D.1 Generic vec­tor iden­ti­ties

The rules of en­gage­ment are as fol­lows:

• The Carte­sian axes are num­bered us­ing an in­dex $i$, with $i$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, 2, and 3 for $x$, $y$, and $z$ re­spec­tively.
• Also, $r_i$ in­di­cates the co­or­di­nate in the $i$ di­rec­tion, $x$, $y$, or $z$.
• De­riv­a­tives with re­spect to a co­or­di­nate $r_i$ are in­di­cated by a sim­ple sub­script $i$.
• If the quan­tity be­ing dif­fer­en­ti­ated is a vec­tor, a comma is used to sep­a­rate the vec­tor in­dex from dif­fer­en­ti­a­tion ones.
• In­dex ${\overline{\imath}}$ is the num­ber im­me­di­ately fol­low­ing $i$ in the cyclic se­quence ...123123...and ${\overline{\overline{\imath}}}$ is the num­ber im­me­di­ately pre­ced­ing $i$.

The first iden­tity to be de­rived in­volves the “vec­to­r­ial triple prod­uct:”

$$\nabla\times\nabla\times\vec v = \nabla(\nabla\cdot\vec v) - \nabla^2 \vec v %$$ (D.1)

To do so, first note that the $i$-​th com­po­nent of $\nabla$ $\times$ $\vec{v}$ is given by

$$v_{{\overline{\overline{\imath}}},{\overline{\imath}}} - v_{{\overline{\imath}},{\overline{\overline{\imath}}}}$$

Re­peat­ing the rule, the $i$-​th com­po­nent of $\nabla$ $\times$ $\nabla$ $\times$ $\vec{v}$ is

$$(v_{{\overline{\imath}},i} - v_{i,{\overline{\imath}}})_{{\... ...rline{\overline{\imath}}},i})_{{\overline{\overline{\imath}}}}$$

That writes out to

$$v_{i,ii} + v_{{\overline{\imath}},{\overline{\imath}}i} + v... ...{\overline{\overline{\imath}}}{\overline{\overline{\imath}}}}$$

since the first and fourth terms can­cel each other. The first three terms can be rec­og­nized as the $i$-​th com­po­nent of $\nabla(\nabla\cdot\vec{v})$ and the last three as the $i$-​th com­po­nent of $-\nabla^2\vec{v}_i$.

A sec­ond iden­tity to be de­rived in­volves the “scalar triple prod­uct:”

$$(\vec a \times \vec b)\cdot \vec c = \vec a \cdot (\vec b\times \vec c) %$$ (D.2)

This is eas­i­est de­rived from sim­ply writ­ing it out. The left hand side is

$$a_yb_zc_x - a_zb_yc_x + a_zb_xc_y - a_xb_zc_y + a_xb_yc_z - a_yb_xc_z$$

while the right hand side is

$$a_xb_yc_z - a_xb_zc_y + a_yb_zc_x - a_yb_xc_z + a_zb_xc_y - a_zb_yc_x$$

In­spec­tion shows it to be the same terms in a dif­fer­ent or­der. Note that since no or­der changes oc­cur, the three vec­tors may be non­com­mut­ing op­er­a­tors.