Laplace operator
Use in physics

An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow: ρ (∂ v ∂ t + (v ⋅ ∇) v) = ρ f − ∇ p + μ (∇ 2 v), {\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla)\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} \right),} where the term with the vector Laplacian of the velocity field μ (∇ 2 v) {\displaystyle \mu \left(\nabla ^{2}\mathbf {v} \right)} represents the viscous stresses in the fluid.

Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents: ∇ 2 E − μ 0 ϵ 0 ∂ 2 E ∂ t 2 = 0. {\displaystyle \nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0.}

This equation can also be written as: ◻ E = 0, {\displaystyle \Box \,\mathbf {E} =0,} where ◻ ≡ 1 c 2 ∂ 2 ∂ t 2 − ∇ 2, {\displaystyle \Box \equiv {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2},} is the D'Alembertian, used in the Klein–Gordon equation.
language:response_language[German], oulipo:avoid_letter[e]