Hola





Ecuación de Laplace

∂^2U/∂x^2 +∂^2U/∂u^2 +∂^2U/∂z^2 = 0







U(x,y,z)=1/√(x^2+y^2+z^2 )



U = (x^2 + y^2 + z^2)^(-1/2)





∂U/∂x = (-1/2) (x^2 + y^2 + z^2)^(-3/2) ( 2 x)

∂U/∂x = -x (x^2 + y^2 + z^2)^(-3/2)





∂^2U/∂x^2 = -(1) (x^2 + y^2 + z^2)^(-3/2) -

- x (-3/2) (x^2 + y^2 + z^2)^(-5/2) ( 2 x)



∂^2U/∂x^2 = - (x^2 + y^2 + z^2)^(-3/2) -

+ (3 x^2) (x^2 + y^2 + z^2)^(-5/2)



∂^2U/∂x^2 = - (x^2 + y^2 + z^2) (x^2 + y^2 + z^2)^(-5/2) -

+ (3 x^2) (x^2 + y^2 + z^2)^(-5/2)



∂^2U/∂x^2 = (-x^2 - y^2 - z^2 + 3 x^2) (x^2 + y^2 + z^2)^(-5/2)



a) ∂^2U/∂x^2 = (2 x^2 - y^2 - z^2) (x^2 + y^2 + z^2)^(-5/2)

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Observamos que U

es función simétrica de x;y;z

y no varía con el intercambio de variables



Deducimos

b) ∂^2U/∂y^2 = (- x^2 + 2 y^2 - z^2) (x^2 + y^2 + z^2)^(-5/2)

c) ∂^2U/∂z^2 = (- x^2 - y^2 + 2 z^2) (x^2 + y^2 + z^2)^(-5/2)



Si sumamos todos



∂^2U/∂x^2 +∂^2U/∂u^2 +∂^2U/∂z^2 =

= (2 x^2 - x^2 - x^2 -

- y^2 + 2 y^2 - y^2 -

- z^2 - z^2 + 2 z^2) * (x^2 + y^2 + z^2)^(-5/2)



Toda esta expresión se anula idénticamente



Concluímos que

el laplaciano de U es cero



∇^2 U = ∂^2U/∂x^2 +∂^2U/∂u^2 +∂^2U/∂z^2 = 0

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Saludos