Returned 94 matches (100 formulae, 64 docs)
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[ formulas ] [ documents ] [ documents-by-formula ]

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
0.4474, 0.5191
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
0.3405, 0.3405
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Strategic_complements.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 4
0.2981, 0.2981
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Polar_curve.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 5
0.2939, 0.2939
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Fourier_series.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 6
0.2350, 0.2350
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Cartesian_tensor.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 7
0.2230, 0.7034
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.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
0.2111, 0.2111
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
0.1940, 0.2764
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 7
0.2230, 0.7034
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.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 10
0.1383, 0.1383
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Pitchfork_bifurcation.html
ε = [ v 1 S 1 v 1 S m v n S 1 v n S m ] .
Doc 11
0.1334, 0.1334
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Elasticity_coefficient.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 12
0.1312, 0.2509
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Hessian_matrix.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 12
0.1312, 0.2509
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Hessian_matrix.html
J = [ Δ P θ Δ P | V | Δ Q θ Δ Q | V | ]
Doc 13
0.1153, 0.1153
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Power-flow_study.html
ϕ ( 𝐗 ) 𝐗 = [ ϕ x 1 , 1 ϕ x 1 , q ϕ x n , 1 ϕ x n , q ]
Doc 14
0.1137, 1.1164
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 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.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 15
0.1105, 0.1769
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Infinitesimal_strain_theory.html
𝐅 𝐗 = [ 𝐅 X 1 , 1 𝐅 X n , 1 𝐅 X 1 , m 𝐅 X n , m ] ,
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.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 16
0.1082, 0.1082
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Kronecker_delta.html
Ω = | x 11 x 1 n x n 1 x n n | .
Doc 17
0.1078, 0.1078
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Cayley's_Ω_process.html
J = [ u u u v v u v v ] .
Doc 18
0.1073, 0.3415
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.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 7
0.2230, 0.7034
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.html
D f ( x , y ) = [ u x u y v x v y ]
Doc 19
0.1026, 0.1026
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Cauchy–Riemann_equations.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 7
0.2230, 0.7034
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.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 20
0.0960, 0.0960
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Implicit_function_theorem.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 21
0.0899, 0.0899
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Fisher_information.html
[ d u d v ] = [ u u u v v u v v ] [ d u d v ]
Doc 18
0.1073, 0.3415
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.html
𝐉 ( r , φ ) = [ x r x φ y r y φ ] = [ cos φ - r sin φ sin φ r cos φ ]
Doc 7
0.2230, 0.7034
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Jacobian_matrix_and_determinant.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 22
0.0890, 0.0890
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Linear_seismic_inversion.html
( f 1 x f 1 y f 2 x f 2 y ) ( δ x δ y ) = ( - f 1 - f 2 )
Doc 23
0.0847, 0.0847
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Intersection_(Euclidean_geometry).html
f = f 𝐱 = [ f x 1 f x 2 f x 3 ] .
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
| U Y U Z V Y V Z | = | Z Y 0 1 | = | Z | .
Doc 24
0.0829, 0.0829
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Probability_density_function.html
det [ f x f y g x g y ]
Doc 9
0.1940, 0.2764
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Invariant_of_a_binary_form.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 25
0.0794, 0.0794
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Constraint_algorithm.html
V = f = ( f x 1 , f x 2 , f x 3 , , f x n ) .
Doc 26
0.0773, 0.0773
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
0.1073, 0.3415
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 27
0.0719, 0.0719
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Trimaximal_mixing.html
F = ( F x , F y , F z )
Doc 2
0.4474, 0.5191
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Mean_curvature.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 14
0.1137, 1.1164
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 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 14
0.1137, 1.1164
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 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.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 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 14
0.1137, 1.1164
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 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
x u = | - F u F y - G u G y | | F x F y G x G y | .
Doc 28
0.0694, 0.0694
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Cramer's_rule.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
0.1073, 0.3415
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Metric_tensor.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 15
0.1105, 0.1769
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Infinitesimal_strain_theory.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 29
0.0659, 0.0659
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Capillary_surface.html
( Φ x Φ y Φ z ) ( f φ x f φ y f z φ ) .
Doc 30
0.0622, 0.0622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mild-slope_equation.html
h = ( h x , h y , h z ) = h x 𝐢 + h y 𝐣 + h z 𝐤
Doc 31
0.0617, 0.0617
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Hydraulic_head.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 ) ( θ ) | θ = θ *
Doc 32
0.0615, 0.0615
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Observed_information.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 33
0.0611, 0.0611
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Divergence.html
= ( x , y , z ) ,
Doc 34
0.0604, 0.0604
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Notation_for_differentiation.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 35
0.0604, 0.0604
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Tribimaximal_mixing.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 36
0.0551, 0.0551
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Experimental_uncertainty_analysis.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 37
0.0550, 0.0550
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Capelli's_identity.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
= ( x , y , z ) = e x x + e y y + e z z
Doc 38
0.0515, 0.0515
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Del.html
a = [ 1 c t , x , y , z ]
Doc 39
0.0510, 0.0510
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Lorentz_covariance.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 40
0.0488, 0.0488
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Variance_function.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 41
0.0468, 0.0468
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Special_relativity.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 42
0.0449, 0.0449
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Vorticity.html
y 𝐱 = [ y x 1 y x 2 y x n ] .
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
𝐲 x = [ y 1 x y 2 x y m x ] .
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
Doc 14
0.1137, 1.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Matrix_calculus.html
2 f x i x j = 2 f x j x i .
Doc 43
0.0417, 0.0417
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 44
0.0415, 0.0415
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 45
0.0402, 0.0402
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Potential_flow.html
2 f x i x j ( a 1 , , a n ) = 2 f x j x i ( a 1 , , a n ) .
Doc 46
0.0386, 0.0386
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Symmetry_of_second_derivatives.html
( 2 S y x ) = ( 2 S x y ) : ( 2 V y x ) = ( 2 V x y )
Doc 47
0.0370, 0.1032
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.html
2 x k x j H = 2 x j x k H .
Doc 48
0.0368, 0.0368
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Eikonal_equation.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 49
0.0359, 0.0359
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Finite_strain_theory.html
u i t + u i u j x j = - P x i + ν 2 u i x j x j + f i
Doc 50
0.0350, 0.0350
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 51
0.0348, 0.0348
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 52
0.0346, 0.1581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.html
u i ¯ t + u i u j x j ¯ = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j .
Doc 52
0.0346, 0.1581
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 53
0.0334, 0.0620
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 52
0.0346, 0.1581
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 54
0.0324, 0.0324
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Saint-Venant's_compatibility_condition.html
R i k m = 1 2 ( 2 g i m x k x + 2 g k x i x m - 2 g i x k x m - 2 g k m x i x )
Doc 55
0.0320, 0.0320
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/List_of_formulas_in_Riemannian_geometry.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
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 56
0.0315, 0.0315
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Reynolds_stress.html
2 y x i x j = k ( y u k 2 u k x i x j ) + k , ( 2 y u k u u k x i u x j ) .
Doc 57
0.0315, 0.0315
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Chain_rule.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 52
0.0346, 0.1581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Large_eddy_simulation.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 53
0.0334, 0.0620
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Reynolds-averaged_Navier–Stokes_equations.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 58
0.0267, 0.1595
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 58
0.0267, 0.1595
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 58
0.0267, 0.1595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.html
σ x = 2 Φ y y z z + 2 Φ z z y y - 2 2 Φ y z y z
Doc 58
0.0267, 0.1595
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 58
0.0267, 0.1595
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 58
0.0267, 0.1595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Stress_functions.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 52
0.0346, 0.1581
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 59
0.0261, 0.0261
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Second_derivative.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 60
0.0257, 0.0257
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Method_of_steepest_descent.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 61
0.0240, 0.0240
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Pfaffian.html
y ( z x ) y = x ( z y ) x = 2 z y x = 2 z x y
Doc 47
0.0370, 0.1032
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.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 62
0.0231, 0.0231
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Partial_differential_equation.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 63
0.0213, 0.0213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Acoustic_theory.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 47
0.0370, 0.1032
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 47
0.0370, 0.1032
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Maxwell_relations.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 64
0.0204, 0.0204
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Elliptic_partial_differential_equation.html