9. Calculus identities

Here some calculus identities are listed and proved, often using Cartesian coordinates and index notation. While these identities are independent on the choice of the coordinates - as calculus is - a generic set of coordinates can be used to prove them, and Cartesian coordinates are the most convenint choice.

(9.1)\[\Delta \vec{v} = \nabla (\nabla \cdot \vec{v}) - \nabla \times \nabla \times \vec{v} \ .\]

Proof.

Uusing the identity

\[\varepsilon_{ijk} \varepsilon_{ilm} = \delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl}\]

and index notation for Cartesian coordinates,

\[\begin{split}\begin{aligned} \big( \nabla \times \nabla \times \vec{v} \big)_i & = \hat{e}_i \cdot \big( \nabla \times \nabla \times \vec{v} \big) = \\ & = \varepsilon_{ijk} \, \partial_j ( \varepsilon_{klm} \, \partial_l v_m ) = \\ & = \varepsilon_{kij} \, \varepsilon_{klm} \partial_{jl} v_m = \\ & = ( \delta_{il} \, \delta_{jm} - \delta_{im} \, \delta_{jl} ) \, \partial_{jl} v_m = \\ & = \partial_{ij} \, v_j - \partial_{jj} \, v_i = \\ & = \hat{e}_i \cdot \big( \nabla (\nabla \cdot \vec{v}) - \Delta \vec{v} \big) = & = \big( \nabla (\nabla \cdot \vec{v}) - \Delta \vec{v} \big)_i \ . \end{aligned}\end{split}\]