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Laplacian

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1: 12.17 Physical Applications
§12.17 Physical Applications
where k is a constant, and 2 is the Laplacian
2: 30.14 Wave Equation in Oblate Spheroidal Coordinates
§30.14(iii) Laplacian
3: 23.21 Physical Applications
The Laplacian operator 2 1.5(ii)) is given by …
4: 30.13 Wave Equation in Prolate Spheroidal Coordinates
§30.13(iii) Laplacian
5: 1.5 Calculus of Two or More Variables
The Laplacian is given by
1.5.13 2 f = 2 f x 2 + 2 f y 2 = 2 f r 2 + 1 r f r + 1 r 2 2 f ϕ 2 .
1.5.15 2 f = 2 f x 2 + 2 f y 2 + 2 f z 2 = 2 f r 2 + 1 r f r + 1 r 2 2 f ϕ 2 + 2 f z 2 .
The Laplacian is given by
1.5.17 2 f = 2 f x 2 + 2 f y 2 + 2 f z 2 = 1 ρ 2 ρ ( ρ 2 f ρ ) + 1 ρ 2 sin 2 θ 2 f ϕ 2 + 1 ρ 2 sin θ θ ( sin θ f θ ) .
6: 3.4 Differentiation
Laplacian
Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …