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, 52 docs)
Lookup 193.083 ms, Re-ranking 36111.555 ms
Found 552996 tuple postings, 174581 formulae, 20949 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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
testing/wikipedia/v3/21960.html
Doc 2
0.1582, -12.0000, 9.0000, 0.3109
testing/wikipedia/v3/23338.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, 8.8772
testing/wikipedia/v3/21960.html
u
s
t
+
1
=
(
1
-
ε
)
f
(
u
s
t
)
+
ε
2
(
f
(
u
s
+
1
t
)
+
f
(
u
s
-
1
t
)
)
t
∈
ℕ
,
ε
∈
[
0
,
1
]
Doc 3
0.1582, -32.0000, 8.0000, 0.1582
testing/wikipedia/v3/23912.html
∂
∂
t
w
(
t
,
ξ
)
=
-
∂
∂
ξ
w
(
t
,
ξ
)
+
u
(
t
)
,
Doc 4
0.1527, -13.0000, 6.0000, 0.1527
testing/wikipedia/v3/12267.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, 8.8772
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, 8.8772
testing/wikipedia/v3/21960.html
Doc 2
0.1582, -12.0000, 9.0000, 0.3109
testing/wikipedia/v3/23338.html
f
y
(
𝐲
)
≜
[
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
0
]
Doc 1
0.8590, -13.0000, 58.0000, 8.8772
testing/wikipedia/v3/21960.html
σ
˙
=
∂
σ
∂
𝐱
𝐱
˙
︷
d
𝐱
d
t
=
∂
σ
∂
𝐱
(
f
(
𝐱
,
t
)
+
B
(
𝐱
,
t
)
𝐮
)
︷
𝐱
˙
Doc 5
0.1466, -24.0000, 5.0000, 0.2370
testing/wikipedia/v3/04357.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, 8.8772
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, 8.8772
testing/wikipedia/v3/21960.html
x
n
i
(
p
,
m
i
)
=
-
∂
v
i
(
p
,
m
i
)
∂
p
n
∂
v
i
(
p
,
m
i
)
∂
m
i
=
∂
f
i
(
p
)
∂
p
n
+
∂
g
(
p
)
∂
p
n
⋅
m
-
f
i
(
p
)
g
(
p
)
Doc 6
0.1386, -54.0000, 6.0000, 0.1386
testing/wikipedia/v3/11308.html
{
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
z
1
z
˙
1
=
u
1
Doc 1
0.8590, -13.0000, 58.0000, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
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, 8.8772
testing/wikipedia/v3/21960.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, 8.8772
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, 8.8772
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, 8.8772
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 7
0.1297, -27.0000, 3.0000, 0.3838
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 8
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 7
0.1297, -27.0000, 3.0000, 0.3838
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 9
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 7
0.1297, -27.0000, 3.0000, 0.3838
testing/wikipedia/v3/06206.html
𝐱
˙
=
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
Doc 1
0.8590, -13.0000, 58.0000, 8.8772
testing/wikipedia/v3/21960.html
𝐱
˙
=
F
(
𝐱
)
where
F
(
𝐱
)
≜
f
x
(
𝐱
)
+
g
x
(
𝐱
)
u
x
(
𝐱
)
Doc 1
0.8590, -13.0000, 58.0000, 8.8772
testing/wikipedia/v3/21960.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, 8.8772
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, 8.8772
testing/wikipedia/v3/21960.html
d
f
(
𝒂
)
(
𝒗
)
=
∂
f
∂
x
1
(
𝒂
)
v
1
+
⋯
+
∂
f
∂
x
n
(
𝒂
)
v
n
.
Doc 10
0.1153, -21.0000, 5.0000, 0.1153
testing/wikipedia/v3/01637.html
d
d
s
u
(
x
(
s
)
,
t
(
s
)
)
=
∂
u
∂
x
d
x
d
s
+
∂
u
∂
t
d
t
d
s
Doc 11
0.1153, -25.0000, 4.0000, 0.1153
testing/wikipedia/v3/06475.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, 8.8772
testing/wikipedia/v3/21960.html
∂
ln
B
(
α
,
β
)
∂
β
=
-
∂
ln
Γ
(
α
+
β
)
∂
β
+
∂
ln
Γ
(
α
)
∂
β
+
∂
ln
Γ
(
β
)
∂
β
=
-
ψ
(
α
+
β
)
+
0
+
ψ
(
β
)
Doc 12
0.1102, -44.0000, 6.0000, 0.2205
testing/wikipedia/v3/02956.html
∂
ln
B
(
α
,
β
)
∂
α
=
-
∂
ln
Γ
(
α
+
β
)
∂
α
+
∂
ln
Γ
(
α
)
∂
α
+
∂
ln
Γ
(
β
)
∂
α
=
-
ψ
(
α
+
β
)
+
ψ
(
α
)
+
0
Doc 12
0.1102, -44.0000, 6.0000, 0.2205
testing/wikipedia/v3/02956.html
m
∂
2
x
(
t
)
∂
t
2
=
-
6
π
R
η
∂
x
(
t
)
∂
t
-
F
l
x
(
t
)
+
f
(
t
)
Doc 13
0.1051, -24.0000, 4.0000, 0.1051
testing/wikipedia/v3/16266.html
∂
g
t
(
z
)
∂
t
=
g
t
(
z
)
ζ
(
t
)
+
g
t
(
z
)
ζ
(
t
)
-
g
t
(
z
)
.
Doc 14
0.1045, -23.0000, 4.0000, 0.1907
testing/wikipedia/v3/14856.html
Ψ
=
z
u
b
(
ξ
)
-
z
3
3
!
u
b
′′
(
ξ
)
+
z
5
5
!
u
b
iv
(
ξ
)
+
⋯
,
Doc 15
0.1008, -23.0000, 4.0000, 0.1008
testing/wikipedia/v3/24009.html
∂
∂
t
(
∇
2
ψ
)
+
∂
ψ
∂
y
∂
∂
x
(
∇
2
ψ
)
-
∂
ψ
∂
x
∂
∂
y
(
∇
2
ψ
)
=
ν
∇
4
ψ
Doc 16
0.1008, -32.0000, 6.0000, 0.1772
testing/wikipedia/v3/01538.html
1
2
σ
(
r
)
2
∂
2
P
∂
r
2
+
[
a
(
r
)
+
σ
(
r
)
+
φ
(
r
,
t
)
]
∂
P
∂
r
+
∂
P
∂
t
-
r
P
=
0
Doc 17
0.1008, -36.0000, 4.0000, 0.1008
testing/wikipedia/v3/06588.html
i
∂
f
∂
x
(
z
0
)
=
∂
f
∂
y
(
z
0
)
,
Doc 18
0.0960, -11.0000, 5.0000, 0.0960
testing/wikipedia/v3/00244.html
∂
u
∂
t
=
u
(
1
-
u
)
+
∂
2
u
∂
x
2
.
Doc 19
0.0960, -12.0000, 6.0000, 0.0960
testing/wikipedia/v3/21952.html
Doc 20
0.0960, -12.0000, 6.0000, 0.0960
testing/wikipedia/v3/09397.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 21
0.0960, -54.0000, 5.0000, 0.0960
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 22
0.0960, -55.0000, 5.0000, 0.0960
testing/wikipedia/v3/22820.html
∂
π
(
x
,
t
)
∂
x
=
∂
(
x
p
(
x
)
-
C
(
x
)
)
∂
x
-
t
=
0
Doc 23
0.0904, -20.0000, 3.0000, 0.1808
testing/wikipedia/v3/05933.html
σ
T
︷
∂
V
∂
σ
σ
˙
︷
d
σ
d
t
︸
d
V
d
t
<
0
(i.e.,
d
V
d
t
<
0
)
Doc 5
0.1466, -24.0000, 5.0000, 0.2370
testing/wikipedia/v3/04357.html
∂
2
(
x
p
(
x
)
-
C
(
x
)
)
∂
2
x
=
∂
2
π
(
x
,
t
)
∂
x
2
,
Doc 23
0.0904, -20.0000, 3.0000, 0.1808
testing/wikipedia/v3/05933.html
∂
∂
t
Q
i
(
t
)
=
-
1
Δ
x
(
f
(
q
(
t
,
x
i
+
1
/
2
)
)
-
f
(
q
(
t
,
x
i
-
1
/
2
)
)
)
,
Doc 24
0.0904, -34.0000, 4.0000, 0.0904
testing/wikipedia/v3/14299.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 25
0.0901, -51.0000, 4.0000, 0.0901
testing/wikipedia/v3/23998.html
𝓠
=
d
d
t
(
∂
T
∂
𝐪
˙
)
-
∂
T
∂
𝐪
,
Doc 26
0.0862, -12.0000, 4.0000, 0.0862
testing/wikipedia/v3/04933.html
J
(
x
,
t
)
=
i
ℏ
2
m
(
ψ
∂
ψ
*
∂
x
-
∂
ψ
∂
x
ψ
)
Doc 27
0.0862, -18.0000, 2.0000, 0.0862
testing/wikipedia/v3/26729.html
∂
f
t
(
z
)
∂
t
=
-
z
f
t
′
(
z
)
ζ
(
t
)
+
z
ζ
(
t
)
-
z
Doc 14
0.1045, -23.0000, 4.0000, 0.1907
testing/wikipedia/v3/14856.html
∂
ℒ
∂
w
-
∂
∂
t
(
∂
ℒ
∂
w
˙
)
+
∂
2
∂
x
2
(
∂
ℒ
∂
w
x
x
)
=
0
Doc 28
0.0862, -26.0000, 6.0000, 0.0862
testing/wikipedia/v3/09923.html
f
(
x
)
∂
2
u
∂
x
2
+
g
(
x
)
∂
u
∂
x
+
h
(
x
)
u
=
∂
u
∂
t
+
k
(
t
)
u
Doc 29
0.0862, -27.0000, 3.0000, 0.0862
testing/wikipedia/v3/05175.html
(
∂
A
)
S
=
-
(
∂
S
)
A
=
P
C
P
T
(
∂
V
∂
P
)
T
+
P
(
∂
V
∂
T
)
P
2
+
S
(
∂
V
∂
T
)
P
Doc 30
0.0862, -34.0000, 3.0000, 0.3156
testing/wikipedia/v3/08946.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 31
0.0862, -35.0000, 4.0000, 0.0862
testing/wikipedia/v3/08334.html
(
∂
U
∂
y
)
x
=
T
(
∂
S
∂
y
)
x
-
P
(
∂
V
∂
y
)
x
Doc 32
0.0817, -18.0000, 3.0000, 0.0817
testing/wikipedia/v3/06999.html
(
∂
G
)
V
=
-
(
∂
V
)
G
=
-
V
(
∂
V
∂
T
)
P
-
S
(
∂
V
∂
P
)
T
Doc 30
0.0862, -34.0000, 3.0000, 0.3156
testing/wikipedia/v3/08946.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 33
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, 8.8772
testing/wikipedia/v3/21960.html
∇
f
(
a
)
=
(
∂
f
∂
x
1
(
a
)
,
…
,
∂
f
∂
x
n
(
a
)
)
.
Doc 34
0.0763, -20.0000, 3.0000, 0.2108
testing/wikipedia/v3/01666.html
Doc 35
0.0763, -20.0000, 3.0000, 0.0763
testing/wikipedia/v3/00254.html
∂
∂
t
(
∇
2
ψ
)
+
∂
(
ψ
,
∇
2
ψ
)
∂
(
y
,
x
)
=
ν
∇
4
ψ
.
Doc 16
0.1008, -32.0000, 6.0000, 0.1772
testing/wikipedia/v3/01538.html
(
∂
U
)
T
=
-
(
∂
T
)
U
=
T
(
∂
V
∂
T
)
P
+
P
(
∂
V
∂
P
)
T
Doc 30
0.0862, -34.0000, 3.0000, 0.3156
testing/wikipedia/v3/08946.html
1
x
∂
∂
x
(
x
∂
∂
x
)
S
(
x
)
+
(
1
-
ν
2
x
2
)
S
(
x
)
=
0
Doc 36
0.0763, -24.0000, 3.0000, 0.0763
testing/wikipedia/v3/27834.html
u
(
∂
u
∂
s
)
+
v
(
∂
v
∂
y
)
=
ν
(
∂
2
u
∂
y
2
)
+
g
β
(
T
-
T
o
)
Doc 37
0.0763, -27.0000, 3.0000, 0.0763
testing/wikipedia/v3/01781.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
∂
f
(
𝒖
(
𝒙
)
)
∂
x
i
=
∑
k
=
1
p
∂
f
¯
(
𝒖
(
𝒙
)
)
∂
u
k
∂
u
k
(
𝒙
)
∂
x
i
∀
i
=
1
,
…
,
n
Doc 40
0.0763, -38.0000, 3.0000, 0.0763
testing/wikipedia/v3/04050.html
Doc 41
0.0763, -38.0000, 3.0000, 0.0763
testing/wikipedia/v3/24625.html
g
(
v
)
=
g
(
x
)
+
∑
k
=
1
∞
y
k
k
!
(
∂
∂
x
)
k
-
1
(
f
(
x
)
k
g
′
(
x
)
)
Doc 42
0.0748, -27.0000, 4.0000, 0.0748
testing/wikipedia/v3/07087.html
(
∂
S
)
V
=
-
(
∂
V
)
S
=
C
P
T
(
∂
V
∂
P
)
T
+
(
∂
V
∂
T
)
P
2
Doc 30
0.0862, -34.0000, 3.0000, 0.3156
testing/wikipedia/v3/08946.html
d
V
d
h
=
π
r
2
3
︷
∂
V
∂
h
+
2
π
r
h
3
︷
∂
V
∂
r
d
r
d
h
Doc 34
0.0763, -20.0000, 3.0000, 0.2108
testing/wikipedia/v3/01666.html
d
V
d
r
=
2
π
r
h
3
︷
∂
V
∂
r
+
π
r
2
3
︷
∂
V
∂
h
d
h
d
r
Doc 34
0.0763, -20.0000, 3.0000, 0.2108
testing/wikipedia/v3/01666.html
d
f
d
t
=
∂
f
∂
x
d
x
d
t
+
∂
f
∂
y
d
y
d
t
.
Doc 43
0.0672, -22.0000, 4.0000, 0.0672
testing/wikipedia/v3/07457.html
∂
f
∂
t
+
d
q
¯
d
t
⋅
∂
f
∂
q
¯
+
d
p
¯
d
t
⋅
∂
f
∂
p
¯
=
0
,
Doc 44
0.0672, -30.0000, 4.0000, 0.0672
testing/wikipedia/v3/09770.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 45
0.0672, -61.0000, 4.0000, 0.0672
testing/wikipedia/v3/23214.html
(
∂
U
∂
θ
)
-
d
d
z
(
∂
U
∂
(
d
θ
d
z
)
)
=
0
Doc 46
0.0622, -19.0000, 3.0000, 0.0622
testing/wikipedia/v3/25072.html
(
∂
x
∂
y
)
z
=
-
(
∂
z
∂
y
)
x
(
∂
z
∂
x
)
y
Doc 47
0.0622, -19.0000, 3.0000, 0.0622
testing/wikipedia/v3/14685.html
(
∂
y
∂
x
)
z
=
-
(
∂
z
∂
x
)
y
(
∂
z
∂
y
)
x
.
Doc 48
0.0622, -20.0000, 3.0000, 0.0622
testing/wikipedia/v3/04601.html
∂
atan2
(
y
,
x
)
∂
x
=
∂
arctan
(
y
/
x
)
∂
x
=
-
y
x
2
+
y
2
Doc 49
0.0622, -23.0000, 3.0000, 0.0622
testing/wikipedia/v3/12634.html
(
∂
P
∂
T
)
V
=
-
(
∂
V
∂
T
)
P
(
∂
V
∂
P
)
T
=
α
β
T
Doc 50
0.0622, -24.0000, 4.0000, 0.0622
testing/wikipedia/v3/22233.html
u
r
=
+
1
r
2
sin
(
θ
)
∂
Ψ
∂
θ
,
u
θ
=
-
1
r
sin
(
θ
)
∂
Ψ
∂
r
.
Doc 51
0.0622, -29.0000, 3.0000, 0.0622
testing/wikipedia/v3/22292.html
M
(
z
)
=
z
1
-
z
1
-
z
1
¯
z
,
φ
(
z
)
=
f
(
z
1
)
-
z
1
-
f
(
z
1
)
¯
z
.
Doc 52
0.0622, -31.0000, 3.0000, 0.0622
testing/wikipedia/v3/08430.html