Help with vector math please?

[tom c.]

Hello. I'm trying to understand how nabla(A) where A is a vector function can be considered equal to the Jacobian of A. I don't see how the nabla operator can operate on A to produce the Jacobian of A. Also I don't understand what nabla^2(A) where A is a vector function means. Thanks for any help.

 

[Biftheunderstudy]

Simply put, \nabla is a vector which is in 3 dimensions (\partial_x, \partial_y, \partial_z). A is also a vector (assuming 3 dimensions) (A_x, A_y, A_z).

\nabla(A) then is a 3x3 matrix from the multiplication.

\nabla^2 should be read as \nabla \cdot \nabla, so this makes it a scalar function (\partial_x^2 + \partial_y^2 + \partial_z^2). \nabla^2 A should be then :

\partial_x^2 A + \partial_y^2 A + \partial_z^2 A

Hope that helps