(t), y(t)} t ∈ [t1, t2]
z2
(x2 ,y2 )
(x1 ,y1 )
C
x1
{u(x, y(x)) − v(x, y(x))y(x)} dx + i
f (z)
x2
x1
C
R
{v(x, y(x)) + u(x, y(x))y(x)} dx.
f (z)dz = 0.
C
f (z)
R
f (z)dz = 0,
C
f (z)
R
R
f (z)
R b
f (z)dz = F (b)
a
F (z ) a b
F (z ) = f (z)
− F (a),
f (z)
f (z)dz = dF C 0
R
C
f (z)dz = 0 =
f (z)dz +
C 0
C j j = 0, 1, . . . , n
C 1
f (z)dz +
C i i = 1, . . . , n
··· +
C n
f (z)dz,
P (x, y) Q(x, y) C
R
∂Q
P (x, y)dx + Q(x, y)dy =
R ∂x
C
∂P ∂y
−
f (z) a
dxdy.
C
R
R 1 f (a) = 2πi
f (z)
f (z) C
z
− a dz.
a a 1 f (a) =
f (z) dz. 2πi C (z a)2
f (z )
−
f (a) f (a), . . . ,
a a
f (z)
C
|f (z)|
C
f (z) M
|f (z)| < M ∀ z ∈ C
f (z)
P (z) = a 0 + a1 z + z1 n 1 n n
··· + anzn
n
≥1
an = 0 P (z) = (z
−
P (z)
f (z)
− z1)Q(z)
Q(z)
C C 1 2πi
N
f (z) C
f (z)
=0
− α)ng(z)
f (z)
M C
− P,
P α f (z) = (z
h(β )
dz = N
C , |f (n) (a)| ≤ Mn! Rn |f (z)| < M
f (z ) n β g(α) = 0
R
C p f (z) = h(z)/(z
f (z)
− β ) p
z = a
n = 1, 2, . . . C
M
|f (z)|
f (z)
f (z)dz,
C
− x − 3x2i C f (z) = z − 1 C z
f (z) = y C
f (z) = e
z1 = 0 z1 = 0
z3 = i
C
f (z) = 1/z 1 i (1
−
−
C i)/2
z2 = 1 + i z2 = 1 + i z(θ ) = 1 + eiθ z1 = 0 z1 = πi z2 = 1 R
0
1+e
2πi
C
z1 = 0 z2 = 1 z3 = 1 + i
2
(3z + 1)dz = 0,
y que
C
α
z2 = 2
z4 = i
z¯dz = 2i.
C
β β
2zdz = β 2
α
α
− α2,
β
α = β
(2,5)
(3x + y)dx + (2y
(0,1)
(0, 1)
y = x 2 + 1 (0, 5) (0, 5)
− x)dy, (0, 1)
(2, 5)
(2, 5) (0, 1)
(2, 5) 88/3
32
40
24 x
i/2
a)
πz
e
e
e + 1/e
cos bxdx,
π +2i
dz,
b)
0
i
(1 + i)/π
ax
z cos dz, 2
0
∞ sin αx 0
α > 0
eax sin bxdx.
x
dx
α < 0
∞ 0
α
cos αxdx
3
c)
1
(z
− 2)3 dz,
(2, 1)
(2, 1)
y = x3
C
− 3x2 + 4x − 1
1 + i
(12z 2
C
2 + 3i
− 4iz)dz.
−156 + 38i
C
C 1 C 2
|z| = 1 |z − 1| = 1
2πi
z1 = 1
|z|2dz,
z2 = i
dz
√ z
C
C sin3z dz, C z + π/2
2πi
−πi
sin6 z dz, π/6 C z
iz
e C
z3
dz,
−
C
(9
−
|z| = 2 zdz z 2 )(z + i)
πi/32 π/5
1 2πi C
ezt dz = sin t, 2 C z + 1
|z| = 3
(2 + 2i)¯ z
C z2 = 2 + 2i
z1 = 1
z 2 + 2z z¯ + z¯2 = (2
− 2i)z +
z¯2 dz + z 2 d¯ z
C
248/15 C
A 1 A = 2i
z¯dz
C
C
A 1 A = 2
C
xdy
− ydx. x = a cos θ
y = b sin θ 0
≤ θ ≤ 2π
C
A
R A =
dxdy,
R
A =
dxdy =
P (x, y)dx + Q(x, y)dy,
C
R P
Q
P (z) = z 4 z
− z2 − 2z + 2
z
1, 1, 1
− ±i P 3 (z ) = z 3
3i)z 3 + (1 + 2i)z 2 + (1
− 13i)z − 10 + 2i (z − i)(z + 2i)(z − 1 − i)(z + 2 − 3i) z = e iθ
C
ke
− z2 + (3 − i)z − 2 − 2i P 4(z) = z 4 + (1 − (z − i)(z + 2i)(z − 1 − i)
kz
k
dz,
z
C
2πi π
dθek cos θ cos(k sin θ) = π.
0
f (z) z0
C C
f (z) C
z
− z0
dz =
f (z) dz. z 0 )2 C (z
−
f (z)
C
1 a) f (a) =
2π
2π
0
a
F (z, z¯)dz = 2i
C
2π
e−niθ f (a + eiθ )dθ,
0
C
R
dA
f (n) (a) 1 = n! 2π
e−iθ f (a + eiθ )dθ, b)
∂F R ∂ ¯z
dA,
dxdy
P (z, z¯) Q(z, z¯) C
R
P (z, z¯)dz + Q(z, z¯)d¯ z = 2i
C
∂P R ∂ ¯z
−
∂ Q ∂z
dA.
u(r, θ) 1 u(r, θ) = 2π R
2π
0
R2
(R2 r2 )u(R, φ) dφ, 2Rr cos(θ φ) + r2
−
−
−
φ
(r, θ) u(r, θ)
m
a +
∞
sin(mx) dx x(x2 + a2 )
0
.
f (z) = e−z 0
≤ y ≤ 1
∞ − e cos(2x)dx −∞∞ e− dx = √ π x2
f (z )
√ π/2e
x2
|z| < R
|z|=1
|x| ≤
.
0
+
2
R > 1
1 f (z) z + 2 + dz z z
f 1z = 2i
∞ z
n
n2 + n2 (2z)n
n=1
∞ x + 1
.
2
0
√
x4 + 1
dx
π/ 2
∞ 0
z α = eα ln z z = 0 π/2cos(απ/2)
xα dx, 1 + x2
α
∈ R,
0 < α < 1.
||
α
2
z dz −2 (z2 + 1)2
x Im z
x1 =
−2
x2 = 2
≤ 0 z0 = i
∞ 0
√ x 5
x3 + x
dx.
R
f (2) = 3
f (z) f (2) = 4
|z| < 10
|z|=5
z + (z
2
− 2) −
3 2 z (z 2)
−
f (z)dz
f (0) = 1 f (0) = 2
f ( f (z )
z0
∞ f f ( f (z ) =
(n) (z
n=1
0)
n!
(z
− z0)n , z0
z0
f ( f (z )
z0
(n)
f
n! (z0 ) = 2πi
f ( f (z ) dz. z0 )n+1 C (z
z0
−
f ( f (z ) C 1 C 2
∞ f ( f (z ) = a (z (z − z ) n
0
−∞
n
=
z0
· · · + (z −a−zn0)n + · · · + (z −a−z20)2 + za−−1z0 + a0 + a1(z − z0) + a2(z − z0)2 + · · · z
1 an = 2πi
C i
C 1
f ( f (z ) dz , n +1 (z z0 )
−
∞
f ( f (z )
(n = 0, 1, 2, . . .) .)
± ±
(z
0
C 2
− z0) an
an = f = f (n) (z0 )/n! /n!
1/(z z0 ) f ( f (z ) n 0
−
≥
z0
an an
an
n
C 2
n C 1
C 1 C 2
z S n = a 1 + a1 r + a1 r2 +
an
· · · a1rn−1 = a1(1 − rn)/(1 − r) f ( f (z )
z0
f ( f (z )
z0
a−1 a−2 + + z z0 (z z0 )2
−
−
z0
n
· · · (z −a−zn0)n ,
f ( f (z )
f ( f (z )
n=1
z0
l´ımz→z f ( f (z ) f ( f (z0 ) l´ımz→z f ( f (z ) 0
≡ F ( F (w) ≡ f (1 f (1/w /w))
f ( f (z ) w = 1/z w = 0
0
f ( f (z )
z0
f ( f (z ) f ( f (z )
z
n
z0
z
z0
n= n=0
∞ a f (z ) n n
l´ımn→∞ zn = 0 z
| |
a f (z) |a f (z)| < 1, 1, l´ım < 1, 1 , l´ ı m →∞ a − f − (z ) →∞ > 1 z a z z = z |z| < |z | a z = S (z ) a z S (z ) = n n
n
n
n n
n
n 1 n 1
n
|z | n
n
z
n
1
1
n
z
n
n
n
S (z ) =
a (z − z ) n
0
S (z )
n
nan z n−1 f ( f (z )
|z − z0| = R0 z − z0
f ( f (z ) f ( f (z )
z
f ( f (z ) f (z )
z
a z n
f ( f (z )
g (z )
n
= f ( f (z )
b
mz
m
= g( g (z )
a z + b z = a = a + b + (a ( a + b )z + (a ( a + b )z + · · · n
n
m
m
0
0
1
f ( f (z ) + g + g((z )
g (z )
2
m
= g( g (z )
2
2
z
a z n
f ( f (z )
1
n
= f ( f (z )
b
mz
a z b z = a = a b + (a ( a b + a b )z + (a ( a b + a b + a b )z + · · · n
n
m
m
0 0
f ( f (z )g (z )
0 1
1 0
0 2
z f
z0
z0
1 1
2 0
2
∞ (−1) − z
n 1 2n 1
a)
(2n
n=1
−
− 1)!
,
∞ n!z . n
b)
n=1
z0 e2z ; (z 1)3
a)
− z − sin z ;
c)
z0 = 1,
b) (z
z0 = 0,
z3
− 3)sin z +1 2 ;
z ; (z + 1)(z + 2)
d)
z0 = z0 =
−2. −2.
z
∞ 2−| |z , n
a)
n
∞
zn , n +1 3 n=−∞
b)
n=
−∞
∞ 2−
n3
e)
n=
−∞
1
− (z − a) +1
2n
∞ 2−
n2
c)
n
(z + 1) ,
,
−∞ ∞
2z,
f)
n n
g)
n=
∞ 2−
n2 n3
z .
n=
−∞
−∞
− 3)
|z| > 3
zn , 2 +1 n n=−∞
d)
n=
1/(z
|z| < 3
∞
z
z = a z2 z
a) d)
1
− 1 , (a = 1); −1
− cos z (a = 0);
e)
z2
b)
sin z , (a = 0); z
c)
1
1 , (a = 0); − z e − 1 sin z
z , (a = 0); tan z f)
z2 1 , (a = z3 + 1
−
∞).
z = a a) d)
1
−
1 , (a = 0); z
z (a = 0); cos z
b) e)
1 z2 + 1 , (a = i); c) , (a = ); (z 2 + 1)2 z + 1 z , (a = πi); f) tan πz, (a = 1/2, 3/2, . . .). z e +1
∞
±
±
z = a a) ez , (a = d) z 2 cos
∞);
π (a = 0); z
2
b) e−z , (a =
e) etan z , (a = π/2);
e z0 = 1
∞);
1 1−z
.
c)
sin f)
π , (a = 0); z2
sin ez , (a =
∞).
z
z0 = 0
1 < z <
|| ∞
ez z(1 z)
−
n =
zn
−2, −1, 0, 1, 2 z a a) b) c) d) e) f)
D 1
z(z
− 3)2 ,
D
(a = 1, D : 1 < z
| − 1| < 2),
1
(z 2
−a
− 9)z2 ,
(a = 1, D : 1 < z
| − 1| < 2),
z + i , (a = i, i D), z2 z2 1 , (a = 1, 2i D), z2 + 1 2z , (a = 1, 1 D), z 2 2i 1 3 , (a = 0, z(z 1)(z 2) 2
− ∈ ∈ − ∈
−
− −
− ∈ D).
−
f (z) = cos
11 + z . 1
−z
0 < z
| − (π + 2)/(π − 2)| < R
(1 + z) p
p z0 = 0 (1 + z) p = 1 +
p p( p 1) 2 z + z + 1! 2!
−
··· + p( p − 1) ···n!( p − n + 1) zn + ··· z0 = 0
|z| > 1 z0 = 0
z (2i + z)−1/2
z3 (z + 1)(z
− 2) , (z + 1)
0 < z + 1 < 3
|
|
|z + 1| > 3
(z + 1) z
1/(1 1 (1
z
− z)2
∞ = nz − ,
1
n 1
(1
n=1
z0 = 0
∞ n(n − 1) = z − ,...
1
n 2
− z)3
2
n=2
1/z 1/z 3
1/z 2
−1
− z) (1
− z)k
∞ n = z −
n k+1
n=k
k
z0 = 1
z0 z , (z0 = 2). (z + 2) 2 z 1 , (z0 = 0). sin2 z eiz , (z0 = ib, b > 0). z 2 + b2 (z 2 + 1)2 , (z0 = ). z 2 + b2
a)
−
−
b) c) d)
−2/(z + 2)2 + 1/(z + 2)
∞
−1/z2 + 1/z
−ie−b/2b(z − ib) f (z) = 1/z(z 2 − 1)
1 < z
| − 1| < 2
z2
z = 1
f (z) =
(z
sin3 (πz)
z = 2
f (z) =
− 1)2
0 < z
| − 1| < 2
z 2πi 2πi + ez 1 z
− −
f (z )
ln(1 + z) zm z0 = 0
f (z) = m
−z,
m
| − 1| < 1
log (1 + z)
− 1)(z − 2)3 .
f (z) = 1/(z 2
0 < z
∈ N. f (z)
f (z ) = f
1 z 2 sinh z
z
0 < z < R
||
R
f (z)dz
y
|z|=1
z2 =
f (z)dz
.
|z|=5
f (z) = z1 = 2πi
ez + 1 , ez 1
−
−4πi
(z2 =
z3 = 0 f (z) z sinh 1 1z (z 1)2 z 3
−
−
−4πi)
z0 Resz=z
0
f (z0 )
1 f (z) = l´ım f (z)dz = Res f (z0 ), R→0 2πi |z −z |=R
0
R > 0 R z0
f (z)
Res f ( ) =
∞ −
f (z ) zi i = 1, . . . , k k
z0 1 l´ım f (z)dz. R→∞ 2πi |z |=R
R
C
C
→0
R k
(Resf (z )). f (z)dz = 2πi i
i=1
z0 Resf (z0 ) = l´ım (z z
z0
0
− z0)f (z).
n Resf (z0 ) =
z0
→z
1 (n
l´ım
− 1)! z→z
0
dn−1 ((z dz n−1
− z0)nf (z)) .
a−1
f (z)
−a−1
z f (z)
z0
a)
2z + 1 ; z2 z 2
z + 1
b)
− −
z
;
−1
sin z ; z2
c)
1 ; cosh z
e)
1 cos x dx = 2
π
d)
cot z;
e−1/z sinh(1/z)dz,
C
C
|z| = 1
∞ sin x 0
∞
π dx = , x 2
2
sin x dx =
0
0
(z 3
C
∞
C
sin z z)(z
−
2
2
,
− i) dz
|z − 1| = 1 |z| = 2 [π(i − 1)sin1]/2 πi(sin 1 − sinh1) sin z ez i/z a) Resz=∞ 2 , b) Resz=∞ e , c) Resz=1 , z (z 1)2 π d) Resz=∞ z 2 sin , e) Resz=∞ z n ea/z , f ) Resz =1 ze − . z i e π 3 /6 an+1 /(n + 1)! 3/2
−
1
z
−1
−
−
a) d)
a)
1
1 , 6 z (z 2)
−
d)
1 , z + z 3
b)
sin πz , (z 1)3
1 + z8 , z 6 (z + 2)
1 sin z sin , z
c)
1 , sin z 2
e)
−
b)
z2 , 1 + z4
f)
1 , + 1)3
ez
1 + z 2n , z n (z a)
c)
cos z , + 1)2
e)
(z 2
a = 0, n = 1, 2, . . .
−
(z 2
1 . +1
sin z . + 1)2
f)
(z 2
C a)
dz , (D : |z − 1| < 1), 1+z dz , (D : |z − 1 − i| < 2), (z − 1) (z + 1) sin z dz, (D : x +y < 2 ), (z + 1) dz , (D : 2 < |z | < 4), (z − 1) (z − 3) z 4
C
b)
2
C
c)
2/3
C
e)
2
2/3
3
C
d)
D
2
C z + 3
2
e1/3z dz,
2
(D : z > 4).
||
2/3
∞ (ln u)
2
I =
du =
1 + u2
0
π3 . 8
OX
2π
0
dθ = a + b sin θ 2π
0
√ a22π− b2 ,
→0
R
→∞
a> b.
||
cos3θ π dθ = . 5 4cos θ 12
−
z f (z) =
z2 = 0
1
z 2 + 2(1
f (z)dz f (z)dz
− i)z − 2i , A z1 =
−2
D ABCD
f (z ) f (z) f (z)
z0 =
z
E z0 = c
a b a > b z(θ) = a cos θ + ib sin θ 0 zE (θ) = a cos θ c + ib sin θ 0 θ π
−
2π
a c cos θ 2π dθ = H (a a2 + c2 2ac cos θ a
≤ ≤
− − c), − 0 2π a − c cos θ 2π I 2 = 2 dθ = b H (a − c), 2 2 2 2 c + a cos θ − 2ac cos θ + b sin θ 0 I 1 =
−1 + i
≤θ≤π
H (x) H (x) = 0,
Im
x < 0,
dz E
2π
0
z
= bI 2 = 2πH (a
dθ , a + b cos θ
2π
(5
0
√ − b2
2π/ a2
H (x) = 1,
−
x > 1.
− c).
a> b.
||
dθ . 3cos θ)2
5π/32 a > 0
∞ x sin ax −∞ x4 + 4
πe −a sin a/2
I =
∞
dx
dx , (x2 + a2 )5
0
a > 0.
35π/256a9 +
∞ sin x I = dx x(x + 1) 2 √ π 6
0
Sol. I =
1
2
− 3√ e cos(
z0
z0 z1 = 1
3/2)
−
z0 m
− 2i
n
− i)4
+
∞
dx −∞ (x2 + 4x + 5)2 a > 0
∞ (x − b )sin ax 2
−π/2 + πe−ab
f (z) , f (z)
z0 f (z ) F (z) f 2 (z ) = 1/(z
0
1 3e
F (z) F (z) =
f (z )
.
b > 0
2
(x2 + b2 )x
dx.
f 1 (z) = (z z2 = i
f (z )
− 1 + 2i)3
z8 f (z) = + z 2 sin (z 1)4
−
1 z + 1
f (z)dz
|z|=2
4π
2cos θ dθ 5 + 4 sin(θ/2)
−
0
2π
2π
2
cos θdθ,
0
sin3 θdθ,
0
2π
0
2
−
dθ sin2 (θ/2)
.
x1 , . . . , xn
R
y1 , . . . , yn
∂ (y , . . . , y ) dy dy ··· dy = dx dx ··· dx , ∂ (x , . . . , x ) 1
n
2
1
n
1
n
1
n
2
|∂ (y1, . . . , yn)/∂ (x1, . . . , xn)|
(x, y) (u, v)
∂ (u, v) = ∂ (x, y)
∂u ∂x ∂v ∂x
∂u ∂y ∂v ∂y
f (z) = 0
∂u ∂v ∂u ∂v ∂u ∂u = ∂x ∂y − ∂y ∂x = ∂x + ∂y = |f (x, y)| . 2
2
2
f (z) = 0
w0 = f (z0 )
z0 z0 ∆z
∆w/∆z = f (z0 )
∆w
w0
arg ∆w = arg ∆z + arg f (z0 )
arg f (z0 )
∆w
f (z)
∆z
z0
w0 ∆w| = |∆z ||f (z0 )| | |f (z0)|2
z0
|f (z0)|2
arg f (z0 )
z0
w = u + iv = f (z)
≡ f (x + iy)
H (x, y) g(w) g(u + iv) u v L(u, v) = H (x(u, v), y(u, v)) H (x, y) = C z = f (w)
≡
u(x, y) =
v(x, y) = f (z) =0
x y x(u, v) y(u, v)
z=
L(u, v) = C f (w) =0
dxdy rdrdθ
w
r
β , ≡ T (z) = αz + γz + δ
αδ
− γβ = 0.
αδ
− γβ = 0
δw β γw + α
z =
2
−
−
×2
SL(2, C)
α,β,γ δ z w 2w w =
w(z) 3(w )2
(a) w = i
z + 1 , z 1
|z| < 1,
−
(c) w =
(b) w =
z
z
z
− 1, w =
z =
z
− 1,
0 < arg z < π/4,
0 < x < 1.
2z 5 z + 4
−
−1 ± 2i 3 + 4i (a) z = 1,
(b) z
| − 1| = 1,
||
(c) z
T (z) 1 T (z) a
T (z) = z + A
a =
∞
−a
=
1
z
− a + A, A
| − i| = 2.
a
T (z)
a
b
a = b
T (z) T (z)
k z1 z2
− a = k z − a , −b z−b
z3
w1 w2
w3 wi = f (zi )
w − w w − w z − z z − z w
1
3
1
− w2
w3
− w2
=
z
1
3
1
− z2
z3
− z2
zi
wi
−1 (a) 0, 1,
∞,
(b) 1,
.
∞, 0,
i
1+i
(c) 2, 3, 4. 0 1+i
(a) 4, 2 + 2i, 0,
(b) 0, 2 + 2i, 4,
(c) 0,
2i
∞, 2i.
|z − i| < 1 i, (d) 2, 1 + i, 0, w = 2iz + 4 w
(e) 0, 1,
∞,
(f)
Re z > 0 2 < 2 w = 2zi/(iz + 1) w
| − 2| > 2
| − |
Re w > 0
∞, −i − 2, 2i, 2, w = (i
− 1)z/(z − 1 − i)
w = f (z) a w = f (z) = f (a) + f (a)(z f (k) = 0
(n)
− a) + ··· + f n!(a) (z − a)n + ··· k = 1, . . . , n − 1 f (n) =0 a
n
z
w z
x = F (t) y = G(t) t
∈ [t1, t2] a
w
z = F (w) + iG(w)
[t1 , t2 ]
b
H (x, y) w F (u, v) = H (x(u, v), y(u, v))
z = f (w) x + iy = f (u + iv) x = x(u, v) y = y(u, v) u v
C
R R R
C
R C
R
C
R
R
h(φ)
1 F (r, θ) = 2π
≡
2π
0
C F (r, θ) F (1, φ)
w = f (z)
C
(1 r2 )h(φ)dφ . 1 2r cos(θ φ) + r2
−
−
−
F (x, y) Im(z) > 0
h(x)
≡ F (x, 0)
1 ∞ yh(t)dt F (x, y) = . 2 π −∞ y + (x t)2
−
Γ(z ) Γ(n + 1) = n!
∞ −
e t tz−1 dt, (Re(z) > 0).
Γ(z) =
0
Re(z )
≤0
z = 0, 1, 2, . . .
− −
1
B(r, s) =
0
r
tr−1 (1
− t)s−1dt,
(r > 0, s > 0).
s B(r, s) = ζ (z) ζ (z)
≡
1 1 1 + + + 1z 2z 3z
Γ(r)Γ(s) . Γ(r + s) Re(z) > 1
∞ 1 , ··· = k=1
kz
Re(z) > 1.
1 ζ (z) = Γ(z)
∞ t −
z 1
0
et + 1
dt, (Re(z) > 0),
z = 1
∇2V = −ρ/0 ρ = 0
2
, ∇2f (x,y,t) = c12 ∂ f (x,y,t) ∂t 2 f
dρ +ρ dt
∇ · v = 0,
ρ
v
dρ/dt = 0
v = 0 v =
∇2Φ = 0
∇Φ
I (t) = I 0 cos ωt R V 0 = I 0 R
V (t) = I (t)R = I 0 R cos ωt = V 0 cos ωt V (t) = (I 0 /Cω)cos(ωt π/2) L V (t) = (I 0 Lω) cos(ωt +π/2)
C
−
I (t) V (t) I (t) I (t) = I 0 eiωt ,
R
→ R,
V (t)
−i , → X C = Cω
C
L
→ X L = iLω,
Re I (t) = I 0 cos ωt, Z
V (t) = I (t)Z
I (t)
I 0 ωt Z = R + iX V (t) = I (t)Z = I 0
Re I (t)
R + X e 2
ω
√ R2 + X 2
2 i(ωt +α) ,
α
t tan α = X/R
Re V (t) = V 0 cos(ωt + α),
V 0 = I 0
R + X . 2
2
(r, θ) r u(x, 0) = 0
x > 0
u(x, 0) = 0
u(x, 0) = 1 x < 0
Im(z ) > 0 x > 0
u(x, 0) = 1
x > 0
F (x, y)
x, y w = u + iv = f (z) f (x + iy)
≡
∂ 2 F (x, y) ∂ 2 F (x, y) + = f (z) 2 2 2 ∂x ∂y
|
|
2
∂ F (x(u, v), y(u, v)) ∂u 2
F
∂ 2 F (x(u, v), y(u, v)) + ∂v 2
x, y
.
u, v
f (z) =0
f (z)
F (x, y) y ∞ F (x, y) = π −∞ (t
−
1 x)2 + y 2
−
1 h(t)dt, (t + x)2 + y 2
F (0, y) = 0
y > 0
y > 0, x
≥ 0,
F (x, +0) = h(x) x > 0
h(x) z=
−a
∓2πk
z=a
V (z) = log((z + a)/(z Ω(z) = U + iV V =
±a
− a)) U
F = F x i + F y j
v=
∇U
F = F x + iF y ¯ = F x F
−
iρ iF y = 2
dΩ(z) C
2
dz
dz,
ρ p
v
y
p + ρv 2 /2 + ρgy =
ρgy L C V (t) = V 0 cos ωt
R a
b a
b
ω 2ω V 0
C
R3
r(t) = x 1 (t)e1 + x2 (t)e2 + x3 (t)e3
≡ x(t)i + y(t) j + z(t)k, xi (t)
ei
t dr/dt = x (t)e1 + x (t)e2 + x (t)e3 1
2
x (t) =0
3
xi (t) t = t(x)
t y = f (x),
F 1 (x,y,z) = 0,
x(t)
z = g(x)
F 2 (x,y,z) = 0, R3
ds =
dx + dy + dz = 1 + y(x) + z(x) dx = x(t) + y(t) + z(t) dt, 2
2
2
2
2
2
2
2
ds/dt = 0
s
s
≡ r˙ (s)
t(s) = dr(s)/ds
t˙ (s)
≡ κ(s)n(s)
n(s)
κ(s) R(s) = 1/ κ(s)
|
n b(s) = t(s)
|
× n(s)
θ OX κ =
dθ , ds
θ = arctan y (x),
⇒
κ =
y (x) (1 + y (x)2 )3/2
=
y (t)x (t)
− y(t)x (t) .
(x (t)2 + y (t)2 )3/2
t(s) n(s) b(s) t˙ (s) = κ(s)n(s),
˙ (s) = n
t˙ = ω
× t,
b˙ (s) =
−κ(s)t(s) + τ (s)b(s), ˙ = ω n
b˙ = ω
× n,
−τ (s)n(s).
× b,
ω = τ t +κb
s κ
τ (s) b
|κ| = |r |r×|3r | ,
κ(s)
τ =
(r
× r ) · r . |r × r|2 τ (s)
t(s) = d r/ds τ (s) = 0 r(s)
κ(s) dθ/ds = κ(s)
r(θ) =
t(θ)ds =
θ(s)
ds (cos θ e1 + sin θ e2 ) dθ + c = dθ
(cos θ e1 + sin θ e2 )
1 dθ + c, κ(θ)
c
θ(s)
r(t)
t r (t) r (t) = d r(t)/dt r (t) = d 2 r(t)/dt2 t n b
v(t) = r (t) = ds/dt
|
|
r (t) =
dr dr ds = = v(t)r˙ (t) dt ds dt
≡ v(t)t(t).
dv(t) dv(t) dv(t) v(t)2 2˙ 2 r (t) = t(t) + v(t) t(t) = t(t) + v(t) κ(t)n(t) = t(t) + n(t). dt
dt
dt
R(t)
aT = dv/dt aN = v 2 /R
R(t) = 1/κ(t)
r(t, λ)
λ λ ∂ (x, y)/∂ (t, λ) = 0
λ F (x,y,λ+∆λ) = 0 ∆λ) F (x,y,λ))/∆λ = 0
λ = λ(t) F (x,y,λ) = 0 ∂F (x,y,λ)/∂λ = 0 (x, y)
e(t) = r (t, λ(t))
F (x,y,λ) = 0
λ (F (x,y,λ+
−
r ∗ (t) = r (t) + R(t)n(t)
r (t) r(t)
R(t)
(x c1 )2 + (y y(x) y (x)
(c1 , c2 )
−
x
c1 = x
c1 = x(t)
n(t)
−
−
1 + y (x)2 y (x), y (x)
− c2)2 = R2 y (x)
1 + y (x)2 c2 = y(x) + , y (x)
(1 + y (x)2 )3/2 1 R = = . y (x) κ
(x (t)2 + y (t)2 )y (t) (x (t)2 + y (t)2 )x (t) (x (t)2 + y (t)2 )3/2 , c2 = y(t) + , R = . y (t)x (t) y (t)x (t) y (t)x (t) y (t)x (t) y (t)x (t) y (t)x (t)
−
−
r(t)
−
t0
λ
r∗ (λ, t0 ) = r (t0 ) + λt(t0 ) r (t)
t0 b
( p
n
≡ (X , Y , Z)
− r(t0)) · b(t0) = 0.
r(t) t
p
t
t0
n
b
p
( p
− r(t0)) · t(t0) = 0. r (t)
n
t0
t
b
p
( p
− r(t0)) · n(t0) = 0. s=0 e1 = t e2 = n e3 = b
r (s) = r (0) + r˙ (0)s + ¨ r(0)
s2 s3 + r (0) + 2! 3!
r (s)
··· ,
−κ2t + κ˙ n + κτ b ¨(0) = κ(0)e2 , r (0) = −κ2 (0)e1 + κ(0) r ˙ e2 + κ(0)τ (0)e3 ,
¨(s) = κ n r (s) = κ˙ n + κn˙ = r r (0) = 0,
x(s) = s
r˙ (0) = e 1 ,
− 16 κ(0)2s3 + o(s3),
y(s) =
1 1 κ(0)s2 + ˙κ(0)s3 + o(s3 ), 2 6
z(s) =
1 κ(0)τ (0)s3 + o(s3 ). 6
x = s,
y =
1 2 κs , 2
z =
1 κτ s3 , 6
s y = κx 2 /2 Y Z
XY XZ
z2
− 2τ 2y3/9κ = 0
z = κτ x3 /6
u(t)
v (t)
t d du dv (u v ) = , v + u dt dt dt
·
·
·
d (u dt
× v) = ddtu × v + u × ddtv , d u d|u| 1 du2 u· = |u| = . dt
dt
2 dt
x(θ) = cos θ(2cos θ + 1) y(θ) = sin θ(2cos θ + 1) 0
≤ θ ≤ 2π
θ
x(θ) = (r0 + r)cos θ
− r cos
r + r 0
θ ,
r
y(θ) = (r0 + r)sin θ
x(θ) = cos3 (θ) y(θ) = sin3 (θ) 0
0
r
θ .
≤ θ ≤ 2π
(0, 1) x2 + (y
− r sin
r + r
y = cosh x
− 2)2 = 1 r (t) = (et cos t)e1 + (et sin t)e2
y = f (x)
F (x, y) = 0 r = r(θ)
x = x(t) y = y(t)
F F F κ =
xx yx x
F xy F yy F y
F x F y 0
(F x 2 + F y 2 )3/2
,
κ =
f , (1 + f 2 )3/2
r(θ) = a cos θ e1 + b sin θ e2
x x
y y κ = 2 , (x + y 2 )3/2
κ =
r2 + 2r2 rr . (r2 + r2 )3/2
−
θ = π/6 AB x2/3
+
y 2/3
l
= l 2/3 r = caθ ln a = a π/2
a y 2 = 4 px x2 /a2 + y 2 /b2 = 1 r (θ) = a(θ sin θ)e1 + a(1
−
(x
√ y = ±x/ 3 κ = 6/t(4 + 9t2 )3/2 t2 e2 + (1 + t3 )e3
− cos θ)e2 − 2a)2 + y 2 = a2
x(t) = t2 y(t) = t3 t = 0 r (t) = (1 + t)e1
t = 1
−
y = x 4
(0, 0) r(s)
r(n) r(1) r(2) = 0,
·
n
n
= d r/ds
r (2) r (2) = κ 2 ,
·
r (1) r (3) =
·
−κ2,
r (2) r (3) = κ κ, ˙
r (3) r(3) = κ 4 + κ2 τ 2 + κ˙ 2 .
·
·
[a, b, c]
3 a b r(t) = (3t
v˙ 1 = ω
× v1 ,
v˙ 2 = ω
c
3 κ2 τ = [ r˙ , r¨, r ]
×
− t3)e1 + 3t2e2 + (3t + t3)e3
× v2 ,
v˙ 3 = ω
× v3 ,
ω
k(s) = (1/2as)1/2 τ (s) = 0 a > 0 s > 0 κ/τ = k(s) = (1/s) τ (s) = 0 s > 0 r(t) = at e1 + bt2 e2 + t3 e3
2b2 = 3a
a = e 1 + e3
(a) x2 = 3y, 2xy = 9z,
(b) x = 2t, y = ln t, z = t 2 , (0, 1, 1) x = at y = bt2 z = ct3
κ(s)
τ (s) τ
a
2
2
1 κ˙ κ
+
κ2 τ
= a 2 . (r
h(r)v v
k(r)
× r) · r v
×v κ =0
x = x(t) y = y(t) z = z(t) κ =
r 2 r 2
τ =
− (r · r )2 1 = , ρ |r|3
ρ2
|r |6 (r × r ) · r .
r (t) = cos t e1 + sin t e2 +
t 2 e3 , 2
t = π/2 x(t) = a cos t y(t) = b sin t
t
∈ [0, 2π]
r = p/(1+ e cos θ) κ = 1/p
(a) x(t) = a cosh t, (b) x(t) = ct,
y(t) = a sinh t,
y(t) =
√
2c ln t,
r(t) = 4cos t e1 + 3t e2
z(t) = at,
z(t) = c/t,
− 4sin t e3,
t = π t = π/2 t1 = π/2
r(t)
r (t) = 45 cos ti + (1
t2 = 5π/2
− sin t) j − 35 cos tk
a = b
r (t)
≡ te1 + cosh te2,
R3
r(t) = (1 + t)e1 + (1
− 31 t3)e2 + √ 12 t2e3 t = 2
t = 3 XOY z =
XO Y
√ 8 t = 0
R3
r(t) = (3t
− t3)e1 + 3t2e2 + (3t + t3)e3,
t = 2 R3
r(t) = (3t
− t3)e1 + 3t2e2 + (3t + t3)e3, t = 3
x + y + z = 0 t = 1
F (x,y,z) = 0 z = f (x, y) r(u, v) = x 1 (u, v)e1 + x2 (u, v)e2 + x3 (u, v)e3
≡ x(u, v)i + y(u, v) j + z(u, v)k,
ru
x = x(u, v) y = y(u, v) rv ∂ r /∂v u =
≡ ∂ r/∂u
v =
u
v
≡
ru n(u, v) = r u
u v z = z(u, v)
×
rv
×
r u rv r v / ru rv T (X , Y , Z ) N (λ) = r 0 + λn0
| × | ≡
r0 n0 (T
· − r0) = 0
C C C
r (u1 , u2 )
≡ ru
ri
gij = r i rj = g ji
·
ds = dr = r1 du1 + r2 du2 =
| | |
|
g11 du21 + 2g12 du1 du2 + g22 du22 , t
ds =
t2
l =
t1
t2
ds =
{u1(t), u2(t)}
g11 u1 2 + 2g12 u1 u2 + g22 u2 2 dt,
t1
i
g11 (t)u1 2 (t) + 2g12 (t)u1 (t)u2 (t) + g22 (t)u2 2 (t) dt.
gij (u1 , u2 ) dr = r 1 du1 + r2 du2
cos θ =
dr = r 1 dv1 + r2 dv2
g11 du1 dv1 + g12du1 dv2 + g21 du2 dv1 + g22du2 dv2
g
2 11 du1 + 2g12 du1 du2 +
g22 du22
g
2 11 dv1 +
2g12 dv1 dv2 + g22 dv22
.
gij
dA = g
11 g22
g11 g22
− g122 du1du2.
− g122 = |r1 × r2|2 > 0 g11 = 1 g12 = g 21 = 0 ds =
dx + dy , 2
2
g22 = 1
dA = dxdy
gij
g
≡ det gij > 0
gij u =
2
×2
gij = gji
v =
−π/2 ≤
θ ≤ π/2
0 ≤ φ ≤ 2π θ = gθθ = 1 gθφ = 0 gφφ = 1
φ = θ = ±π/2
φ
φ =
n(u1 , u2 ) =
r(u + du,v + dv) = r (u, v) + ru du + rv dv +
(r(u + du,v + dv)
× r2 |r1 × r2| r1
r(u+du,v +dv)
1 ruu (du2 ) + 2 r uv dudv + rvv (dv2 ) + 2!
− r(u, v)) · n = 2!1 Luu(du2) + 2Luv dudv + Lvv (dv2) Lij = r ij n
r
y2
2L12 xy + L 22 L11 L22 L 212 > 0
−
··· ,
·
z = L11 x2 +
L11 L22 L11 L22
− L2122 < 0 − L12 = 0
Lij Lij = 0
t˙
n
Luu (du/dt)2 + 2Luv (du/dt)(dv/dt) + Lvv (dv/dt)2 κn = t˙ n = . guu (du/dt)2 + 2guv (du/dt)(dv/dt) + gvv (dv/dt)2
·
du
r
ds ds dt = r u du+ rv dv = r˙ dt = t dt, dt dt
r
dt = t˙ 2
ds dt
2
dt2 + t
dv
d2 s 2 dt = r uu du2 +2 ruv dudv+ rvv dv2 , dt2
t· n
=0
u v du = cos θ
dv = sin θ
κn = Luu cos2 θ + 2Luv cos θ sin θ + Lvv sin2 θ,
|κn | =
±1
1/r2 x = r cos θ y = r sin θ
Luu x2 + 2Luv xy + L vv y 2 =
(x, y) R1 R1 = R2
R1 R1
|κn|
R2
R2
R2
R1
R2 du
(g11 g22
− g122 )κ2 − (g11L22 + g22L11 − 2g12L12)κ + (L11L22 − L212) = 0, κ2
− 2Hκ + K = 0. κ(du,dv)
κ(du,dv)(guu du2 + 2guv dudv + gvv dv2 ) − (Luu du2 + 2Luv dudv + Lvv dv2 ) = 0, du dv κ
κ
(Luu − κguu )du + (Luv − κguv )dv = 0,
(Luv − κguv )du + (Lvv − κgvv )dv = 0, du
dv κ
κ1
κ2
(du, dv)
1 H = 2
g
11 L22 +
1
g22 L11 2g12 L12 1 = 2 g11 g22 g12 2
−
−
1 1 + = (κ1 + κ2 ). R1 R2 2
dv
K =
L11 L22 g11 g22
− L212 = 1 = κ1κ2. − g122 R1R2
K = 0 r1 r1
r2
r2
n
n
rij
∂ ri = Γk ij rk + Lij n, ≡ ∂u j Li k
Γk ij gij
g
k
Γ
ij
kl
g11 g22
= g rij
·
gij Lij g gki = δ ij
jk
g22
g 11 =
∂ n = −Li k r k , ≡ ∂u i
ni
−
jk
2 , g12
1 rl = gkl 2
−g12 , 2 g11 g22 − g12
g 12 = g 21 =
∂g
il ∂u j
∂ glj + ∂u i
Rijkl = Lik Ljl
−
∂ gij ∂u l
,
g 22 =
g11 g11 g22
Lij = r ij n,
·
− g122 .
Li k = g kj Lij .
Ri jkl = g im Rmjkl .
− Lil Ljk ,
K = L1 1 L2 2
2H = g ij Lij
− L12L21
s1
l = g (u , u )u˙ ˙u ds ij
1
2
i j
s0
u1 (s)
∂g ij u˙ i ˙uj ∂u k
− dsd (gik ˙ui + gkj ˙uj ) = 0, u ¨k + Γ k ij ˙ui ˙uj = 0,
k
Γ s
s0 ij
=0
u2 (s)
d ∂g ik gik = u˙ l , ds ∂u l k = 1, 2,
s1 u1 (s) = as + b u2 (s) = cs + d u2 = mu 1 + h
r (t) = t e1 + t2 e2 + t3 e3
x = u 2 + v 2 y = u 3 z = v 3 x+y
− z = 0
x+y
− z = 8/27
(a) ds2 = du 2 + 4dudv + dv 2 ;
(b) ds2 = du2 + 4dudv + 4dv2 ;
(c) ds2 = du 2
(d) ds2 = du2 + 4dudv
− 4dudv + 6dv2;
− 2dv2;
gij dA = Hdudv =
g
− guv2 dudv f (x − az,y − bz) = 0 uu gvv
z = xφ(y/x)
x = a
1 u2 v 2 , 1 + u2 + v 2
− −
y =
2bu , 1 + u2 + v2
z =
2cv , 1 + u2 + v2
x2 y 2 z 2 + 2 + 2 = 1. a2 b c x2 + 3y 2 + 2z 2 = 9 x2 + y 2 + z 2 (2, 1, 1)
x = u cos v,
y = u sin v,
z = u,
0 < u <
− 8x − 8y − 6z + 24 = 0
∞, v ∈ (0, 2π), π/4 z
u3 u x sin u
− y cos u + kz − au = 0
− x = 0
− 3u2x + 3uy − z = 0
a =
x = a sin u cos v,
y = b sin u sin v, u =
z = c cos u v = x = 1 z = 2y
OZ
x2 + y 2
− z2/4 = 1
r (t) = a cos te1 + a sin te2 + bte3 .
r(u, v) =
u+v u v e1 + e2 + uv e3 . 2 2
−
z 2 = x2 u =
v =
d ds
κ˙ κ2 τ
− y2
− κτ = 0.
z = f (y)
ZY
r(u, v) = u cos v e 1 + u sin v e 2 + f (u) e3 ,
z = f (y)
OZ OX
30◦
(a, 0, 0)
r (θ, φ) = (b + a sin φ)cos θ e 1 + (b + a sin φ)sin θ e 2 + a cos φ e3 ,
x 2y + z
− 3 = 0? r
2h
θ OZ r(u1 , u2 )
ri
gij = r i rj
·
n(u1 , u2 ) =
≡ ru
i
× r2 2 11 g22 − g12
r1
g
A = θ
r1
cos θ =
√ gg1112g22 .
r (u, v) = u e1 + v e2 + (u2 + v 3 )e3 ,
v > 0
v < 0
θ = 0
v = 0
g
r2
S
11 g22
− g122 du1du2.
−
II = d 2 r n =
I = d r dr,
·
·
−dr · dn. III = dn d n
·
III H
− 2HII + KI = 0,
K z = y 2 + 2
Y OZ OY
(11, 3, 0) z = f (x, y) K =
2 f xx f yy f xy (1 + f x2 + f y2 )2
−
OZ r(t, θ) = f (t)cos θ e1 + f (t)sin θe2 + te3 ,
f (t) > 0
OZ
OZ s r (s) R
r∗ (s, u) = r (s) + R(v 2 (s)cos u + v 3 (s)sin u), v 2 (s)
R > 0,
0
≤ u ≤ 2π.
v 3 (s)
s n(s, u) u = v 2 (s)cos u + v 3 (s)sin u r(s)
r (u, v) = cosh v cos ue1 + cosh v sin ue2 + v e3 ,
u, v
∈ R.
r(0, 0) R (t) = r (t, t)
t = 0
x(θ, φ) = (b + a cos θ)cos φ,
y(θ, φ) = (b + a cos θ)sin φ,
z(θ, φ) = a sin θ.
4
1 0.5 0 -0.5 -1 -4 4
2
0 -2 -2
0 2 4
b rθ = ( a sin θ cos φ,
−
-4
a
−a sin θ sin φ, a cos θ), rφ = (−(b + a cos θ)sin φ, (b + a cos θ)cos φ, 0), |rθ × rφ| = a(b + a cos θ) = 0, a < b. n = (− cos θ cos φ, − cos θ sin φ, − sin θ). rθθ = ( a cos θ cos φ,
−a cos θ sin φ, −a sin θ), r φφ = (−(b + a cos θ)cos φ, −(b + a cos θ)sin φ, 0), rθφ = (a sin θ sin φ, −a sin θ cos φ, 0). −
gθθ
≡ g11 = a2,
≡ g22 = (b + a cos θ)2,
gφφ
Lθθ
≡ L11 = a, H =
Lθφ
gθφ
≡ g12 = 0,
≡ L12 = 0,
(b + 2a cos θ) , 2a(b + a cos θ)
det g = a 2 (b + a cos θ)2 = 0,
Lφφ
≡ L22 = (b + a cos θ)cos θ
K =
cos θ . a(b + a cos θ)
a < b.
x2 y2 z2 + + =1 a2 b2 c2 x(θ, φ) = a cos θ cos φ,
y(θ, φ) = b cos θ sin φ,
z(θ, φ) = c sin θ.
2
1
0
-1
-2 1
0.5
0
-0.5
-1 -1 -0.5 0 0.5 1
r θ = ( a sin θ cos φ,
−b sin θ sin φ, c cos θ), rφ = (−a cos θ sin φ, b cos θ cos φ, 0), |rθ × rφ| = g11g22 − g122 = det g. r θθ = (−a cos θ cos φ, −b cos θ sin φ, −c sin θ), r φφ = (−a cos θ cos φ, −b cos θ sin φ, 0), r θφ = (a sin θ sin φ, −b sin θ cos φ, 0), 2 2 2 2 2 2 2 2 2 n = (−bc cos θ cos φ, −ac cos θ sin φ, −ab sin θ)/ a b sin θ + c cos θ(a sin φ + b cos φ) −
gθθ
≡ g11 = (a2 cos2 φ + b2 sin2 φ)sin2 θ + c2 cos2 θ, gφφ ≡ g22 = (a2 sin2 φ + b2 cos2 φ)cos2 θ, gθφ ≡ g12 = (a2 − b2 )sin θ cos θ sin φ cos φ, det g = cos2 θ a2 b2 sin2 θ + c2 cos2 θ(a2 sin2 φ + b2 cos2 φ) .
Lθθ
≡ L11 =
abc
a2 b2
Lφφ
2
sin θ +
≡ L22 =
c2
cos2 θ(b2 cos2
φ+
a2 sin2 φ)
,
Lθφ
≡ L12 = 0,
abc cos2 θ
a2 b2 sin2 θ + c2 cos2
θ(b2
cos2 φ
+
a2
2
.
sin φ)
abc((a2 + b2 sin2 θ)sin2 φ + (b2 + a2 sin2 θ)cos2 φ + c2 cos2 θ) H = , 2(a2 b2 sin2 θ + c2 cos2 θ(b2 cos2 φ + a2 sin2 φ))3/2 K =
a2 b2 c2 . (a2 b2 sin2 θ + c2 cos2 θ(b2 cos2 φ + a2 sin2 φ))2
x2 y2 z2 + =1 a2 b2 c2 x(θ, φ) = a cosh θ cos φ, y(θ, φ) = b cosh θ sin φ,
−
z(θ, φ) = c sinh θ.
2 0 -2
4
2
0
-2
-4
-2 0 2
rθ = (a sinh θ cos φ, b sinh θ sin φ, c cosh θ)
|rθ × rφ
= (−a cosh θ sin φ, b cosh θ cos φ, 0), | = det g. rφ
rθθ = (a cosh θ cos φ, b cosh θ sin φ, c sinh θ) rφφ = ( a cosh θ cos φ,
n = ( bc cosh θ cos φ,
−
gθθ
− −b cosh θ sin φ, 0), rθφ = (−a sinh θ sin φ, b sinh θ cos φ, 0)
−ac cosh θ sin φ,ab sinh θ)/
c2 (a2 sin2 φ + b2 cos2 φ) cosh2 θ + a2 b2 sinh2 θ
≡ g11 = (a2 cos2 φ + b2 sin2 φ)sinh2 θ + c2 cosh2 θ, gφφ ≡ g22 = (a2 sin2 φ + b2 cos2 φ) cosh2 θ, gθφ ≡ g12 = (b2 − a2 )sinh θ cosh θ sin φ cos φ, det g = cosh2 θ c2 (a2 sin2 φ + b2 cos2 φ) cosh2 θ + a2 b2 sinh2 θ .
Lθθ
≡ L11 =
c2 (a2 sin2 φ
−abc
+
≡ L22 =
b2 cos2 φ) cosh2
Lφφ
c2 (a2
H =
K =
abc
2
2
2
θ +
a2 b2 sinh2 θ
−abc cosh2 θ
2
sin φ
2
+ b2 cos2
φ) cosh θ +
2
2
2
2
,
a2 b2
Lθφ
≡ L12 = 0,
2
.
sinh θ
2
2
2
2 2
2
3/2
2
−a sin φ − b cos φ + c cosh θ + (a cos φ + b sin φ)sinh θ , 2 c (a sin φ + b cos φ) cosh θ + a b sinh θ 2
2
2
2
2
−16a2b2c2
2
{[a2 + b2 − (a2 − b2)cos(2φ)]c2(1 + cosh(2θ)) + 2a2b2(1 − cosh(2θ))}2 < 0.
x2 y 2 + 2 a2 b
2
− zc2 = −1
x(θ, φ) = a sinh φ cos θ,
y(θ, φ) = b sinh φ sin θ,
z(θ, φ) =
±c cosh φ.
2 0 -2
2
0
-2
-2 0 2
rθ = ( a sinh φ sin θ, b sinh φ cos θ, 0)
−
= (a cosh φ cos θ, b cosh φ sin θ, c sinh φ) rφ
|rθ × rφ| =
a2 b2 cosh2 φ sinh2 φ + c2 (a2 sin2 θ + b2 cos2 θ)sinh4 φ.
gθθ gφφ
Lθθ
≡ g11 = (a2 sin2 θ + b2 cos2 θ)sinh2 φ,
≡ g22 = (a2 cos2 θ + b2 sin2 θ) cosh2 φ + c2 sinh2 φ, gθφ ≡ g12 = (b2 − a2 )sin θ cos θ sinh φ cosh φ, det g = |rθ × rφ |2 .
≡ L11 =
a2 b2
2
2
cosh φ sinh φ Lθφ
Lφφ
≡ L22 =
a2 b2
H =
−abc sinh3 φ
2
+ c2 (b2 cos2
θ +
a2 sin2 θ) sinh4
θ +
a2
≡ L12 = 0, −abc sinh φ 2
cosh φ sinh φ
+ c2 (b2 cos2
2
4
, φ
.
sin θ) sinh φ
−abc(b2 cos2 θ + a2 sin2 θ + cosh2 φ(a2 cos2 θ + b2 sin2 θ) + c2 sinh2 φ) , 2(a2 b2 cosh2 φ + c2 (b2 cos2 θ + a2 sin2 θ)sinh2 φ)3/2
a2 b2 c2 K = a2 b2 cosh2 φ + c2 (a2 sin2 θ + b2 cos2 θ)sinh2 φ
x2 y2 + a2 b2 x(θ, φ) = a sinh φ cos θ,
2
− zc2 = 0
y(θ, φ) = b sinh φ sin θ,
z(θ, φ) = c sinh φ.
2 0 -2
2
0
-2
-2 0 2
rθ = ( a sinh φ sin θ, b sinh φ cos θ, 0)
−
rφ = (a cosh φ cos θ, b cosh φ sin θ, c cosh φ)
|rθ × rφ| =
cosh2 φ sinh2 φ(b2 c2 cos2 θ + a2 (b2 + c2 sin2 θ)).
≡ g11 = (b2 cos2 θ + a2 sin2 θ)sinh2 φ, gφφ ≡ g22 = (c2 + a2 cos2 θ + b2 sin2 θ) cosh2 φ, gθφ ≡ g12 = (b2 − a2 )cos θ sin θ cosh φ sinh φ, det g = |rθ × rφ |2 . gθθ
Lθθ
≡ L11 =
b2 c2
−abc sinh φ cos2 θ +
a2 (b2
+
c2
2
,
sin θ)
Lθφ
≡ L12 = 0, Lφφ ≡ L22 = 0. H =
2 θ + b2 sin2 θ) −abc(c2 + a2 cos , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (b c + a (2b + c ) − (a − b )c cos2θ) (b c cos θ + a (b + c sin θ)) sinh φ
K = 0
z = x(θ, φ) = aφ cos θ,
x2 y 2 + 2 a2 b z(θ, φ) = φ 2 .
y(θ, φ) = bφ sin θ, 2 1
0 -1 -2 2 4
3
2
1
0 -2 -1 0 1 2
rθ = ( aφ sin θ,bφ cos θ, 0)
−
= (a cos θ, b sin θ, 2φ) rφ
|rθ × rφ| =
a2 φ2 (b2 + 4φ2 sin2 θ) + 4b2 φ4 cos2 θ.
≡ g11 = (a2 sin2 θ + b2 cos2 θ)φ2, gφφ ≡ g22 = 4φ2 + a2 cos2 θ + b2 sin2 θ, gθφ ≡ g12 = (b2 − a2 )φ sin θ cos θ, det g = |rθ × rφ |2 . gθθ
Lθθ
−2abφ3
≡ L11 =
4b2 φ4 cos2 θ + a2 φ2 (b2 + 4φ2 sin2 θ) Lθφ
Lφφ
≡ L22 =
≡ L12 = 0, −2abφ
4b2 φ4 cos2
H =
θ +
a2 φ2 (b2
+ 4φ2 sin2 θ)
,
.
+ 4φ2 ) −abφ(a2 + b2 , 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 (2b φ + a (b + 2φ ) − 2(a − b )φ cos2θ) 4b φ cos θ + a φ (b + 4φ sin θ) K =
4a2 b2 (2b2 φ2 + a2 (b2 + 2φ2 ) 2(a2
−
− b2)φ2 cos2θ)2
z = x(θ, φ) = aφ cosh θ,
x2 a2
2
− yb2
z(θ, φ) = φ 2 .
y(θ, φ) = bφ sinh θ, 4 2
0 -2 -4
10
0
-10
-4 -2 0 2 4
r θ = (aφ sinh θ,bφ cosh θ, 0)
= (a cosh θ, b sinh θ, 2φ) rφ
|rθ × rφ| =
4b2 φ4 cosh2 θ + a2 φ2 (b2 + 4φ2 sinh2 θ).
≡ g11 = φ2(b2 cosh2 θ + a2 sinh2 θ), gφφ ≡ g22 = 4φ2 + a2 cosh2 θ + b2 sinh2 θ, gθφ ≡ g12 = (a2 + b2 )φ cosh θ sinh θ, det g = |rθ × rφ |2 . gθθ
Lθθ
2abφ3
≡ L11 =
4b2 φ4 cosh2 Lθφ
Lφφ
≡ L22 =
≡ L12 = 0, −2abφ
4b2 φ4 cosh2 abφ(a2
H = (2b2 φ2
+ a2 (b2
−
2φ2 ) + 2(a2
K =
+
b2 )φ2
(2b2 φ2 + a2 (b2
θ +
a2 φ2 (b2
θ +
a2 φ2 (b2
− b2 + 4φ2)
cosh2θ)
−4a2b2
+ 4φ2 sinh2 θ)
+ 4φ2 sinh2 θ)
4b2 φ4 cosh2
θ +
,
.
a2 φ2 (b2
− 2φ2) + 2(a2 + b2)φ2 cosh 2θ)2 ,
+ 4φ2 sinh2
, θ)
z = x(θ, φ) = θ,
−x2
y(θ, φ) = φ,
z(θ, φ) =
4 2 0 -2 -4 4 0
-5
-10
-15 -4 -2 0 2 4
rθ = (1, 0,
−2θ)
r φ = (0, 1, 0)
|rθ × rφ| =
gθθ
1 + 4θ . 2
≡ g11 = 1 + 4θ2, gφφ ≡ g22 = 1, gθφ ≡ g12 = 0,
det g = 1 + 4θ2 .
Lθθ Lθφ
≡ L11 = √ 1−+24θ2 ,
≡ L12 = 0, Lφφ ≡ L22 = 0. H =
−1
(1 + 4θ 2 )3/2 K = 0.
,
−θ2.
r(θ) = a cos θ e1 + a sin θe2 + bθe3 ,
r(θ, λ) = r (θ) + λr (θ) r (θ, λ) = a(cos θ rθ rλ rθθ r θλ rλλ
= =
− λ sin θ)e1 + a(sin θ + λ cos θ)e2 + b(θ + λ)e3,
−a(sin θ + λ cos θ)e1 + a(cos θ − λ sin θ)e2 + be3, −a sin θe1 + a cos θe2 + be3, −a(cos θ − λ sin θ)e1 − a(sin θ + λ cos θ)e2, −a cos θe1 − a sin θe2,
= = = 0.
20
10 4 0
2
0
-4 -2 -2
0 2 4
n =
-4
× rλ = (−b sin θe1 + b cos θe2 − ae3) / a2 + b2 |rθ × rλ | rθ
gθθ = b 2 + a2 (1 + λ2 ), Lθθ =
gθλ = a 2 + b2 ,
a + b ,
−abλ/ H =
2
2
Lθλ = 0,
−b , √ 2aλ a2 + b2
gλλ = a 2 + b2 . Lλλ = 0.
K = 0.
r(u, v) = u cos u(4 + cos(u + v))e1 + u sin u(4 + cos(u + v))e2 + u sin(u + v)e3 ,
u
∈ (0, 6π) v ∈ (0, 2π)
10 0 -10
50
0 -50 0
-50 50
r (u, v) = u cos v e1 + u sin v e2 + u3 cos(3v)e3 ,
u
∈ R+ v ∈ [0, 2π] 0.2 0.1 0 -0.1 -0.2 .2 0.004
0.002
0
-0.002
-0.004 -0.2 0.2 -0.1 0 0.1 0.2
