The vector Laplacian of a vector field 
in Euclidean space is defined
by
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(1)
|
where the notation 
is sometimes used to distinguish the vector Laplacian from the scalar Laplacian 
(Moon and Spencer 1988, p. 3).
In tensor notation, 
is written 
, and the identity becomes
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(2)
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(3)
|
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(4)
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A tensor Laplacian may be similarly defined.
In cylindrical coordinates 
, with 
the azimuth, write 
in the orthonormal
basis. The physical components of the vector Laplacian are
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(5)
|
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(6)
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(7)
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where 
on the right-hand sides is the scalar Laplacian
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(8)
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In spherical coordinates 
, with 
the azimuth and 
the polar angle, write 
in
the orthonormal basis. The physical components
of the vector Laplacian, ordered as 
, 
, and 
, are given by
![del ^2v=[1/r(partial^2(rv_r))/(partialr^2)+1/(r^2)(partial^2v_r)/(partialphi^2)+1/(r^2sin^2phi)(partial^2v_r)/(partialtheta^2)+(cotphi)/(r^2)(partialv_r)/(partialphi)-2/(r^2)(partialv_phi)/(partialphi)-2/(r^2sinphi)(partialv_theta)/(partialtheta)-(2v_r)/(r^2)-(2cotphi)/(r^2)v_phi; 1/r(partial^2(rv_theta))/(partialr^2)+1/(r^2)(partial^2v_theta)/(partialphi^2)+1/(r^2sin^2phi)(partial^2v_theta)/(partialtheta^2)+(cotphi)/(r^2)(partialv_theta)/(partialphi)+2/(r^2sinphi)(partialv_r)/(partialtheta)+(2cotphi)/(r^2sinphi)(partialv_phi)/(partialtheta)-(v_theta)/(r^2sin^2phi); 1/r(partial^2(rv_phi))/(partialr^2)+1/(r^2)(partial^2v_phi)/(partialphi^2)+1/(r^2sin^2phi)(partial^2v_phi)/(partialtheta^2)+(cotphi)/(r^2)(partialv_phi)/(partialphi)+2/(r^2)(partialv_r)/(partialphi)-(2cotphi)/(r^2sinphi)(partialv_theta)/(partialtheta)-(v_phi)/(r^2sin^2phi)].](/images/equations/VectorLaplacian/NumberedEquation3.svg) |
(9)
|
The scalar Laplacian is
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(10)
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