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H
i
j
=
[
∂
2
V
i
j
∂
x
i
∂
x
j
∂
2
V
i
j
∂
x
i
∂
y
j
∂
2
V
i
j
∂
x
i
∂
z
j
∂
2
V
i
j
∂
y
i
∂
x
j
∂
2
V
i
j
∂
y
i
∂
y
j
∂
2
V
i
j
∂
y
i
∂
z
j
∂
2
V
i
j
∂
z
i
∂
x
j
∂
2
V
i
j
∂
z
i
∂
y
j
∂
2
V
i
j
∂
z
i
∂
z
j
]
Search
Returned 96 matches (100 formulae, 64 docs)
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H
i
j
=
[
∂
2
V
i
j
∂
x
i
∂
x
j
∂
2
V
i
j
∂
x
i
∂
y
j
∂
2
V
i
j
∂
x
i
∂
z
j
∂
2
V
i
j
∂
y
i
∂
x
j
∂
2
V
i
j
∂
y
i
∂
y
j
∂
2
V
i
j
∂
y
i
∂
z
j
∂
2
V
i
j
∂
z
i
∂
x
j
∂
2
V
i
j
∂
z
i
∂
y
j
∂
2
V
i
j
∂
z
i
∂
z
j
]
Doc 1
1.0000, 1.0855
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Anisotropic_Network_Model.html
Hess
(
F
)
=
(
∂
2
F
∂
x
2
∂
2
F
∂
x
∂
y
∂
2
F
∂
x
∂
z
∂
2
F
∂
x
∂
y
∂
2
F
∂
y
2
∂
2
F
∂
y
∂
z
∂
2
F
∂
x
∂
z
∂
2
F
∂
y
∂
z
∂
2
F
∂
z
2
)
.
Doc 2
0.3754, 0.4243
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Mean_curvature.html
[
d
x
1
*
d
p
1
d
x
2
*
d
p
1
d
y
2
*
d
p
1
]
=
[
∂
2
U
x
∂
x
1
∂
x
1
∂
2
U
x
∂
x
1
∂
x
2
∂
2
U
x
∂
x
1
∂
y
2
∂
2
U
x
∂
x
1
∂
x
2
∂
2
U
x
∂
x
2
∂
x
2
∂
2
U
x
∂
y
2
∂
x
2
∂
2
U
y
∂
x
1
∂
y
2
∂
2
U
y
∂
x
2
∂
y
2
∂
2
U
y
∂
y
2
∂
y
2
]
-
1
[
-
∂
2
U
x
∂
p
1
∂
x
1
-
∂
2
U
x
∂
p
1
∂
x
2
-
∂
2
U
y
∂
p
1
∂
y
2
]
Doc 3
0.2765, 0.2765
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Strategic_complements.html
[
∂
x
1
∂
x
∂
x
1
∂
y
∂
x
1
∂
z
∂
x
2
∂
x
∂
x
2
∂
y
∂
x
2
∂
z
∂
x
3
∂
x
∂
x
3
∂
y
∂
x
3
∂
z
]
Doc 4
0.2484, 0.2484
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Fourier_series.html
H
(
f
)
=
[
∂
2
f
∂
x
2
∂
2
f
∂
x
∂
y
∂
2
f
∂
x
∂
z
∂
2
f
∂
y
∂
x
∂
2
f
∂
y
2
∂
2
f
∂
y
∂
z
∂
2
f
∂
z
∂
x
∂
2
f
∂
z
∂
y
∂
2
f
∂
z
2
]
,
Doc 5
0.2478, 0.2478
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Polar_curve.html
𝐉
𝐅
(
r
,
θ
,
φ
)
=
[
∂
x
∂
r
∂
x
∂
θ
∂
x
∂
φ
∂
y
∂
r
∂
y
∂
θ
∂
y
∂
φ
∂
z
∂
r
∂
z
∂
θ
∂
z
∂
φ
]
=
[
sin
θ
cos
φ
r
cos
θ
cos
φ
-
r
sin
θ
sin
φ
sin
θ
sin
φ
r
cos
θ
sin
φ
r
sin
θ
cos
φ
cos
θ
-
r
sin
θ
0
]
.
Doc 6
0.1928, 0.5490
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
(
x
¯
1
x
¯
2
x
¯
3
)
=
(
∂
x
¯
1
∂
x
1
∂
x
¯
1
∂
x
2
∂
x
¯
1
∂
x
3
∂
x
¯
2
∂
x
1
∂
x
¯
2
∂
x
2
∂
x
¯
2
∂
x
3
∂
x
¯
3
∂
x
1
∂
x
¯
3
∂
x
2
∂
x
¯
3
∂
x
3
)
(
x
1
x
2
x
3
)
Doc 7
0.1774, 0.1774
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Cartesian_tensor.html
𝐉
=
[
∂
x
1
∂
q
1
∂
x
1
∂
q
2
∂
x
1
∂
q
3
∂
x
2
∂
q
1
∂
x
2
∂
q
2
∂
x
2
∂
q
3
∂
x
3
∂
q
1
∂
x
3
∂
q
2
∂
x
3
∂
q
3
]
,
𝐉
-
1
=
[
∂
q
1
∂
x
1
∂
q
1
∂
x
2
∂
q
1
∂
x
3
∂
q
2
∂
x
1
∂
q
2
∂
x
2
∂
q
2
∂
x
3
∂
q
3
∂
x
1
∂
q
3
∂
x
2
∂
q
3
∂
x
3
]
Doc 8
0.1631, 0.1631
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Curvilinear_coordinates.html
H
(
f
)
=
[
∂
2
f
∂
x
2
∂
2
f
∂
x
∂
y
∂
2
f
∂
y
∂
x
∂
2
f
∂
y
2
]
.
Doc 9
0.1568, 0.2170
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Invariant_of_a_binary_form.html
𝐉
𝐅
(
x
1
,
x
2
,
x
3
)
=
[
∂
y
1
∂
x
1
∂
y
1
∂
x
2
∂
y
1
∂
x
3
∂
y
2
∂
x
1
∂
y
2
∂
x
2
∂
y
2
∂
x
3
∂
y
3
∂
x
1
∂
y
3
∂
x
2
∂
y
3
∂
x
3
∂
y
4
∂
x
1
∂
y
4
∂
x
2
∂
y
4
∂
x
3
]
=
[
1
0
0
0
0
5
0
8
x
2
-
2
x
3
cos
x
1
0
sin
x
1
]
.
Doc 6
0.1928, 0.5490
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
H
(
f
,
g
)
=
[
0
∂
g
∂
x
1
∂
g
∂
x
2
⋯
∂
g
∂
x
n
∂
g
∂
x
1
∂
2
f
∂
x
1
2
∂
2
f
∂
x
1
∂
x
2
⋯
∂
2
f
∂
x
1
∂
x
n
∂
g
∂
x
2
∂
2
f
∂
x
2
∂
x
1
∂
2
f
∂
x
2
2
⋯
∂
2
f
∂
x
2
∂
x
n
⋮
⋮
⋮
⋱
⋮
∂
g
∂
x
n
∂
2
f
∂
x
n
∂
x
1
∂
2
f
∂
x
n
∂
x
2
⋯
∂
2
f
∂
x
n
2
]
Doc 10
0.1029, 0.1995
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Hessian_matrix.html
[
ε
x
x
ε
x
y
ε
x
z
ε
y
x
ε
y
y
ε
y
z
ε
z
x
ε
z
y
ε
z
z
]
=
[
∂
u
x
∂
x
1
2
(
∂
u
x
∂
y
+
∂
u
y
∂
x
)
1
2
(
∂
u
x
∂
z
+
∂
u
z
∂
x
)
1
2
(
∂
u
y
∂
x
+
∂
u
x
∂
y
)
∂
u
y
∂
y
1
2
(
∂
u
y
∂
z
+
∂
u
z
∂
y
)
1
2
(
∂
u
z
∂
x
+
∂
u
x
∂
z
)
1
2
(
∂
u
z
∂
y
+
∂
u
y
∂
z
)
∂
u
z
∂
z
]
Doc 11
0.1006, 0.1601
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Infinitesimal_strain_theory.html
H
=
[
∂
2
f
∂
x
1
2
∂
2
f
∂
x
1
∂
x
2
⋯
∂
2
f
∂
x
1
∂
x
n
∂
2
f
∂
x
2
∂
x
1
∂
2
f
∂
x
2
2
⋯
∂
2
f
∂
x
2
∂
x
n
⋮
⋮
⋱
⋮
∂
2
f
∂
x
n
∂
x
1
∂
2
f
∂
x
n
∂
x
2
⋯
∂
2
f
∂
x
n
2
]
.
Doc 10
0.1029, 0.1995
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Hessian_matrix.html
∂
f
∂
x
(
0
,
r
o
)
=
0
,
∂
2
f
∂
x
2
(
0
,
r
o
)
=
0
,
∂
3
f
∂
x
3
(
0
,
r
o
)
≠
0
,
∂
f
∂
r
(
0
,
r
o
)
=
0
,
∂
2
f
∂
r
∂
x
(
0
,
r
o
)
≠
0.
Doc 12
0.0922, 0.0922
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Pitchfork_bifurcation.html
∂
ϕ
(
𝐗
)
∂
𝐗
=
[
∂
ϕ
∂
x
1
,
1
⋯
∂
ϕ
∂
x
1
,
q
⋮
⋱
⋮
∂
ϕ
∂
x
n
,
1
⋯
∂
ϕ
∂
x
n
,
q
]
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
D
f
(
x
,
y
)
=
[
∂
u
∂
x
∂
u
∂
y
∂
v
∂
x
∂
v
∂
y
]
Doc 14
0.0794, 0.0794
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Cauchy–Riemann_equations.html
ε
=
[
∂
v
1
∂
S
1
⋯
∂
v
1
∂
S
m
⋮
⋱
⋮
∂
v
n
∂
S
1
⋯
∂
v
n
∂
S
m
]
.
Doc 15
0.0788, 0.0788
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Elasticity_coefficient.html
𝐉
=
d
𝐟
d
𝐱
=
[
∂
𝐟
∂
x
1
⋯
∂
𝐟
∂
x
n
]
=
[
∂
f
1
∂
x
1
⋯
∂
f
1
∂
x
n
⋮
⋱
⋮
∂
f
m
∂
x
1
⋯
∂
f
m
∂
x
n
]
Doc 6
0.1928, 0.5490
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
∂
𝐅
∂
𝐗
=
[
∂
𝐅
∂
X
1
,
1
⋯
∂
𝐅
∂
X
n
,
1
⋮
⋱
⋮
∂
𝐅
∂
X
1
,
m
⋯
∂
𝐅
∂
X
n
,
m
]
,
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
𝐅
(
𝐗
)
∂
𝐗
=
[
∂
f
1
,
1
∂
𝐗
⋯
∂
f
1
,
p
∂
𝐗
⋮
⋱
⋮
∂
f
m
,
1
∂
𝐗
⋯
∂
f
m
,
p
∂
𝐗
]
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
𝐉
𝐟
(
x
,
y
)
=
[
∂
f
1
∂
x
∂
f
1
∂
y
∂
f
2
∂
x
∂
f
2
∂
y
]
=
[
2
x
y
x
2
5
cos
y
]
Doc 6
0.1928, 0.5490
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
Ω
=
|
∂
∂
x
11
⋯
∂
∂
x
1
n
⋮
⋱
⋮
∂
∂
x
n
1
⋯
∂
∂
x
n
n
|
.
Doc 16
0.0719, 0.0719
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Cayley's_Ω_process.html
δ
=
1
4
π
∬
R
s
t
|
x
y
z
∂
x
∂
s
∂
y
∂
s
∂
z
∂
s
∂
x
∂
t
∂
y
∂
t
∂
z
∂
t
|
(
x
2
+
y
2
+
z
2
)
x
2
+
y
2
+
z
2
d
s
d
t
.
Doc 17
0.0697, 0.0697
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Kronecker_delta.html
J
=
[
∂
u
∂
u
′
∂
u
∂
v
′
∂
v
∂
u
′
∂
v
∂
v
′
]
.
Doc 18
0.0659, 0.2323
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
U
=
[
1
3
1
3
1
3
ω
3
1
3
ω
¯
3
ω
¯
3
1
3
ω
3
]
⇒
(
|
U
i
α
|
2
)
=
[
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
]
Doc 19
0.0655, 0.0655
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Trimaximal_mixing.html
J
=
[
∂
Δ
P
∂
θ
∂
Δ
P
∂
|
V
|
∂
Δ
Q
∂
θ
∂
Δ
Q
∂
|
V
|
]
Doc 20
0.0652, 0.0652
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Power-flow_study.html
[
d
u
d
v
]
=
[
∂
u
∂
u
′
∂
u
∂
v
′
∂
v
∂
u
′
∂
v
∂
v
′
]
[
d
u
′
d
v
′
]
Doc 18
0.0659, 0.2323
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
|
∂
U
∂
Y
∂
U
∂
Z
∂
V
∂
Y
∂
V
∂
Z
|
=
|
Z
Y
0
1
|
=
|
Z
|
.
Doc 21
0.0628, 0.0628
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Probability_density_function.html
𝐉
(
r
,
φ
)
=
[
∂
x
∂
r
∂
x
∂
φ
∂
y
∂
r
∂
y
∂
φ
]
=
[
cos
φ
-
r
sin
φ
sin
φ
r
cos
φ
]
Doc 6
0.1928, 0.5490
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
[
|
U
e
1
|
2
|
U
e
2
|
2
|
U
e
3
|
2
|
U
μ
1
|
2
|
U
μ
2
|
2
|
U
μ
3
|
2
|
U
τ
1
|
2
|
U
τ
2
|
2
|
U
τ
3
|
2
]
=
[
2
3
1
3
0
1
6
1
3
1
2
1
6
1
3
1
2
]
.
Doc 22
0.0604, 0.0604
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Tribimaximal_mixing.html
∇
f
=
∂
f
∂
𝐱
=
[
∂
f
∂
x
1
∂
f
∂
x
2
∂
f
∂
x
3
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
det
[
∂
f
∂
x
∂
f
∂
y
∂
g
∂
x
∂
g
∂
y
]
Doc 9
0.1568, 0.2170
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Invariant_of_a_binary_form.html
ε
i
j
=
1
2
(
u
i
,
j
+
u
j
,
i
)
=
[
ε
11
ε
12
ε
13
ε
21
ε
22
ε
23
ε
31
ε
32
ε
33
]
=
[
∂
u
1
∂
x
1
1
2
(
∂
u
1
∂
x
2
+
∂
u
2
∂
x
1
)
1
2
(
∂
u
1
∂
x
3
+
∂
u
3
∂
x
1
)
1
2
(
∂
u
2
∂
x
1
+
∂
u
1
∂
x
2
)
∂
u
2
∂
x
2
1
2
(
∂
u
2
∂
x
3
+
∂
u
3
∂
x
2
)
1
2
(
∂
u
3
∂
x
1
+
∂
u
1
∂
x
3
)
1
2
(
∂
u
3
∂
x
2
+
∂
u
2
∂
x
3
)
∂
u
3
∂
x
3
]
Doc 11
0.1006, 0.1601
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Infinitesimal_strain_theory.html
(
∂
f
1
∂
x
∂
f
1
∂
y
∂
f
2
∂
x
∂
f
2
∂
y
)
(
δ
x
δ
y
)
=
(
-
f
1
-
f
2
)
Doc 23
0.0576, 0.0576
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Intersection_(Euclidean_geometry).html
J
=
[
∂
x
(
R
,
θ
)
∂
R
∂
x
(
R
,
θ
)
∂
θ
∂
y
(
R
,
θ
)
∂
R
∂
y
(
R
,
θ
)
∂
θ
]
=
[
cos
θ
-
R
sin
θ
sin
θ
R
cos
θ
]
.
Doc 24
0.0563, 0.0563
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Implicit_function_theorem.html
V
=
∇
f
=
(
∂
f
∂
x
1
,
∂
f
∂
x
2
,
∂
f
∂
x
3
,
…
,
∂
f
∂
x
n
)
.
Doc 25
0.0561, 0.0561
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Vector_field.html
[
E
′
F
′
F
′
G
′
]
=
[
∂
u
∂
u
′
∂
u
∂
v
′
∂
v
∂
u
′
∂
v
∂
v
′
]
T
[
E
F
F
G
]
[
∂
u
∂
u
′
∂
u
∂
v
′
∂
v
∂
u
′
∂
v
∂
v
′
]
Doc 18
0.0659, 0.2323
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
∂
2
V
i
j
∂
x
i
∂
y
j
=
-
γ
s
i
j
2
(
x
j
-
x
i
)
(
y
j
-
y
i
)
Doc 1
1.0000, 1.0855
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Anisotropic_Network_Model.html
𝐀
=
[
∂
F
1
(
q
→
)
∂
p
1
∂
F
1
(
q
→
)
∂
p
2
⋯
∂
F
1
(
q
→
)
∂
p
n
∂
F
2
(
q
→
)
∂
p
1
⋯
∂
F
2
(
q
→
)
∂
p
n
-
1
∂
F
2
(
q
→
)
∂
p
n
⋮
∂
F
j
(
q
→
)
∂
p
i
⋮
⋮
∂
F
m
(
q
→
)
∂
p
1
∂
F
m
(
q
→
)
∂
p
2
⋯
∂
F
m
(
q
→
)
∂
p
n
]
Doc 26
0.0531, 0.0531
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Linear_seismic_inversion.html
∂
x
∂
u
=
|
-
∂
F
∂
u
∂
F
∂
y
-
∂
G
∂
u
∂
G
∂
y
|
|
∂
F
∂
x
∂
F
∂
y
∂
G
∂
x
∂
G
∂
y
|
.
Doc 27
0.0507, 0.0507
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Cramer's_rule.html
∂
y
∂
𝐗
=
[
∂
y
∂
x
11
∂
y
∂
x
12
⋯
∂
y
∂
x
1
q
∂
y
∂
x
21
∂
y
∂
x
22
⋯
∂
y
∂
x
2
q
⋮
⋮
⋱
⋮
∂
y
∂
x
p
1
∂
y
∂
x
p
2
⋯
∂
y
∂
x
p
q
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
y
∂
𝐗
=
[
∂
y
∂
x
11
∂
y
∂
x
21
⋯
∂
y
∂
x
p
1
∂
y
∂
x
12
∂
y
∂
x
22
⋯
∂
y
∂
x
p
2
⋮
⋮
⋱
⋮
∂
y
∂
x
1
q
∂
y
∂
x
2
q
⋯
∂
y
∂
x
p
q
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
𝐘
∂
x
=
[
∂
y
11
∂
x
∂
y
12
∂
x
⋯
∂
y
1
n
∂
x
∂
y
21
∂
x
∂
y
22
∂
x
⋯
∂
y
2
n
∂
x
⋮
⋮
⋱
⋮
∂
y
m
1
∂
x
∂
y
m
2
∂
x
⋯
∂
y
m
n
∂
x
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∇
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
,
Doc 28
0.0505, 0.0505
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Notation_for_differentiation.html
∂
𝐲
∂
𝐱
=
[
∂
y
1
∂
x
1
∂
y
1
∂
x
2
⋯
∂
y
1
∂
x
n
∂
y
2
∂
x
1
∂
y
2
∂
x
2
⋯
∂
y
2
∂
x
n
⋮
⋮
⋱
⋮
∂
y
m
∂
x
1
∂
y
m
∂
x
2
⋯
∂
y
m
∂
x
n
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
𝐲
∂
𝐱
=
[
∂
y
1
∂
x
1
∂
y
2
∂
x
1
⋯
∂
y
m
∂
x
1
∂
y
1
∂
x
2
∂
y
2
∂
x
2
⋯
∂
y
m
∂
x
2
⋮
⋮
⋱
⋮
∂
y
1
∂
x
n
∂
y
2
∂
x
n
⋯
∂
y
m
∂
x
n
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
Σ
∂
θ
m
=
[
∂
Σ
1
,
1
∂
θ
m
∂
Σ
1
,
2
∂
θ
m
⋯
∂
Σ
1
,
N
∂
θ
m
∂
Σ
2
,
1
∂
θ
m
∂
Σ
2
,
2
∂
θ
m
⋯
∂
Σ
2
,
N
∂
θ
m
⋮
⋮
⋱
⋮
∂
Σ
N
,
1
∂
θ
m
∂
Σ
N
,
2
∂
θ
m
⋯
∂
Σ
N
,
N
∂
θ
m
]
.
Doc 29
0.0496, 0.0496
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Fisher_information.html
∇
F
=
(
∂
F
∂
x
,
∂
F
∂
y
,
∂
F
∂
z
)
Doc 2
0.3754, 0.4243
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Mean_curvature.html
D
φ
=
[
∂
φ
1
∂
x
1
∂
φ
1
∂
x
2
…
∂
φ
1
∂
x
n
∂
φ
2
∂
x
1
∂
φ
2
∂
x
2
…
∂
φ
2
∂
x
n
⋮
⋮
⋱
⋮
∂
φ
m
∂
x
1
∂
φ
m
∂
x
2
…
∂
φ
m
∂
x
n
]
.
Doc 18
0.0659, 0.2323
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
σ
i
j
=
-
(
p
0
0
0
p
0
0
0
p
)
+
μ
(
2
∂
u
∂
x
∂
u
∂
y
+
∂
v
∂
x
∂
u
∂
z
+
∂
w
∂
x
∂
v
∂
x
+
∂
u
∂
y
2
∂
v
∂
y
∂
v
∂
z
+
∂
w
∂
y
∂
w
∂
x
+
∂
u
∂
z
∂
w
∂
y
+
∂
v
∂
z
2
∂
w
∂
z
)
=
-
p
I
+
μ
(
∇
𝐯
+
(
∇
𝐯
)
T
)
Doc 30
0.0469, 0.0469
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Capillary_surface.html
=
-
(
∂
2
∂
θ
1
2
∂
2
∂
θ
1
∂
θ
2
⋯
∂
2
∂
θ
1
∂
θ
n
∂
2
∂
θ
2
∂
θ
1
∂
2
∂
θ
2
2
⋯
∂
2
∂
θ
2
∂
θ
n
⋮
⋮
⋱
⋮
∂
2
∂
θ
n
∂
θ
1
∂
2
∂
θ
n
∂
θ
2
⋯
∂
2
∂
θ
n
2
)
ℓ
(
θ
)
|
θ
=
θ
*
Doc 31
0.0451, 0.0451
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Observed_information.html
X
=
|
x
11
x
12
x
13
⋯
x
1
n
x
12
x
22
x
23
⋯
x
2
n
x
13
x
23
x
33
⋯
x
3
n
⋮
⋮
⋮
⋱
⋮
x
1
n
x
2
n
x
3
n
⋯
x
n
n
|
,
D
=
|
2
∂
∂
x
11
∂
∂
x
12
∂
∂
x
13
⋯
∂
∂
x
1
n
∂
∂
x
12
2
∂
∂
x
22
∂
∂
x
23
⋯
∂
∂
x
2
n
∂
∂
x
13
∂
∂
x
23
2
∂
∂
x
33
⋯
∂
∂
x
3
n
⋮
⋮
⋮
⋱
⋮
∂
∂
x
1
n
∂
∂
x
2
n
∂
∂
x
3
n
⋯
2
∂
∂
x
n
n
|
Doc 32
0.0437, 0.0437
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Capelli's_identity.html
∂
a
=
[
1
c
∂
∂
t
,
∂
∂
x
,
∂
∂
y
,
∂
∂
z
]
Doc 33
0.0432, 0.0432
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Lorentz_covariance.html
∇
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
=
e
→
x
∂
∂
x
+
e
→
y
∂
∂
y
+
e
→
z
∂
∂
z
Doc 34
0.0430, 0.0430
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Del.html
𝐉
=
(
∂
σ
1
∂
λ
1
∂
σ
1
∂
λ
2
…
∂
σ
1
∂
λ
n
∂
σ
2
∂
λ
1
∂
σ
2
∂
λ
2
…
∂
σ
2
∂
λ
n
⋮
⋮
⋱
⋮
∂
σ
n
∂
λ
1
∂
σ
n
∂
λ
2
…
∂
σ
n
∂
λ
n
)
.
Doc 35
0.0424, 0.0424
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Constraint_algorithm.html
∇
h
=
(
∂
h
∂
x
,
∂
h
∂
y
,
∂
h
∂
z
)
=
∂
h
∂
x
𝐢
+
∂
h
∂
y
𝐣
+
∂
h
∂
z
𝐤
Doc 36
0.0421, 0.0421
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Hydraulic_head.html
(
∂
Φ
∂
x
∂
Φ
∂
y
∂
Φ
∂
z
)
≈
(
f
∂
φ
∂
x
f
∂
φ
∂
y
∂
f
∂
z
φ
)
.
Doc 37
0.0421, 0.0421
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mild-slope_equation.html
∂
2
f
∂
x
i
∂
x
j
=
∂
2
f
∂
x
j
∂
x
i
.
Doc 38
0.0399, 0.0399
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Partial_derivative.html
∂
2
u
∂
x
i
∂
y
j
-
∂
2
u
∂
y
i
∂
x
j
=
0
Doc 39
0.0397, 0.0397
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/John's_equation.html
∂
2
Φ
∂
x
i
∂
x
i
-
M
i
M
j
∂
2
Φ
∂
x
i
∂
x
j
=
0.
Doc 40
0.0385, 0.0385
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Potential_flow.html
Δ
z
≈
(
∂
z
∂
x
1
∂
z
∂
x
2
∂
z
∂
x
3
⋯
∂
z
∂
x
p
)
(
Δ
x
1
Δ
x
2
Δ
x
3
⋮
Δ
x
p
)
𝐄𝐪
(
𝟖
)
Doc 41
0.0372, 0.0372
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Experimental_uncertainty_analysis.html
∂
2
f
∂
x
i
∂
x
j
(
a
1
,
…
,
a
n
)
=
∂
2
f
∂
x
j
∂
x
i
(
a
1
,
…
,
a
n
)
.
Doc 42
0.0369, 0.0369
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Symmetry_of_second_derivatives.html
∂
2
∂
x
k
∂
x
j
H
=
∂
2
∂
x
j
∂
x
k
H
.
Doc 43
0.0341, 0.0341
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Eikonal_equation.html
(
1
c
∂
ϕ
∂
t
′
∂
ϕ
∂
x
′
∂
ϕ
∂
y
′
∂
ϕ
∂
z
′
)
=
(
1
c
∂
ϕ
∂
t
∂
ϕ
∂
x
∂
ϕ
∂
y
∂
ϕ
∂
z
)
(
γ
-
β
γ
0
0
-
β
γ
γ
0
0
0
0
1
0
0
0
0
1
)
.
Doc 44
0.0334, 0.0334
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Special_relativity.html
∂
u
i
∂
t
+
∂
u
i
u
j
∂
x
j
=
-
∂
P
∂
x
i
+
ν
∂
2
u
i
∂
x
j
∂
x
j
+
f
i
Doc 45
0.0327, 0.0327
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Discretization_of_Navier–Stokes_equations.html
ρ
(
∂
u
i
∂
t
+
u
j
∂
u
i
∂
x
j
)
=
-
∂
p
∂
x
i
+
μ
∂
2
u
i
∂
x
j
∂
x
j
+
f
i
Doc 46
0.0325, 0.0325
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Computational_Fluid_Dynamics_for_Phase_Change_Materials.html
∂
u
i
∂
t
+
∂
u
i
u
j
∂
x
j
=
-
1
ρ
∂
p
∂
x
i
+
ν
∂
2
u
i
∂
x
j
∂
x
j
.
Doc 47
0.0323, 0.1475
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
∂
2
V
i
j
∂
x
i
2
=
∂
2
V
i
j
∂
x
j
2
=
γ
s
i
j
2
(
x
j
-
x
i
)
2
Doc 1
1.0000, 1.0855
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Anisotropic_Network_Model.html
ω
→
=
∇
×
v
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
×
(
v
x
,
v
y
,
v
z
)
=
(
∂
v
z
∂
y
-
∂
v
y
∂
z
,
∂
v
x
∂
z
-
∂
v
z
∂
x
,
∂
v
y
∂
x
-
∂
v
x
∂
y
)
Doc 48
0.0318, 0.0318
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Vorticity.html
∂
u
i
¯
∂
t
+
∂
u
i
u
j
∂
x
j
¯
=
-
1
ρ
∂
p
¯
∂
x
i
+
ν
∂
2
u
i
¯
∂
x
j
∂
x
j
.
Doc 47
0.0323, 0.1475
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
∂
u
i
∂
t
+
u
j
∂
u
i
∂
x
j
=
f
i
-
1
ρ
∂
p
∂
x
i
+
ν
∂
2
u
i
∂
x
j
∂
x
j
Doc 49
0.0311, 0.0579
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Reynolds-averaged_Navier–Stokes_equations.html
∂
u
i
∂
t
¯
+
∂
u
i
u
j
∂
x
j
¯
=
-
1
ρ
∂
p
∂
x
i
¯
+
ν
∂
2
u
i
∂
x
j
∂
x
j
¯
.
Doc 47
0.0323, 0.1475
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
W
i
j
k
l
=
∂
2
F
i
j
∂
x
k
∂
x
l
+
∂
2
F
k
l
∂
x
i
∂
x
j
-
∂
2
F
i
l
∂
x
j
∂
x
k
-
∂
2
F
j
k
∂
x
i
∂
x
l
Doc 50
0.0310, 0.0310
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Saint-Venant's_compatibility_condition.html
(
∂
2
S
∂
y
∂
x
)
=
(
∂
2
S
∂
x
∂
y
)
:
(
∂
2
V
∂
y
∂
x
)
=
(
∂
2
V
∂
x
∂
y
)
Doc 51
0.0306, 0.0952
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
div
→
(
ϵ
¯
¯
)
=
[
∂
ϵ
x
x
∂
x
+
∂
ϵ
x
y
∂
y
+
∂
ϵ
x
z
∂
z
∂
ϵ
y
x
∂
x
+
∂
ϵ
y
y
∂
y
+
∂
ϵ
y
z
∂
z
∂
ϵ
z
x
∂
x
+
∂
ϵ
z
y
∂
y
+
∂
ϵ
z
z
∂
z
]
Doc 52
0.0302, 0.0302
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Divergence.html
∂
2
y
∂
x
i
∂
x
j
=
∑
k
(
∂
y
∂
u
k
∂
2
u
k
∂
x
i
∂
x
j
)
+
∑
k
,
ℓ
(
∂
2
y
∂
u
k
∂
u
ℓ
∂
u
k
∂
x
i
∂
u
ℓ
∂
x
j
)
.
Doc 53
0.0302, 0.0302
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Chain_rule.html
ρ
[
∂
u
i
¯
∂
t
+
∂
u
i
¯
u
j
¯
∂
x
j
+
∂
u
i
′
¯
u
j
′
¯
∂
x
j
]
=
-
∂
p
¯
∂
x
i
+
μ
∂
2
u
i
¯
∂
x
j
∂
x
j
.
Doc 54
0.0294, 0.0294
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Reynolds_stress.html
∂
𝐲
∂
x
=
[
∂
y
1
∂
x
∂
y
2
∂
x
⋮
∂
y
m
∂
x
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
y
∂
𝐱
=
[
∂
y
∂
x
1
∂
y
∂
x
2
⋮
∂
y
∂
x
n
]
.
Doc 13
0.0883, 0.7909
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
∂
u
i
¯
∂
t
+
u
j
¯
∂
u
i
¯
∂
x
j
=
-
1
ρ
∂
p
¯
∂
x
i
+
ν
∂
2
u
i
¯
∂
x
j
∂
x
j
-
∂
τ
i
j
∂
x
j
.
Doc 47
0.0323, 0.1475
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
∂
G
i
j
∂
x
k
=
(
∂
2
X
α
∂
x
i
∂
x
k
∂
X
β
∂
x
j
+
∂
X
α
∂
x
i
∂
2
X
β
∂
x
j
∂
x
k
)
g
α
β
+
∂
X
α
∂
x
i
∂
X
β
∂
x
j
∂
g
α
β
∂
x
k
Doc 55
0.0279, 0.0279
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Finite_strain_theory.html
D
⏟
n
×
p
=
[
∂
μ
1
∂
β
1
⋯
⋯
∂
μ
1
∂
β
p
∂
μ
2
∂
β
1
⋯
⋯
∂
μ
2
∂
β
p
⋮
⋮
∂
μ
m
∂
β
1
⋯
⋯
∂
μ
m
∂
β
p
]
V
⏟
n
×
n
=
diag
(
V
(
μ
1
)
,
V
(
μ
2
)
,
…
,
…
,
V
(
μ
n
)
)
Doc 56
0.0277, 0.0277
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Variance_function.html
∂
u
i
¯
∂
t
+
u
j
¯
∂
u
i
¯
∂
x
j
+
u
j
′
∂
u
i
′
∂
x
j
¯
=
f
i
¯
-
1
ρ
∂
p
¯
∂
x
i
+
ν
∂
2
u
i
¯
∂
x
j
∂
x
j
.
Doc 49
0.0311, 0.0579
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Reynolds-averaged_Navier–Stokes_equations.html
∂
2
S
(
s
y
m
b
o
l
ϕ
(
y
)
)
∂
y
i
∂
y
j
=
∑
l
,
k
=
1
n
∂
2
S
(
z
)
∂
z
k
∂
z
l
|
z
=
s
y
m
b
o
l
ϕ
(
y
)
∂
ϕ
k
∂
y
i
∂
ϕ
l
∂
y
j
+
∑
k
=
1
n
∂
S
(
z
)
∂
z
k
|
z
=
s
y
m
b
o
l
ϕ
(
y
)
∂
2
ϕ
k
∂
y
i
∂
y
j
Doc 57
0.0245, 0.0245
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Method_of_steepest_descent.html
∂
u
i
¯
∂
t
+
∂
u
i
¯
u
j
¯
∂
x
j
=
-
1
ρ
∂
p
¯
∂
x
i
+
ν
∂
2
u
i
¯
∂
x
j
∂
x
j
-
(
∂
u
i
u
j
∂
x
j
¯
-
∂
u
i
¯
u
j
¯
∂
x
j
)
.
Doc 47
0.0323, 0.1475
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
∂
2
f
∂
x
∂
y
,
∂
2
f
∂
x
∂
z
,
and
∂
2
f
∂
y
∂
z
.
Doc 58
0.0244, 0.0244
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Second_derivative.html
1
pf
(
A
)
∂
2
pf
(
A
)
∂
x
i
∂
x
j
=
1
2
tr
(
A
-
1
∂
2
A
∂
x
i
∂
x
j
)
-
1
2
tr
(
A
-
1
∂
A
∂
x
i
A
-
1
∂
A
∂
x
j
)
+
1
4
tr
(
A
-
1
∂
A
∂
x
i
)
tr
(
A
-
1
∂
A
∂
x
j
)
.
Doc 59
0.0229, 0.0229
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Pfaffian.html
σ
x
=
∂
2
Φ
y
y
∂
z
∂
z
+
∂
2
Φ
z
z
∂
y
∂
y
-
2
∂
2
Φ
y
z
∂
y
∂
z
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ
y
=
∂
2
Φ
x
x
∂
z
∂
z
+
∂
2
Φ
z
z
∂
x
∂
x
-
2
∂
2
Φ
z
x
∂
z
∂
x
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ
z
=
∂
2
Φ
y
y
∂
x
∂
x
+
∂
2
Φ
x
x
∂
y
∂
y
-
2
∂
2
Φ
x
y
∂
x
∂
y
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
∂
∂
y
(
∂
z
∂
x
)
y
=
∂
∂
x
(
∂
z
∂
y
)
x
=
∂
2
z
∂
y
∂
x
=
∂
2
z
∂
x
∂
y
Doc 51
0.0306, 0.0952
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
(
∂
2
U
∂
x
∂
y
)
=
(
∂
T
∂
x
)
y
(
∂
S
∂
y
)
x
+
T
(
∂
2
S
∂
x
∂
y
)
-
(
∂
P
∂
x
)
y
(
∂
V
∂
y
)
x
-
P
(
∂
2
V
∂
x
∂
y
)
Doc 51
0.0306, 0.0952
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
(
∂
2
U
∂
y
∂
x
)
=
(
∂
T
∂
y
)
x
(
∂
S
∂
x
)
y
+
T
(
∂
2
S
∂
y
∂
x
)
-
(
∂
P
∂
y
)
x
(
∂
V
∂
x
)
y
-
P
(
∂
2
V
∂
y
∂
x
)
Doc 51
0.0306, 0.0952
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
R
i
k
ℓ
m
=
1
2
(
∂
2
g
i
m
∂
x
k
∂
x
ℓ
+
∂
2
g
k
ℓ
∂
x
i
∂
x
m
-
∂
2
g
i
ℓ
∂
x
k
∂
x
m
-
∂
2
g
k
m
∂
x
i
∂
x
ℓ
)
Doc 61
0.0209, 0.0209
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/List_of_formulas_in_Riemannian_geometry.html
σ
x
y
=
-
∂
2
Φ
x
y
∂
z
∂
z
-
∂
2
Φ
z
z
∂
x
∂
y
+
∂
2
Φ
y
z
∂
x
∂
z
+
∂
2
Φ
z
x
∂
y
∂
z
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ
y
z
=
-
∂
2
Φ
y
z
∂
x
∂
x
-
∂
2
Φ
x
x
∂
y
∂
z
+
∂
2
Φ
z
x
∂
y
∂
x
+
∂
2
Φ
x
y
∂
z
∂
x
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ
z
x
=
-
∂
2
Φ
z
x
∂
y
∂
y
-
∂
2
Φ
y
y
∂
z
∂
x
+
∂
2
Φ
x
y
∂
z
∂
y
+
∂
2
Φ
y
z
∂
x
∂
y
Doc 60
0.0225, 0.1280
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
∇
\cdotsymbol
τ
≡
∂
s
i
j
∂
x
i
=
μ
[
∂
∂
x
i
(
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
)
]
+
λ
[
∂
∂
x
i
(
∂
u
k
∂
x
k
)
]
δ
i
j
=
μ
∂
2
u
i
∂
x
i
∂
x
j
+
μ
∂
2
u
j
∂
x
i
∂
x
i
+
λ
∂
2
u
k
∂
x
k
∂
x
j
=
(
μ
+
λ
)
∂
2
u
i
∂
x
i
∂
x
j
+
μ
∂
2
u
j
∂
x
i
2
≡
(
μ
+
λ
)
∇
(
∇
⋅
𝐮
)
+
μ
∇
2
𝐮
.
Doc 62
0.0201, 0.0201
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Acoustic_theory.html
a
∂
2
u
∂
x
2
+
b
∂
2
u
∂
x
∂
y
+
c
∂
2
u
∂
y
2
+
d
∂
2
u
∂
y
∂
z
+
e
∂
2
u
∂
z
2
+ (lower-order terms)
=
0
,
Doc 63
0.0190, 0.0190
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Elliptic_partial_differential_equation.html
f
(
x
1
,
…
,
x
n
,
u
,
∂
u
∂
x
1
,
…
,
∂
u
∂
x
n
,
∂
2
u
∂
x
1
∂
x
1
,
…
,
∂
2
u
∂
x
1
∂
x
n
,
…
)
=
0.
Doc 64
0.0185, 0.0185
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Partial_differential_equation.html