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62
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s cos θ
S
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t
e
r
m
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a
r
dz dt
o
dw dt
u
.
3
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√t
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g
r
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∂z ∂t
e
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e
,
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a
,
r
E
p
z = arctan(2x + y ) x = s2 t y = s ln t
)
c
a
t z = 1 + 2t
−
,
∂z 4st + ln t = ∂s 1 + (2x + y )2 (
i
z = x2 + xy + y2 x = s + t y = st
)
b
e
− sin2x√sint y
∂z = 2x + y + xt + 2yt ∂s (
d
+ 2xy )t2
,
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r
a
2
− 3(x
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)
e
a
z = sin x cos y x = πt y =
dw =e dt
E
C
,
dz = π cos x cos y dt c
a
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)
(
d
z = x2 y + xy 2 x = 2 + t4 y = 1
)
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a
e
,
2s2 + st ∂z = ∂t 1 + (2x + y )2
e
√ θ= s
2
− √s
t
2
,
−4
,
sin θ
+ t2
z = f (x, y )
o
+ t2
n
d
f x (2, 7) = 6
,
e
e
f
∂z = er ∂t
e
− √s
t cos θ
s
2
sin θ
+ t2
x = g (t) y = h(t) g(3) = 2 g (3) = 5 dz t=3 dt
é
d
i
f
e
r
f y (2, 7) =
e
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36, 24, 30
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p
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(
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o
S
l
e
.
:
∂z = ∂r
∂z ∂x
z = f (x
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m
t
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d
a
s
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s
x
e
r
c
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c
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2
8
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j
a
x = r cos θ
i
t
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p
a
r
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e
r
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c
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2
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r
c
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s
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d
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c
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d
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u = 2, v = 1, w = 0
q
u
a
n
d
o
x=y=1
S
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j
a
q
u
a
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d
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p = 2, r = 3, θ = 0
t
e
r
m
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a
o
r
.
y
l
.
:
− y) = xe sin(x − y ) dy = dx sin(x − y ) − xe
∂z ∂x
y e
∂z ∂y
e
.
2 2
u
n
ç
õ
e
s
d
a
d
a
y = r sin θ
e
2
s
s
,
d
ã
e
o
t
2
s
t
r
e
q
u
e
∂z ∂y
+
∂z ∂y
cos θ +
o
y
− 1 + y z+ y z
e
d
r
i
m
sin θ
z = xye
x y
f
e
i
r
n
e
e
n
2
c
i
∂z ∂r
á
v
e
i
s
.
∂z ∂θ
e
e
m
o
s
∂z = ∂θ
e
V
e
r
+
∂z ∂x
1
r2
t
r
e
q
u
e
2
∂z ∂θ
∂z ∂y
r sin θ +
r cos θ
.
∂z ∂z + =0 ∂x ∂y
.
∂z ∂r
=
x2 y2 x2 + y 2
−
,
s
i
q
s
u
e
q
u
e
x
e
(x, y ) = (0, 0)
.
0,
E
p
−
f (x, y ) =
c
f
xy
E
í
∂z = ∂y
e
∂z ∂x
S
s
− −
∂z 1 + y2 z 2 = ∂x 1 + y + y2 z 2
o
a
3xz 2y ∂z = ∂y 2z 3xy
e
− z = arctan(yz )
x
)
S
E
o
x2 + y 2 + z 2 = 3xyz 3yz 2x ∂z = ∂x 2z 3xy
)
S
(
i
d
cos(x
)
− √ −
c
d
a
∂R ∂R , ∂x ∂y
dy dx
4(xy ) dy y = dx x 2x2 xy
n
v
.
3 2
S
i
∂u ∂u ∂u , , ∂p ∂r ∂θ
;
√xy = 1 + x y
)
r
a
;
2
(
e
.
,
o
a
∂z ∂z ∂z , , ∂u ∂v ∂w
;
u = x2 + yx x = pr cos θ y = pr sin θ z = p + r
)
n
.
,
S
E
4
R = ln(u2 + v 2 + w2 ) u = x + 2 y v = 2x
S
c
2
,
)
(
o
z = x2 + xy 3 x = uv 2 + w3 y = u + ve w
)
S
(
i
e
C
l
c
u
l
(x, y ) = (0, 0)
∂ 3 z ∂x 3
∂ 3 z ∂y∂x2
6
a
+y
=0
.
e
∂ 2 f (0, 0) ∂x∂y
e
∂ 2 f (0, 0) ∂y∂x
.