tangent
u
1
(
𝐱𝟎
,
z
1
)
=
v
1
+
u
˙
x
︸
x1
=
-
∂
V
x
∂
x2
g
x
(
𝐱𝟑
)
-
k
1
(
z
1
-
u
x
(
𝐱
)
︸
x4
)
︷
v
1
+
x5
∂
x6
(
f
x7
(
x8
)
+
g
x
(
𝐱𝟗
)
z
1
︸
x10
(i.e.,
d
𝐱𝟏𝟏
x12
t
?x13
)
︷
x14
x
(i.e.,
d
u
x15
d
t
)
Search
Returned 97 matches (100 formulae, 48 docs)
Lookup 1356.520 ms, Re-ranking 40255.016 ms
Found 1729858 tuple postings, 206841 formulae, 22235 documents
[ formulas ]
[ documents ]
[ documents-by-formula ]
u
1
(
𝐱
,
z
1
)
=
v
1
+
u
˙
x
︸
By definition of
v
1
=
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
(
z
1
-
u
x
(
𝐱
)
︸
e
1
)
︷
v
1
+
∂
u
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
︸
𝐱
˙
(i.e.,
d
𝐱
d
t
)
)
︷
u
˙
x
(i.e.,
d
u
x
d
t
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
1
(
𝐱
,
z
1
)
=
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
(
z
1
-
u
x
(
𝐱
)
)
+
∂
u
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
a
1
(
𝐱
,
z
1
)
=
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
(
z
1
-
u
x
(
𝐱
)
)
+
∂
u
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
1
(
𝐱
,
z
1
)
=
1
g
1
(
𝐱
,
z
1
)
(
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
(
z
1
-
u
x
(
𝐱
)
)
+
∂
u
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
)
︷
u
a
1
(
𝐱
,
z
1
)
-
f
1
(
𝐱
,
z
1
)
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
i
(
𝐱
,
z
1
,
z
2
,
…
,
z
i
︷
≜
𝐱
i
)
=
-
∂
V
i
-
1
∂
𝐱
i
-
1
g
i
-
1
(
𝐱
i
-
1
)
-
k
i
(
z
i
-
u
i
-
1
(
𝐱
i
-
1
)
)
+
∂
u
i
-
1
∂
𝐱
i
-
1
(
f
i
-
1
(
𝐱
i
-
1
)
+
g
i
-
1
(
𝐱
i
-
1
)
z
i
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
2
(
𝐱
,
z
1
,
z
2
)
=
-
∂
V
1
∂
𝐱
1
g
1
(
𝐱
1
)
-
k
2
(
z
2
-
u
1
(
𝐱
1
)
)
+
∂
u
1
∂
𝐱
1
(
f
1
(
𝐱
1
)
+
g
1
(
𝐱
1
)
z
2
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
3
(
𝐱
,
z
1
,
z
2
,
z
3
)
=
-
∂
V
2
∂
𝐱
2
g
2
(
𝐱
2
)
-
k
3
(
z
3
-
u
2
(
𝐱
2
)
)
+
∂
u
2
∂
𝐱
2
(
f
2
(
𝐱
2
)
+
g
2
(
𝐱
2
)
z
3
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
u
i
(
𝐱
,
z
1
,
z
2
,
…
,
z
i
︷
≜
𝐱
i
)
=
1
g
i
(
𝐱
i
)
(
-
∂
V
i
-
1
∂
𝐱
i
-
1
g
i
-
1
(
𝐱
i
-
1
)
-
k
i
(
z
i
-
u
i
-
1
(
𝐱
i
-
1
)
)
+
∂
u
i
-
1
∂
𝐱
i
-
1
(
f
i
-
1
(
𝐱
i
-
1
)
+
g
i
-
1
(
𝐱
i
-
1
)
z
i
)
︷
Single-integrator stabilizing control
u
a
i
(
𝐱
i
)
-
f
i
(
𝐱
i
-
1
)
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
˙
1
=
V
˙
x
(
𝐱
)
+
1
2
(
2
e
1
e
˙
1
)
=
V
˙
x
(
𝐱
)
+
e
1
e
˙
1
=
V
˙
x
(
𝐱
)
+
e
1
v
1
︷
e
˙
1
=
∂
V
x
∂
𝐱
𝐱
︸
˙
(i.e.,
d
𝐱
d
t
)
︷
V
˙
x
(i.e.,
d
V
x
d
t
)
+
e
1
v
1
=
∂
V
x
∂
𝐱
(
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
+
g
x
(
𝐱
)
e
1
)
︸
𝐱
˙
︷
V
˙
x
+
e
1
v
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
v
1
=
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
e
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
˙
x
=
∂
V
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
≤
-
W
(
𝐱
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
˙
1
=
-
W
(
𝐱
)
+
∂
V
x
∂
𝐱
g
x
(
𝐱
)
e
1
+
e
1
(
-
∂
V
x
∂
𝐱
g
x
(
𝐱
)
-
k
1
e
1
)
︷
v
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
˙
1
=
∂
V
x
∂
𝐱
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
︷
≤
-
W
(
𝐱
)
+
∂
V
x
∂
𝐱
g
x
(
𝐱
)
e
1
+
e
1
v
1
≤
-
W
(
𝐱
)
+
∂
V
x
∂
𝐱
g
x
(
𝐱
)
e
1
+
e
1
v
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
3
(
𝐱
,
z
1
,
z
2
,
z
3
)
=
V
2
(
𝐱
2
)
+
1
2
(
z
3
-
u
2
(
𝐱
2
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
˙
1
=
-
W
(
𝐱
)
+
∂
V
x
∂
𝐱
g
x
(
𝐱
)
e
1
-
e
1
∂
V
x
∂
𝐱
g
x
(
𝐱
)
︷
0
-
k
1
e
1
2
=
-
W
(
𝐱
)
-
k
1
e
1
2
≤
-
W
(
𝐱
)
<
0
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
1
(
𝐱
,
z
1
)
=
V
x
(
𝐱
)
+
1
2
(
z
1
-
u
x
(
𝐱
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
Doc 2
0.1582, -12.0000, 9.0000, 0.3109
testing/wikipedia/v3/23338.html
V
1
(
𝐱
,
z
1
)
≜
V
x
(
𝐱
)
+
1
2
(
z
1
-
u
x
(
𝐱
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
i
(
𝐱
i
)
=
V
i
-
1
(
𝐱
i
-
1
)
+
1
2
(
z
i
-
u
i
-
1
(
𝐱
i
-
1
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
2
(
𝐱
,
z
1
,
z
2
)
=
V
1
(
𝐱
1
)
+
1
2
(
z
2
-
u
1
(
𝐱
1
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
V
2
(
𝐱
,
z
1
,
z
2
)
=
V
1
(
𝐱
,
z
1
)
+
1
2
(
z
2
-
u
1
(
𝐱
,
z
1
)
)
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
Doc 2
0.1582, -12.0000, 9.0000, 0.3109
testing/wikipedia/v3/23338.html
{
[
𝐱
˙
z
˙
1
]
︷
≜
𝐱
˙
1
=
[
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
0
]
︷
≜
f
1
(
𝐱
1
)
+
[
𝟎
1
]
︷
≜
g
1
(
𝐱
1
)
z
2
( by Lyapunov function
V
1
,
subsystem stabilized by
u
1
(
𝐱
1
)
)
z
˙
2
=
u
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
[
𝐱
˙
z
˙
1
z
˙
2
]
︷
≜
𝐱
˙
2
=
[
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
2
z
2
0
]
︷
≜
f
2
(
𝐱
2
)
+
[
𝟎
0
1
]
︷
≜
g
2
(
𝐱
2
)
z
3
( by Lyapunov function
V
2
,
subsystem stabilized by
u
2
(
𝐱
2
)
)
z
˙
3
=
u
3
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
A
f
(
x
)
=
∑
i
b
i
(
x
)
∂
f
∂
x
i
(
x
)
+
1
2
∑
i
,
j
(
σ
σ
⊤
)
i
,
j
(
x
)
∂
2
f
∂
x
i
∂
x
j
(
x
)
.
Doc 3
0.1440, -37.0000, 7.0000, 0.1440
testing/wikipedia/v3/17507.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
u
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
u
a
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
z
2
z
˙
2
=
u
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
+
g
x
(
𝐱
)
e
1
e
˙
1
=
v
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
z
2
z
˙
2
=
z
3
z
˙
3
=
u
3
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
+
g
x
(
𝐱
)
e
1
e
˙
1
=
u
1
-
u
˙
x
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐲
˙
=
f
y
(
𝐲
)
+
g
y
(
𝐲
)
z
2
( where this
𝐲
subsystem is stabilized by
z
2
=
u
1
(
𝐱
,
z
1
)
)
z
˙
2
=
u
2
.
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
f
1
(
𝐱
,
z
1
)
+
g
1
(
𝐱
,
z
1
)
u
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
+
(
g
x
(
𝐱
)
u
x
(
𝐱
)
-
g
x
(
𝐱
)
u
x
(
𝐱
)
)
︸
0
z
˙
1
=
u
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
f
′′′
(
x
0
)
≈
-
1
2
f
(
x
-
2
)
+
f
(
x
-
1
)
-
f
(
x
+
1
)
+
1
2
f
(
x
+
2
)
h
x
3
+
O
(
h
x
2
)
Doc 4
0.1325, -36.0000, 5.0000, 0.1325
testing/wikipedia/v3/25461.html
{
x
˙
=
(
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
)
︸
F
(
𝐱
)
+
g
x
(
𝐱
)
(
z
1
-
u
x
(
𝐱
)
)
︸
z
1
error tracking
u
x
z
˙
1
=
u
1
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
[
𝐱
˙
1
z
˙
2
]
︷
𝐱
˙
2
=
[
f
1
(
𝐱
1
)
+
g
1
(
𝐱
1
)
z
2
0
]
︷
f
2
(
𝐱
2
)
+
[
𝟎
1
]
︷
g
2
(
𝐱
2
)
z
3
( by Lyapunov function
V
2
,
subsystem stabilized by
u
2
(
𝐱
2
)
)
z
˙
3
=
u
3
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
f
1
(
𝐱
,
z
1
)
+
g
1
(
𝐱
,
z
1
)
1
g
1
(
𝐱
,
z
1
)
(
u
a
1
-
f
1
(
𝐱
,
z
1
)
)
︷
u
1
(
𝐱
,
z
1
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
( by Lyapunov function
V
x
,
subsystem stabilized by
u
x
(
𝐱
)
)
z
˙
1
=
z
2
z
˙
2
=
z
3
⋮
z
˙
i
=
z
i
+
1
⋮
z
˙
k
-
2
=
z
k
-
1
z
˙
k
-
1
=
z
k
z
˙
k
=
u
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
=
𝔼
θ
[
V
(
x
(
θ
)
,
θ
)
-
u
¯
(
θ
0
)
-
1
-
P
(
θ
)
p
(
θ
)
∂
V
∂
θ
-
c
(
x
(
θ
)
)
]
Doc 5
0.1297, -27.0000, 3.0000, 0.6432
testing/wikipedia/v3/06206.html
∂
2
𝐱
i
(
t
)
∂
t
2
m
i
=
-
∂
∂
𝐱
i
[
V
(
𝐱
i
(
t
)
)
+
∑
k
=
1
n
λ
k
σ
k
(
t
)
]
,
i
=
1
…
N
.
Doc 6
0.1297, -34.0000, 4.0000, 0.1297
testing/wikipedia/v3/16262.html
H
=
(
V
(
x
,
θ
)
-
u
¯
(
θ
0
)
-
1
-
P
(
θ
)
p
(
θ
)
∂
V
∂
θ
(
x
,
θ
)
-
c
(
x
)
)
p
(
θ
)
+
ν
(
θ
)
∂
x
∂
θ
Doc 5
0.1297, -27.0000, 3.0000, 0.6432
testing/wikipedia/v3/06206.html
∫
ϕ
2
(
x
)
ϕ
1
(
x
)
(
∂
V
∂
x
(
x
,
θ
)
-
1
-
P
(
θ
)
p
(
θ
)
∂
2
V
∂
θ
∂
x
(
x
,
θ
)
-
∂
c
∂
x
(
x
)
)
d
θ
=
0
Doc 5
0.1297, -27.0000, 3.0000, 0.6432
testing/wikipedia/v3/06206.html
∂
ν
∂
θ
=
-
∂
H
∂
x
=
-
(
∂
V
∂
x
(
x
,
θ
)
-
1
-
P
(
θ
)
p
(
θ
)
∂
2
V
∂
θ
∂
x
(
x
,
θ
)
-
∂
c
∂
x
(
x
)
)
p
(
θ
)
Doc 5
0.1297, -27.0000, 3.0000, 0.6432
testing/wikipedia/v3/06206.html
ψ
(
𝐫
1
,
𝐫
2
)
=
1
2
(
u
A
(
𝐫
1
)
u
B
(
𝐫
2
)
+
u
B
(
𝐫
1
)
u
A
(
𝐫
2
)
)
Doc 7
0.1244, -24.0000, 3.0000, 0.1244
testing/wikipedia/v3/00649.html
𝔼
θ
[
V
(
x
(
θ
)
,
θ
)
-
u
¯
(
θ
0
)
-
∫
θ
0
θ
∂
V
∂
θ
~
d
θ
~
-
c
(
x
(
θ
)
)
]
Doc 5
0.1297, -27.0000, 3.0000, 0.6432
testing/wikipedia/v3/06206.html
∂
t
E
(
u
)
+
∂
x
F
(
u
)
=
0
where
E
(
u
)
=
S
(
u
)
-
1
2
κ
(
u
x
,
u
)
,
F
(
u
)
=
1
2
κ
(
u
t
,
u
)
.
Doc 8
0.1198, -35.0000, 4.0000, 0.2287
testing/wikipedia/v3/29918.html
∂
Π
(
n
,
k
)
∂
n
=
1
2
(
k
2
-
n
)
(
n
-
1
)
(
E
(
k
)
+
1
n
(
k
2
-
n
)
K
(
k
)
+
1
n
(
n
2
-
k
2
)
Π
(
n
,
k
)
)
Doc 9
0.1198, -42.0000, 5.0000, 0.1198
testing/wikipedia/v3/00354.html
{
𝐱
˙
1
=
f
1
(
𝐱
1
)
+
g
1
(
𝐱
1
)
z
2
( by Lyapunov function
V
1
,
subsystem stabilized by
u
1
(
𝐱
1
)
)
z
˙
2
=
u
2
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
{
𝐱
˙
2
=
f
2
(
𝐱
2
)
+
g
2
(
𝐱
2
)
z
3
( by Lyapunov function
V
2
,
subsystem stabilized by
u
2
(
𝐱
2
)
)
z
˙
3
=
u
3
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
d
d
s
u
(
x
(
s
)
,
t
(
s
)
)
=
∂
u
∂
x
d
x
d
s
+
∂
u
∂
t
d
t
d
s
Doc 10
0.1153, -25.0000, 4.0000, 0.1153
testing/wikipedia/v3/06475.html
ε
x
x
=
∂
u
x
∂
x
=
-
z
∂
φ
∂
x
;
ε
x
z
=
1
2
(
∂
u
x
∂
z
+
∂
u
z
∂
x
)
=
1
2
(
-
φ
+
∂
w
∂
x
)
Doc 11
0.1121, -41.0000, 7.0000, 0.1121
testing/wikipedia/v3/17975.html
u
1
(
𝐱
,
z
1
)
=
1
g
1
(
𝐱
,
z
1
)
(
u
a
1
-
f
1
(
𝐱
,
z
1
)
)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
Ψ
(
x
1
,
…
x
n
)
=
1
n
!
|
ψ
1
(
x
1
)
…
ψ
n
(
x
1
)
⋮
⋮
ψ
1
(
x
n
)
…
ψ
n
(
x
n
)
|
Doc 12
0.1102, -31.0000, 2.0000, 0.1102
testing/wikipedia/v3/03194.html
∂
ρ
∂
t
+
1
r
∂
∂
r
(
ρ
r
u
r
)
+
1
r
∂
(
ρ
u
ϕ
)
∂
ϕ
+
∂
(
ρ
u
z
)
∂
z
=
0.
Doc 13
0.1102, -32.0000, 5.0000, 0.1102
testing/wikipedia/v3/01538.html
v
1
𝐞
1
+
v
2
𝐞
2
=
𝐞
1
h
2
h
3
∂
∂
x
2
(
h
3
ψ
3
)
-
𝐞
2
h
3
h
1
∂
∂
x
1
(
h
3
ψ
3
)
Doc 14
0.1102, -36.0000, 3.0000, 0.1102
testing/wikipedia/v3/16864.html
e
˙
=
d
d
t
H
(
x
)
-
d
d
t
H
(
x
^
)
=
d
d
t
H
(
x
)
-
M
(
x
^
)
sgn
(
V
(
t
)
-
H
(
x
^
(
t
)
)
)
,
Doc 15
0.1102, -38.0000, 4.0000, 0.1102
testing/wikipedia/v3/08059.html
∂
t
I
(
u
)
+
∂
x
G
(
u
)
=
0
where
I
(
u
)
=
1
2
ω
(
u
x
,
u
)
,
G
(
u
)
=
S
(
u
)
-
1
2
ω
(
u
t
,
u
)
.
Doc 8
0.1198, -35.0000, 4.0000, 0.2287
testing/wikipedia/v3/29918.html
L
u
(
x
)
=
∑
i
,
j
=
1
n
(
a
i
j
(
x
)
u
x
i
)
x
j
+
∑
i
=
1
n
b
i
(
x
)
u
x
i
(
x
)
+
c
(
x
)
u
(
x
)
Doc 16
0.1051, -35.0000, 3.0000, 0.1051
testing/wikipedia/v3/16787.html
μ
z
≈
z
(
μ
1
,
μ
2
)
+
1
2
{
∂
2
z
∂
x
1
2
σ
1
2
+
∂
2
z
∂
x
2
2
σ
2
2
}
+
∂
z
2
∂
x
1
∂
x
2
σ
12
𝐄𝐪
(
𝟏𝟒
)
Doc 17
0.1051, -44.0000, 3.0000, 0.2963
testing/wikipedia/v3/23828.html
V
1
=
2
R
[
u
]
(
∫
x
1
x
2
[
p
(
x
)
u
′
(
x
)
v
′
(
x
)
+
q
(
x
)
u
(
x
)
v
(
x
)
-
λ
u
(
x
)
v
(
x
)
]
d
x
+
a
1
u
(
x
1
)
v
(
x
1
)
+
a
2
u
(
x
2
)
v
(
x
2
)
)
,
Doc 18
0.1051, -59.0000, 5.0000, 0.1051
testing/wikipedia/v3/02611.html
ℒ
:=
1
2
μ
(
∂
w
∂
t
)
2
-
1
2
E
I
(
∂
2
w
∂
x
2
)
2
+
q
(
x
)
w
(
x
,
t
)
=
μ
2
w
˙
2
-
E
I
2
w
x
x
2
+
q
w
≡
ℒ
(
x
,
t
,
w
,
w
˙
,
w
x
x
)
Doc 19
0.1051, -60.0000, 3.0000, 0.1051
testing/wikipedia/v3/09923.html
log
(
x
0
)
+
1
x
0
(
x
-
x
0
)
-
1
x
0
2
(
x
-
x
0
)
2
2
+
⋯
.
Doc 21
0.1045, -23.0000, 4.0000, 0.1045
testing/wikipedia/v3/01026.html
∂
g
t
(
z
)
∂
t
=
g
t
(
z
)
ζ
(
t
)
+
g
t
(
z
)
ζ
(
t
)
-
g
t
(
z
)
.
Doc 20
0.1045, -23.0000, 4.0000, 0.1045
testing/wikipedia/v3/14856.html
𝒜
f
(
x
)
=
∑
i
b
i
(
x
)
∂
f
∂
x
i
(
x
)
+
1
2
∑
i
,
j
(
σ
(
x
)
σ
(
x
)
⊤
)
i
,
j
∂
2
f
∂
x
i
∂
x
j
(
x
)
.
Doc 22
0.1008, -42.0000, 5.0000, 0.2017
testing/wikipedia/v3/17512.html
A
f
(
x
)
=
∑
i
b
i
(
x
)
∂
f
∂
x
i
(
x
)
+
1
2
∑
i
,
j
(
σ
(
x
)
σ
(
x
)
⊤
)
i
,
j
∂
2
f
∂
x
i
∂
x
j
(
x
)
,
Doc 22
0.1008, -42.0000, 5.0000, 0.2017
testing/wikipedia/v3/17512.html
Doc 23
0.1008, -42.0000, 5.0000, 0.1008
testing/wikipedia/v3/17534.html
(
z
-
E
[
z
]
)
2
≈
(
∂
z
∂
x
1
)
2
(
x
1
-
x
¯
1
)
2
+
(
∂
z
∂
x
2
)
2
(
x
2
-
x
¯
2
)
2
+
2
(
∂
z
∂
x
1
)
(
∂
z
∂
x
2
)
(
x
1
-
x
¯
1
)
(
x
2
-
x
¯
2
)
Doc 17
0.1051, -44.0000, 3.0000, 0.2963
testing/wikipedia/v3/23828.html
d
d
x
(
∫
f
1
(
x
)
f
2
(
x
)
g
(
t
)
d
t
)
=
g
[
f
2
(
x
)
]
f
2
′
(
x
)
-
g
[
f
1
(
x
)
]
f
1
′
(
x
)
Doc 24
0.0960, -35.0000, 3.0000, 0.0960
testing/wikipedia/v3/10729.html
a
b
¯
=
(
d
x
+
∂
u
x
∂
x
d
x
)
2
+
(
∂
u
y
∂
x
d
x
)
2
=
1
+
2
∂
u
x
∂
x
+
(
∂
u
x
∂
x
)
2
+
(
∂
u
y
∂
x
)
2
d
x
Doc 25
0.0960, -54.0000, 5.0000, 0.1864
testing/wikipedia/v3/02741.html
length
(
a
b
)
=
(
d
x
+
∂
u
x
∂
x
d
x
)
2
+
(
∂
u
y
∂
x
d
x
)
2
=
d
x
1
+
2
∂
u
x
∂
x
+
(
∂
u
x
∂
x
)
2
+
(
∂
u
y
∂
x
)
2
Doc 26
0.0960, -55.0000, 5.0000, 0.0960
testing/wikipedia/v3/22820.html
∂
2
(
x
p
(
x
)
-
C
(
x
)
)
∂
2
x
=
∂
2
π
(
x
,
t
)
∂
x
2
,
Doc 27
0.0904, -21.0000, 3.0000, 0.0904
testing/wikipedia/v3/05933.html
H
z
(
i
)
(
z
1
,
z
2
)
=
A
z
(
i
)
(
z
1
,
z
2
)
B
z
(
i
)
(
z
1
,
z
2
)
Doc 28
0.0904, -25.0000, 4.0000, 0.0904
testing/wikipedia/v3/30220.html
|
f
(
z
1
)
-
f
(
z
2
)
1
-
f
(
z
1
)
¯
f
(
z
2
)
|
≤
|
z
1
-
z
2
1
-
z
1
¯
z
2
|
Doc 29
0.0904, -30.0000, 4.0000, 0.3335
testing/wikipedia/v3/08430.html
|
f
(
z
1
)
-
f
(
z
2
)
f
(
z
1
)
¯
-
f
(
z
2
)
|
≤
|
z
1
-
z
2
|
|
z
1
¯
-
z
2
|
.
Doc 29
0.0904, -30.0000, 4.0000, 0.3335
testing/wikipedia/v3/08430.html
|
f
(
z
1
)
-
f
(
z
2
)
1
-
f
(
z
1
)
¯
f
(
z
2
)
|
≤
|
z
1
-
z
2
1
-
z
1
¯
z
2
|
.
Doc 29
0.0904, -30.0000, 4.0000, 0.3335
testing/wikipedia/v3/08430.html
∂
∂
t
Q
i
(
t
)
=
-
1
Δ
x
(
f
(
q
(
t
,
x
i
+
1
/
2
)
)
-
f
(
q
(
t
,
x
i
-
1
/
2
)
)
)
,
Doc 30
0.0904, -34.0000, 4.0000, 0.0904
testing/wikipedia/v3/14299.html
f
(
ζ
)
=
1
(
2
π
i
)
n
∫
⋯
∬
∂
D
1
×
…
×
∂
D
n
f
(
z
1
,
…
,
z
n
)
(
z
1
-
ζ
1
)
…
(
z
n
-
ζ
n
)
d
z
1
…
d
z
n
Doc 31
0.0904, -46.0000, 3.0000, 0.0904
testing/wikipedia/v3/01998.html
e
r
s
=
1
2
(
∂
u
r
∂
x
s
+
∂
u
s
∂
x
r
-
∂
u
k
∂
x
r
∂
u
k
∂
x
s
)
≈
1
2
(
∂
u
r
∂
x
s
+
∂
u
s
∂
x
r
)
Doc 25
0.0960, -54.0000, 5.0000, 0.1864
testing/wikipedia/v3/02741.html
E
[
∂
2
z
∂
x
1
∂
x
2
(
x
1
-
x
¯
1
)
(
x
2
-
x
¯
2
)
]
=
∂
2
z
∂
x
1
∂
x
2
E
[
(
x
1
-
x
¯
1
)
(
x
2
-
x
¯
2
)
]
=
∂
2
z
∂
x
1
∂
x
2
σ
1
,
2
Doc 17
0.1051, -44.0000, 3.0000, 0.2963
testing/wikipedia/v3/23828.html
M
2
f
=
-
△
n
P
(
f
)
+
n
-
2
x
n
∂
P
(
f
)
∂
x
n
-
(
△
n
Q
(
f
)
-
n
-
2
x
n
∂
Q
(
f
)
∂
x
n
+
n
-
2
x
n
2
Q
(
f
)
)
e
n
Doc 32
0.0901, -51.0000, 4.0000, 0.0901
testing/wikipedia/v3/23998.html
M
11
=
-
D
(
∂
φ
1
∂
x
1
+
ν
∂
φ
2
∂
x
2
)
,
M
22
=
-
D
(
∂
φ
2
∂
x
2
+
ν
∂
φ
1
∂
x
1
)
,
M
12
=
-
D
(
1
-
ν
)
2
(
∂
φ
1
∂
x
2
+
∂
φ
2
∂
x
1
)
Doc 33
0.0901, -64.0000, 3.0000, 0.0901
testing/wikipedia/v3/26549.html
f
(
x
)
∂
2
u
∂
x
2
+
g
(
x
)
∂
u
∂
x
+
h
(
x
)
u
=
∂
u
∂
t
+
k
(
t
)
u
Doc 34
0.0862, -27.0000, 3.0000, 0.0862
testing/wikipedia/v3/05175.html
(
E
c
(
z
)
-
∂
∂
z
ℏ
2
2
m
c
(
z
)
∂
∂
z
+
ℏ
2
𝐤
2
2
m
c
(
z
)
)
f
k
(
z
)
=
E
f
k
(
z
)
Doc 35
0.0862, -35.0000, 4.0000, 0.0862
testing/wikipedia/v3/08334.html
σ
x
=
2
μ
ε
x
+
λ
(
ε
x
+
ε
y
+
ε
z
)
=
2
μ
∂
u
x
∂
x
+
λ
(
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
z
)
Doc 36
0.0862, -42.0000, 4.0000, 0.2389
testing/wikipedia/v3/03795.html
y
˙
=
d
h
(
x
)
d
t
=
d
h
(
x
)
d
x
x
˙
=
d
h
(
x
)
d
x
f
(
x
)
+
d
h
(
x
)
d
x
g
(
x
)
u
Doc 37
0.0817, -38.0000, 3.0000, 0.0817
testing/wikipedia/v3/14293.html
{
𝐱
˙
=
f
x
(
𝟎
︸
𝐱
)
+
(
g
x
(
𝟎
︸
𝐱
)
)
(
0
︸
z
1
)
=
0
+
(
g
x
(
𝟎
)
)
(
0
)
=
𝟎
(i.e.,
𝐱
=
𝟎
is stationary)
z
˙
1
=
0
︷
u
1
(i.e.,
z
1
=
0
is stationary)
Doc 1
0.8590, -13.0000, 58.0000, 8.6262
testing/wikipedia/v3/21960.html
∂
g
∂
x
j
(
x
)
=
-
(
∂
f
∂
y
(
x
,
g
(
x
)
)
)
-
1
∂
f
∂
x
j
(
x
,
g
(
x
)
)
Doc 38
0.0763, -30.0000, 3.0000, 0.0763
testing/wikipedia/v3/05668.html
∂
2
F
(
z
,
w
)
∂
z
∂
w
=
f
′
(
z
)
f
′
(
w
)
(
f
(
z
)
-
f
(
w
)
)
2
-
1
(
z
-
w
)
2
,
Doc 39
0.0763, -34.0000, 3.0000, 0.0763
testing/wikipedia/v3/05547.html
u
(
x
)
=
u
i
+
(
x
-
x
i
)
(
x
i
+
1
-
x
i
)
(
u
i
+
1
-
u
i
)
∀
x
∈
(
x
i
,
x
i
+
1
]
.
Doc 40
0.0763, -35.0000, 3.0000, 0.0763
testing/wikipedia/v3/14309.html
∏
x
x
x
=
C
e
ζ
′
(
-
1
,
x
)
-
ζ
′
(
-
1
)
=
C
e
ψ
(
-
2
)
(
z
)
+
z
2
-
z
2
-
z
2
ln
(
2
π
)
=
C
K
(
x
)
Doc 41
0.0763, -43.0000, 4.0000, 0.0763
testing/wikipedia/v3/23918.html
(
λ
+
μ
)
∂
∂
x
(
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
z
)
+
μ
(
∂
2
u
x
∂
x
2
+
∂
2
u
x
∂
y
2
+
∂
2
u
x
∂
z
2
)
+
F
x
=
0
Doc 36
0.0862, -42.0000, 4.0000, 0.2389
testing/wikipedia/v3/03795.html
∂
∂
x
(
2
μ
∂
u
x
∂
x
+
λ
(
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
z
)
)
+
μ
∂
∂
y
(
∂
u
x
∂
y
+
∂
u
y
∂
x
)
+
μ
∂
∂
z
(
∂
u
z
∂
x
+
∂
u
x
∂
z
)
+
F
x
=
0
Doc 36
0.0862, -42.0000, 4.0000, 0.2389
testing/wikipedia/v3/03795.html
ρ
(
∂
u
→
x
∂
t
+
∇
y
⋅
u
→
x
u
→
y
)
=
-
∇
x
p
+
ν
∇
y
⋅
(
∇
x
(
ρ
u
→
y
)
+
∇
y
(
ρ
u
→
x
)
)
Doc 42
0.0672, -39.0000, 2.0000, 0.0672
testing/wikipedia/v3/13787.html
L
=
-
ρ
{
∫
-
h
(
x
,
y
)
ζ
(
x
,
y
,
t
)
[
∂
Φ
∂
t
+
1
2
(
(
∂
Φ
∂
x
)
2
+
(
∂
Φ
∂
y
)
2
+
(
∂
Φ
∂
z
)
2
)
]
d
z
+
1
2
g
(
ζ
2
-
h
2
)
}
,
Doc 43
0.0672, -61.0000, 4.0000, 0.0672
testing/wikipedia/v3/23214.html
(
∂
z
∂
x
)
y
=
-
(
∂
z
∂
y
)
x
(
∂
y
∂
x
)
z
Doc 44
0.0622, -18.0000, 3.0000, 0.0622
testing/wikipedia/v3/14685.html
(
∂
y
∂
x
)
z
=
-
(
∂
z
∂
x
)
y
(
∂
z
∂
y
)
x
.
Doc 45
0.0622, -20.0000, 3.0000, 0.1866
testing/wikipedia/v3/04601.html
M
(
z
)
=
z
1
-
z
1
-
z
1
¯
z
,
φ
(
z
)
=
f
(
z
1
)
-
z
1
-
f
(
z
1
)
¯
z
.
Doc 29
0.0904, -30.0000, 4.0000, 0.3335
testing/wikipedia/v3/08430.html
d
z
=
[
(
∂
z
∂
x
)
y
(
∂
x
∂
y
)
z
+
(
∂
z
∂
y
)
x
]
d
y
+
(
∂
z
∂
x
)
y
(
∂
x
∂
z
)
y
d
z
,
Doc 45
0.0622, -20.0000, 3.0000, 0.1866
testing/wikipedia/v3/04601.html
[
1
-
(
∂
z
∂
x
)
y
(
∂
x
∂
z
)
y
]
d
z
=
[
(
∂
z
∂
x
)
y
(
∂
x
∂
y
)
z
+
(
∂
z
∂
y
)
x
]
d
y
.
Doc 45
0.0622, -20.0000, 3.0000, 0.1866
testing/wikipedia/v3/04601.html
ξ
=
∫
f
3
(
x
)
E
2
d
x
,
u
=
(
y
+
f
2
(
x
)
3
f
3
(
x
)
)
E
-
1
,
E
=
exp
(
∫
(
f
1
(
x
)
-
f
2
2
(
x
)
3
f
3
(
x
)
)
d
x
)
Doc 46
0.0622, -51.0000, 3.0000, 0.0622
testing/wikipedia/v3/27863.html
x
1
=
f
1
(
t
)
g
1
(
t
)
⋮
x
n
=
f
n
(
t
)
g
n
(
t
)
,
Doc 47
0.0480, -23.0000, 2.0000, 0.0480
testing/wikipedia/v3/04243.html
{
∂
∂
z
1
=
1
2
(
∂
∂
x
1
-
i
∂
∂
y
1
)
⋮
∂
∂
z
n
=
1
2
(
∂
∂
x
n
-
i
∂
∂
y
n
)
,
Doc 48
0.0480, -45.0000, 3.0000, 0.0480
testing/wikipedia/v3/25912.html