Post on 27-Feb-2018
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f(x, y) =x2 +y2 cos(xy)
f(x, y) =
xx2 +y2
f(x, y) = log
x +y
x y
f(x, y) = arctan
x+y
x y
f(x, y) = cos(3x)sin(3y)
f(x, y) = x2 +y2 cos(x2 +y2)
f
x
(x, y) = 2xy3 sin(xy)
f
y
(x, y) = 2y cos(xy)xy2 sin(xy)
f
x(x, y) =
y2
(x2 +y2)3/2
f
y(x, y) = xy
(x2 +y2)3/2
f
x(x, y) =
2y
y2 x2 f
y(x, y) =
2x
x2 y2
f
x(x, y) = y
x2 +y2
f
y(x, y) =
x
x2 +y2
f
x (x, y) = 3sin(3x) sin(3y) f
y (x, y) = 3 cos(3x)cos(3y)
f
x(x, y) = 2x 2xy2 sin(x2 +y2)
f
y(x, y) = 2y cos(x2 +y2) 2y3 sin(x2 +y2)
f(x, y) =exy + sin(x+y)
xD1f(x, y) y D2f(x, y) = (x y) cos(x+y).
g(x,y,z) = cos(
x+y
2z )
xD1g(x,y,z) +y D2g(x,y,z) +z D3g(x,y,z) = 0.
D1f(x, y) =yexy + cos(x+y)
D2f(x, y) = xe
xy + cos(x+y),
xD1f(x, y) y D2f(x, y) =xyexy + cos(x+y)
yxexy + cos(x+y)
= (x y)cos(x+y).
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D1g(x,y,z) = sin(x+y2z )1
2z
D2g(x,y,z) = sin(x+y2z )1
2z
D3g(x,y,z) = sin(x+y2z
)x+y
2z ,
xD1g(x,y,z) +y D2g(x,y,z) +z D3g(x,y,z) =
x
2z y
2z+x+y
2z
sin(x+y
2z ) = 0.
f(x, y) = x2
y2
(1, 1)
60
OX
f
f
x(x, y) = 2x
f
y(x, y) = 2y.
(1, 1)
f
(1, 1)
Duf(1, 1) = f(1, 1) u u
60
OX u = (cos60 , sin 60 ) = (12 ,32 )
Duf(1, 1) = (2,2) (12 ,32
) = 1
3.
f(x, y) =x2xy+ y2
(1, 1)
OX
f
f
x(x, y) = 2x y
f
y(x, y) = x+ 2y
(1, 1)
f
(1, 1)
Duf(1, 1) = f(1, 1) u
u
OX
u= (cos, sin)
u
Duf(1, 1) = f(1, 1) u= f(1, 1) cos(f(1, 1), u).
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Duf(1, 1) cos(f(1, 1), u) = 1 f(1, 1) u.
Duf(1, 1) cos(f(1, 1), u) = 0 f(1, 1) u.
Duf(1, 1) cos(f(1, 1), u) = 1 f(1, 1) u.
f(1, 1) = (1, 1)
Duf(1, 1) (1, 1) u u= ( 12 , 12)
Duf(1, 1) (1, 1) u u = ( 12 , 12) u= ( 1
2, 1
2)
Duf(1, 1) (1, 1) u u= ( 12 , 12)
f(x,y,z) =x3 y3 3xy(x y) + ez
(0, 0, 0)
f
x(x,y,z) = 3(x y)2 f
y(x,y,z) = 3(x y)2 f
z =ez.
f
(0,
0,
0) =f
x (0,
0,
0), f
y (0,
0,
0), f
z (0,
0,
0) f(0, 0, 0) = (0, 0, 1)
(x, y)
10
T(x, y) = 100 (x2 +y2)
(4, 3)
(4, 3)
T(x, y) = (2x,2y)
u =T(4, 3) =
(8,6)
(4, 3)
T(4, 3)
=
(
8,
6)
= (
8)2 + (
6)2 = 10
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Q
x
SO4H2 y
H2O Q=
aybx+y
a
b
x
y
x
x
Q
y(x, y) =
abx
(bx+y)2.
Q
x
(x, y) =
aby
(bx+y)2.
dQ = Q
x(x0, y0)dx+
Qy
(x0, y0)dy
(x0, y0) dx
dy
dx= 0, 05x0
dy = 0, 1y0
dQ= 0, 05aby0x0(bx0+ y0)2
+ 0, 1abx0y0(bx0+ y0)2
=0, 05aby0x0(bx0+ y0)2
.
x0 = 10y0
dQ=0, 05aby010y0(10by0+ y0)2
= 0, 5aby20y20(10b+ 1)
2 = ab
2(10b+ 1)2.
f(x, t) = f1(x+ at) + f2(x at) f1 f2
a
D22f(x, t) =a2D11f(x, t)
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D2f(x, t) =
t
f1(x+at) +f2(x at)
= f1(x+at) a+f2(x at) (a) =
= D22f(x, t) = t
a f1(x+at) a f2(x at)
=a f1 (x+at) a a f2 (x at) (a)=a2
f1 (x+at) +f
2 (x at)
.
D1f(x, t) =
x
f1(x+at) +f2(x at)
= f1(x+at) 1 +f2(x at) 1 =
= D11f(x, t) = x
f1(x+at) +f
2(x at)
= f1 (x+at) 1 +f2 (x at) 1.
D
22f
(x, t
) =a2D
11f
(x, t
)
f(x, y) = alog(x2 +y2) + b
f :=D11f(x, y) +D22f(x, y) = 0.
D1f(x, y) =
xa log(x2 +y2) +b= a
2x
x2
+y2
=
=D11f(x, y) =
x
a
2x
x2 +y2
= a2(x2 +y2) 2x 2x
(x2 +y2)2 =a
2(y2 x2)(x2 +y2)2
D2f(x, y) =
y
a log(x2 +y2) + b
= a
2y
x2 +y2 =
=D22f(x, y) = y
a
2y
x2 +y2
= a2(x2 +y2) 2y 2y
(x2 +y2)2 = a
2(x2 y2)(x2 +y2)2
,
f :=D11f(x, y) +D22f(x, y) =a 2(y2 x2)
(x2 +y2)2 +a2(x2 y2)
(x2 +y2)2 = 0.
f(x, y) =x3y xy3
g(x, y) =x2 +y2 x
h(x,y,z) =xyz3 4x2y2z.
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f(t x) =f(tx, ty) = (tx)3(ty)
(tx)(ty)3 =t4(x3yxy3) =t4 f(x, y) = t4 f(x),
f
f
R2
4 f(x) = f(x) x
f
x(x, y) = 3x2y y3
f
y(x, y) =x3 3xy2,
f(x, y) = (3x2y y3, x3 3xy2)
f(x)
x= (3x2y
y3, x3
3xy2)
(x, y)
= 3x3y xy3 +x3y 3xy3= 4(x3y xy3) = 4 f(x).
g
R2
g
g
x(x, y) = 2x 1
g
y = 2y,
g(x, y) = (2x 1, 2y)
g(x)
x= (2x
1, 2y)
(x, y)
= 2x2 x+ 2y2=r g(x),
r R
r g(x) = rx2 +r2y rx
g
h(t x) = h(tx, ty, tz) = (tx)(ty)(tz)3 4(tx)2(ty)2(tz) =t5(xyz3 x2y2z)= t5 h(x,y,z) =t5 h(x),
h
h
R3
5 h(x) = h(x) x
h
x(x,y,z) =yz3 8xy2z
h
y(x,y,z) =xz3 8x2yz
h
z(x,y,z) = 3xyz2 4x2y2,
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h(x,y,z ) = (yz3 8xy2z,xz3 8x2yz, 3xyz2 4x2y2)
h(x) x= (yz3 8xy2z,xz3 8x2yz, 3xyz2 4x2y2) (x,y,z)=xyz3
8x2y2z+ xyz3
8x2y2z+ 3xyz3
4x2y2z
= 5xyz3 20x2y2z= 5(xyz3 4x2y2z) = 5h(x).
F(x, y) = x2y xy2
F(x, y) = 0
y
x
(1, 1)
y=y(x)
H1 F R
2
F
F
x(x, y) = 2xy y2 F
y(x, y) = x2 2xy,
H2 F(1, 1) = 1
2 1 1 12 = 0.
H3 Fy
(1, 1) =x2 2xy|(1,1) = 1 = 0
(1, 1) F(x, y) = 0
y
x
y = y(x)
x2 y(x) x y(x)2 = 0.
x
2x y(x) +x2 y(x) [y(x) + x 2y(x) y(x)] = 0.
x= 1
2y(1) +y(1) [y(1) + 2y(1) y(1)] = 0,
y(1) = 1
2 +y(1) 1 2y(1) = 0
y(1) = 1
F(x,y,z) = x3 + 2y2 z2
F(x,y,z) = 0
z
x y
(1, 0, 1)
z=z(x, y)
(1, 0)
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H1 F R
3
F
F
x(x,y,z) = 3x2,
F
y(x,y,z ) = 4y,
F
z(x,y,z) = 2z,
H2 F(1, 0, 1) = 13 + 2 02 12 = 0.
H3 F
z(1, 0, 1) = 2z|(1,0,1)= 2 = 0
(1, 0, 1)
F(x,y,z ) = 0
z
x
y
z= z(x, y)
x3 + 2y2 z(x, y)2 = 0.
x
3x2 + 0 2z(x, y) zx
(x, y) = 0.
(x, y) = (1, 0)
3 2z(1, 0) zx
(1, 0) = 0,
z(1, 0) = 1
z
x (1, 0) =
3
2
y
4y 2z(x, y) zy
(x, y) = 0.
(x, y) = (1, 0)
0 2z(1, 0) zx
(1, 0) = 0,
z(1, 0) = 1
z
y(1, 0) = 0
F(x,y,z ) = C
z
x
y
P(x0, y0, z0) C z=z(x, y)
(x0, y0)
F(x,y,z) = log z x
2y
z P(1, 1, 1)
F(x,y,z) = log
10x
y+ z
100
P(10, 2, 200)
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G(x,y,z) :=F(x,y,z )C= log z x2y
zC
G
H1 G G
G
x(x,y,z ) = 2xy
y , G
y(x,y,z) = x
2
z , G
z(x,y,z ) =
1
z+x2y
z2 ,
H2 G(1, 1, 1) = log 1 12 11 C= 0 C= 1
H3 G
z(1, 0, 1) = 1
z+ x
2yz2 (1,0,1)
= 2 = 0
C =1
(1, 1, 1) G(x,y,z ) = 0
F(x,y,z ) =1
z
x y
z=z(x, y)
log z(x, y) x2y
z(x, y)= 1.
x
1
z(x, y)z
x
(x, y)
2xy z(x, y) x2y z
x(x, y)
z(x, y)2 = 0.
(x, y) = (1, 1)
1
z(1, 1) zx
(1, 1) 2 z(1, 1) zx
(1, 1)
z(1, 1)2 = 0,
z(1, 1) = 1
z
x(1, 1) = 1
y
1
z(x, y)z
y (x, y) x2
z(x, y)
x2y
zy
(x, y)
z(x, y)2 = 0.
(x, y) = (1, 1)
1
z(1, 1)zy
(1, 1) z(1, 1) z
y(x, y)
z(1, 1)2 = 0,
z(1, 1) = 1
z
y(1, 0) =
1
2
G(x,y,z ) := F(x,y,z)
C = log 10xy + z100
C
G
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H1 G G
G
x(x,y,z) =
1
x, G
y(x,y,z ) = 1
y, G
z(x,y,z) =
1
100,
H2 G(10, 2, 200) = log
10102
+ 200
100C= 0 C= log(50) + 2
H3 G
z(1, 0, 1) = 1
z+ x
2yz2
(1,0,1)
= 2 = 0
C= log(50) + 2
(1, 2, 200)
G(x,y,z) = 0
F(x,y,z) = log(50) + 2
z
x
y
z=z(x, y)
log
10x
y
+z(x, y)
100 = log(50) + 2.
x
log
10x
y
+z(x, y)
100 = 0.
(x, y) = (1, 1)
1
z(1, 1) zx
(1, 1) 2 z(1, 1) zx
(1, 1)
z(1, 1)2 = 0,
z(1, 1) = 1
z
x(1, 1) = 1
y
1
z(x, y)zy
(x, y) x2 z(x, y) x2y z
y(x, y)
z(x, y)2 = 0.
(x, y) = (1, 1)
1z(1, 1)zy (1, 1)
z(1, 1)
z
y(x, y)
z(1, 1)2 = 0,
z(1, 1) = 1
z
y(1, 0) =
1
2