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f(x)
y = f(x) x= t+ sin(t)
y = 1 + cos(t) x2 +y2 = 1
(x,y,z)
(x1, y1, z1) (x2, y2, z2) (x1x2)2 + (y1
y2)2 + (z1
z2)2
z = f(x, y) f x y (x, y) z = f(x, y) (x, y) z = f(x, y)
z= 0 xy
a x + b y + c z= d a,b,c,d
z= x2 +y2
z= x2 +y2
z = x2 y2 x y
z= x3 3xy2
z =e3
x2+y2/12 cos
x2 +y2
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x2 +y2 +z2 = 1
(0, 0, 0)
x2 +z2 = 1 y
y
n (a1, a2,
, an) v =
a1, a2,
, an
ai
()
v v v
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(0, 0) (1, 1) (1, 1) (2, 2) 1, 1
v=a1, , an w=b1, , bn v + w=a1+b1, , an+bn
r r v=ra1, , ran
1/2
1
v=1, 2, 2 w=3, 0, 4 2w= 6, 0, 8 v+w= 2, 2, 2 v2w= 7, 2, 10
r(t) t
r(t) =
t2, 2t
r1(t) =
et, cos(t), t2 1 r2(t) = t, 0, t2 r1(t)+r2(t) =et +t, cos(t),
1
2r2(t) =
2t, 0, 2t2
r(t) =x(t), y(t) r(t) =x(t), y(t), z(t) t
(x(t), y(t), z(t)) = (t,t,t)
(x(t), y(t), z(t)) = (sin(t), cos(t), t) x2+ y2 = 1 t x y z
0t8
(x(t), y(t), z(t)) = (cos(t), sin(t), cos(t)) z= x x2 +y2 =
1
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n
(a1, . . . , an)
(a1)2 + + (an)2
v=
a1, . . . , an
||v||
= (a1)2 + + (an)2
a1, . . . , an (0, . . . ,0) (a1, . . . , an)
v =1, 2, 2 w =3, 0, 4 ||v|| =
(1)2 + 22 + 22 = 3 ||w|| =32 + 02 + (4)2 = 5
r
||r v||=|r| ||v||
r2 =|r|
v
v
u= v
||v||
v
u
v
v=1, 2, 2 u1=
1
3,2
3,2
3
v
w=3, 0, 4 u2 =
3
5, 0, 4
5
w
v1 =a1, . . . , an v2 =b1, . . . , bn v1v2 = a1b1+a2b2+
+anbn
1, 2 3, 4 (1)(2) + (3)(4) = 14
1, 2, 2 3, 0, 4 (1)(3) + (2)(0) + (2)(4) = 11
v,v1,v2,w r
v w= w v
(v1+ v2) w= (v1w) + (v2w) (r v) w= r (v w)
v v=||v||2
v1 v2 v1v2 =||v1|| ||v2|| cos()
v1 v2 v2v1 ||v2v1||2 =||v1||2+ ||v2||22 ||v1||||v2|| cos()
||v2v1||2 = (v2v1)(v2v1)= (v2v2)(v1v2)(v2v1) + (v1v1)= ||v2||2
2(v1v2) +||v1||2 .
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cos
2
= 0
2
2, 1, 4 3, 2, 1 (2)(3)+(1)(2)+(4)(1) = 0
2, 2, 1 3, 0, 4 10 = 3 5 cos() = cos1
2
3
i=1, 0, 0 j=0, 1, 0 k=0, 0, 1
P0 =x0, y0, z0 P1 =x1, y1, z1 x,y,z l P0 P1 x,y,z = P0+t (P1P0)
t
P0 P1
l x= x0+ t(x1 x0) y= y0+ t(y1 y0) z= z0+t(z1z0)
l P0 t= 0 P0 t= 1 P1
v= P1P0 l
P0 P0+ t (P1P0)
(1, 2, 3) (1, 2, 1) v =(1)1, 22, (1)3 =2, 0, 4
x,y,z
=
1
2t, 2, 3
4t
a x+b y+c z = d n=a,b,c a,b,c
l
n
v = P2P1 P2 =x2, y2, z2 P1 =x1, y1, z1
P1n= a x1+b y1+c z1 = d P1 P2n= d v n= P2n P1n= dd= 0
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(12)2 x y z 2x1x2y1y2
v1 v2 1
2||v1||||v2|| sin()
P1 = (3, 0, 1) P2 = (1, 2, 2) P3 =(2, 1, 4)
v2= P3P1 =5, 1, 5 v1 = P2P1=2, 2, 3
n = v1v2 = 1 52 3
i 5 52 3
j+ 5 12 2
k =7, 5, 8 n v1= (7)(5) + (5)(1) + (8)(5) = 0 n v2 = (7)(2) +
(5)(2) + (8)(3) = 0
7x+ 5y8z = d d P1 d= (7)(3) + 5(0)8(1) =29 7x+ 5y8z =29
v1 v2 v3
V =|v1(v2v3)|
v2 v3 v2v3 ||v1|| sin() v1
sin() = cos() v1
n = 2
V =||v1||||v2v3|| cos() =|v1(v2v3)|
