[Calculus] Vector Laplacian Operator
By: Tao Steven Zheng (鄭濤)
【Problem】
Let be a vector field with components that are continuous second partial derivatives in three dimensional space. The vector Laplacian operator is defined as
. Express the vector Laplacian operator in terms of
.
【Solution】
Calculate :
Calculate :
Consequently,?
By Clairaut's theorem of mixed partials, each bracket in the above vector equal zero; therefore,
This expression can be compactly expressed as
where the left-hand side is the vector Laplacian of , and the right-hand side is a vector of the scalar Laplacian of the components of
.
標簽：
