Instead of plugging π π e , p directly into the expression for the electric field, several simplifications can be made. Begin with the curl of the curl vector identity, ∇ ∇ × × ( ∇ ∇ × × π π e , p ) = ∇ ∇ ( ∇ ∇ ⋅ ⋅ π π e , p ) − − ∇ ∇ 2 π π e , p , therefore, E p = ∇ ∇ × × ∇ ∇ × × π π e − − P ϵ ϵ 0 = ∇ ∇ ( ∇ ∇ ⋅ ⋅ π π e , p ) − − ∇ ∇ 2 π π e , p − − P ϵ ϵ 0 .