# Laplancian vector?

by pivoxa15
Tags: laplancian, vector
 HW Helper P: 2,567 If (u,v,w) is an arbitrary curvilinear coordinate system, then: $$\nabla^2 \vec A = \nabla^2 (\hat e_u A_u + \hat e_v A_v + \hat e_w A_w ) = \nabla^2 (\hat e_u A_u) + \nabla^2 ( \hat e_v A_v) + \nabla^2 ( \hat e_w A_w )$$ In a system where the basis vectors depended on position, there will be extra terms, eg: $$\nabla^2 (\hat e_u A_u) = \hat e_u \nabla^2 A + 2 (\nabla \cdot \hat e_u) \nabla A_u + A_u \nabla^2 \hat e_u$$ If you're asking why we don't just define the Laplacian so that: $$\nabla^2 \vec A = \hat e_u \nabla^2 (A_u) + \hat e_v \nabla^2 (A_v)+ \hat e_w \nabla^2 (A_w)$$ the reason is that the answer we'd get would depend on the coordinate system we're using, which is something we don't want.
 Quote by StatusX If (u,v,w) is an arbitrary curvilinear coordinate system, then: $$\nabla^2 \vec A = \nabla^2 (\hat e_u A_u + \hat e_v A_v + \hat e_w A_w ) = \nabla^2 (\hat e_u A_u) + \nabla^2 ( \hat e_v A_v) + \nabla^2 ( \hat e_w A_w )$$