Definition:Biharmonic Operator

Definition
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $f \left({x_1, x_2, \ldots, x_n}\right)$ denote a real-valued function on $\R^n$.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$.

Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$.

The biharmonic operator on $f$ is defined as:

In $3$ dimensions with the standard ordered basis $\left({\mathbf i, \mathbf j, \mathbf k}\right)$, this is usually rendered: