mult. de lagrange3 ejercicios
TRANSCRIPT
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7/23/2019 Mult. de Lagrange3 Ejercicios
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f(x, y) =x2 +y2 g(x, y) =x2 y2 +xy3 (0, 0, 0) z=x sin(y
x)
(a,b,a sin(
b
a))
a= 0
z=xf( yx
).
x + 2y + 3z= 1
x2
8 + y
2
2+ z2 = 1.
t = g(x, y) g C1 F(t) f(x, y) f(x, y) =F[g(x, y)].
f
x=F [g(x, y)]
g
x
f
y =F [g(x, y)]
g
y
F(t) =esin(t), g(x, y) = cos(x2+y2),
fx
,fy
f(x, y) x y f
x,f
y f
u= (x y)
2 v =
(x+y)
2 , f(u, v) F(x, y)
F
x
F
y
f
u,f
v
u= f(x, y), x= X(s, t), y =Y(s, t), u s t u= F(s, t).
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F
s
F
t f, X Y
2f
xy =
2f
yx
2F
s2 =
f
x
2X
s2 +
2f
x2 (
X
s )2
+ 2
X
s
Y
s
2f
xy +
f
y
2Y
s2 +
2f
y2 (
Y
s )2
.
2F
t2
2F
st
X(s, t) =s+t , Y(s, t) =st
X(s, t) =st , Y(s, t) = st
G(s, t) =f(2s t + 1, s + 3t)
f
(1, 0) f D1f= 1; D2f=
1; D11f= 2; D12f=
2; D22f=
3. D12G(0, 0) D22G(0, 0) u= f(x, y), x= r cos(), y =r sin()
u2x+u
2y ur
u
fxx+fyy
f C2
g : R2 R f(x, y) = g( (x
2 +y2)
2 , xy)
D1g(
52
, 2) = 2, D2g(
52
, 2) = 1, f(1, 2) f (1, 2) (0, 1) (1, 0).
r=
x2 +y2 +z2.
1
r
(x,y,z) =
xr3
, y
r3, z
r3
F(t) = f(x+ht, y+ kt) (x, y) (h, k) f F(t), F(t), F(t).
f : R2 R m N
t
R,
(x, y)
R2f(tx,ty) =tmf(x, y). (I)
f m. f
xD1f(x, y) +yD2f(x, y) =mf(x, y),
t t= 1. f
h(x, y) =f(x+cy) +g(x cy),
f, g R R
c h
hxx 1c2
hyy = 0.
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w= f(x, y) x= u+v y= u v, f C2 2w
uv =
2f
x2
2f
y2.
u(x , y, z, t) = f(t+r)
r +
g(t r)r
f, g
R R r2 =x2 + y2 + z2
uxx+uyy+uzz =utt.
f : R2 R fxx+ fyy = 0.
u(x, y) =f(x2 y2, 2xy)
(x, y) =f
x
x2 +y2,
y
x2 +y2
(r, ) R2 u = 2u
x2+
2u
y2 = 0
2u
r2 +
1
r22u
2 +
1
r
u
r = 0
uxx+uyy +uzz
x = r sin() cos(),
y = r sin() sin(),z = r cos().
f : R3 R3 f(x,y,z) =x +y +z Jf(x,y,z)
f : R3 R3
Jf(x,y,z)
f : R2 R2, g: R3 R2
f(x, y) =ex+2y + sin(y+ 2x) ,
g(u,v,w) = (u+ 2v2 + 3w3) + (2v u2) .
Jf(x, y), Jg(u,v,w)
h(u,v,w) =f[g(u,v,w)]
Jh(1, 1, 1)
f g f g
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f(x,y,z) = (z log(x2y4), cos(y) + x, 1), g(x,y,z) = (xz3, 3yz,xyz) (0, 1, 1) f(x,y,z) = ( xy
xy , zxxz
), g(x,y,z) = (x + y+ z, x + y z, x y z) (1, 2, 3) f(x, y) = (2x, arctg(x+y), y7, xy), g(x,y,z) = ( 1
xy, 1z
) (1, 1, 1)
f(x, y) =ex sin y (0, 0)
f(x,y,z) =x2 3xz+z2 4xy+x4y2
(0, 0, 0)
f(x,y,z) =xyz2 (0, 1, 2)
A= 0,97
15,05 + 3
0,98
f(a+h)=f(a) + f(a) h f(x,y,z) = xy+ 3
z
a= (1, 15, 1).
A 0,2421726...
10 25 0,1
f, g U Rn
p U.
f+g
p. f g
f(x, y) =|x| + |y| (0, 0). f
(0, 0)
f(x, y) =x3 + 3x+y3 +y
f(x,y,z) =x5 +y5 +z3 + 4x+ 2y+ 9z+ 2
f(x,y,z) = arctan(x+ 2y+ 3z)
f(x1,...,xn) =a1x1+...+anxn+b, aj= 0
f(x, y) =ax2 + by2, a, b ab
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f(x, y) = 3x4 4x2y+y2. (0, 0)
f(x, y) = (yx2)(y3x2) f
f : R+ R F : R2 R, F(x, y) = f(1 +x2 +y2)
G(x1,...,xn) =f(1 +x2
1+...+x2
n) Rn
.
f(x, y) =x2 (y 1)2; f(x, y) =x3 +y3;
f(x, y) =x3 +y3 3xy; f(x, y) = 2x2 xy 3y2 3x+ 7y;
f(x,y,z) =x2 +y2 + 3z2 1;
f(x,y,z) = (8x2
6xy+ 3y2)exp(2x+ 3y).
f(x, y) =(x21)2(x2y x1)2
9x2 + 36y2 + 4z2 = 36.
32000 cm3.
y x
x3 xy xy2 y3 1 = 0 x= 0 sin x+ cos y+ 2y = 0 x= 0
1 xy log(x2
+y2
) = 0
x= 0
x2y+ 3x2 2y2 2y= 0 x= 1 z+ x + (y + z)4 = 0 z=f(x, y).
fxx fyy .
g(x, y) = 0
y
x
y = w(x)
f : R2 R h(x) =f(x, w(x)).
h(x)
f
g.
2x2 + 3y2 z2 = 25 x2 +y2 = z2 C (
7, 3, 4) x y
z C z
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T C P
C z
sin(x+y) + sin(y+z) = 1 z x y
z=f(x, y).
D1,2f
x
y
z
f C1 f x0 f. x0 f f(x)
f(x0) x
f : R2 R
f(x, y) =y4 ex2 + 2y2
ex +ex2
(0, 0) f f
y = f(x)
x2
+xy+y2
27 = 0 y x,
f(x, y) =xy(1 x2 y2) {(x, y) R2 / 0x1, 0y1} f(x, y) =x3/3 (3/2)x2 + 2x + y2 2y + 1
x= 0, y = 0, x+y = 1.
f(x, y) = (x2 +y2)2
2(x2 +y2)
{(x, y) / x2 +y2
4
}
f(x, y) =x2y3(1 x y) {(x, y) /|x| + |y| 1}. (2/5,3/5) f 216/3125. (1/2, 1/2) f 1/132
f(x, y) = sin(x) +
cos(y) R= [0, 2] [0, 2] f R 2 (
2, 0) (
2, 2) f R 2
( 32 , )
x,y, z
0 1
6/18,
6/6
6/3.
6/18,
6/6
6/3
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x2+y2 = 1 x+y +z= 0
f(x,y,z) =x2+y2+z2 x2+y2 = 1, x+y+z= 0.
f (1/2, 1/
2,2
2) (1/
2,1/
2, 2
2),
3. f (1/
2,1/
2, 0) (1/
2, 1/
2, 0)
(2, 3, 2) x 1 =(y+ 1) =z+ 1
258/3, (1/3, 1/3,7/3).
x2 xy+y2 z2 = 1yx2 +y2 = 1
f(x,y,z) =x 2y+ 2z
x2 +y2 +z2 = 8
f(x, y) =x2 +xy+y2
D={(x, y) / x2 + y2 1}
f D
x y y= mx + h m h
(x1, y1), (x2, y2), , (xn, yn),
m h y =mx+h
di = yi (mxi+h) (xi, yi) m h
ni=1 d
2i ,
mni=1
xi+hn=ni=1
yi mn
i=1
x2i +hni=1
xi=n
i=1
xiyi
m h.
D(m, h) =ni=1
d2i =n
i=1
[yi (mxi+h)]2;
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m h
D(m, h) =
n
i=1
2[yi
(mxi+h)](
xi),
n
i=1
2[yi
(mxi+h)](
1) = (0, 0) =
=n
i=1[xiyi (mx2i +xih)] = 0ni=1[yi+mxi+h)] = 0
=
=
mn
i=1 x2i +h
ni=1 xi =
ni=1 xiyi
mn
i=1 xi+hn = n
i=1 yi
= ()
HD(m, h) =
2n
i=1 x2i 2
ni=1 xi
2n
i=1 xi hn
detHD = 4
n n
i=1
x2i
ni=1
xi
2
detHD >0
|x, y| x y x, y Rn
|x, y| =ni=1 xiyi y = (1, 1, ..., 1) := 1
n
i=1
xi2
x
2
1
2
12=n
ni=1
xi
2n
ni=1
x2i x Rn
1
4
HD =n
ni=1
x2
i ni=1
xi2
>0 (
xi= 0)
(m, h) ()
Dmm(m, h) = 2n
i=1 x2i >0
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x Rn
D
P(1, 1), Q(4, 2), R(2, 3)
2,9 m
1975 6421976 644
1977 6561978 6671979 6731980 6881981 6961982 6981983 7131984 7171985 7251986 742
1987 757