\begin{definition}[Definition:Biharmonic Operator]
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $\map U {x_1, x_2, \ldots, x_n}$ be a real-valued function on $\R^n$.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.
Let the partial derivative of $V$ with respect to $x_k$ exist for all $x_k$.
The '''biharmonic operator on $U$''' is defined as:
{{begin-eqn}}
{{eqn | l = \nabla^4 U
      | o = :=
      | r = \map {\nabla^2} {\nabla^2 U}
}}
{{eqn | r = \paren {\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} } \paren {\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} } U
      | c = {{Defof|Del Operator}}
}}
{{end-eqn}}
In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:
{{begin-eqn}}
{{eqn | l = \nabla^4 U
      | o = :=
      | r = \map {\nabla^2} {\nabla^2 U}
}}
{{eqn | r = \paren {\dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2} + \dfrac {\partial^2} {\partial z^2} } \paren {\dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2} + \dfrac {\partial^2} {\partial z^2} } U
      | c = {{Defof|Del Operator}}
}}
{{eqn | r = \dfrac {\partial^4 U} {\partial x^4} + \dfrac {\partial^4 U} {\partial y^4} + \dfrac {\partial^4 U} {\partial z^4} + 2 \dfrac {\partial^4 U} {\partial x^2 \partial y^2} + 2 \dfrac {\partial^4 U} {\partial x^2 \partial z^2}
      | c = 
}}
{{end-eqn}}
\end{definition}