### D.1 Generic vec­tor iden­ti­ties

The rules of en­gage­ment are as fol­lows:

• The Carte­sian axes are num­bered us­ing an in­dex , with 1, 2, and 3 for , , and re­spec­tively.
• Also, in­di­cates the co­or­di­nate in the di­rec­tion, , , or .
• De­riv­a­tives with re­spect to a co­or­di­nate are in­di­cated by a sim­ple sub­script .
• If the quan­tity be­ing dif­fer­en­ti­ated is a vec­tor, a comma is used to sep­a­rate the vec­tor in­dex from dif­fer­en­ti­a­tion ones.
• In­dex is the num­ber im­me­di­ately fol­low­ing in the cyclic se­quence ...123123...and is the num­ber im­me­di­ately pre­ced­ing .

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

 (D.1)

To do so, first note that the -​th com­po­nent of is given by

Re­peat­ing the rule, the -​th com­po­nent of is

That writes out to

since the first and fourth terms can­cel each other. The first three terms can be rec­og­nized as the -​th com­po­nent of and the last three as the -​th com­po­nent of .

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

 (D.2)

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

while the right hand side is

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.