Fundamentals of Plasma Physics

516 Appendix A

sign changes. Thus

A·B×C=A×B·C interchange dot and cross, (A.3)
A·B×C=B×C·A permutation maintaining cyclic order, (A.4)
A·B×C=−A·C×B permutation changing cyclic order. (A.5)
Derivation of the vector calculus identities
The basic idea is to replace one of the vectors by the∇operator and then rearrange
terms and if necessary add terms. The criterion for these maneuvers is that, just like getting
the right piece placed in a jigsaw puzzle, here all applicable rules of vector algebra and of
calculus must be simultaneously satisfied. The simple example

∇·(ψQ)=ψ∇·Q+Q·∇ψ (A.6)

illustrates this principle of satisfying the vector algebra and the calculus rules simultane-
ously. Here the dot always goes between the∇and theQin order to satisfy the rules of
vector algebra and the∇operates once onQand once onψin order to satisfy the product
rule(ab)′=ab′+a′bof calculus.
A less trivial example is∇·(B×C).Here the∇must operate on bothBandC
according to the product rule so, neglecting vector issues for now, the result must be of the
formB∇C+C∇B. This basic product rule result is then adjusted to satisfy the vector
dot-cross rules. In particular, the dots and crosses may be interchanged atwill and the sign
is plus if the cyclic order is∇BC,BC∇,orC∇Band the sign is minus if the cyclic order
is∇CB,CB∇,orB∇C.Since the∇is supposed to operate on only one vector at a time,
we pick the arrangement where the∇is in the middle and eitherBorCis to its right
so that each ofBorChas a turn at being acted on by the∇operator. The arrangements
of interest are thusC·∇×Band−B·∇×Cwhere the minus sign has been inserted to
account for the change in cyclic order. The result is

∇·(B×C)=C·∇×B−B·∇×C (A.7)

which satisfies both the vector algebra dot-cross rule and the product rule of calculus.
Next consider the triple productA×(∇×C).Here the∇operates only on theCand
the vector triple product generates two terms as indicated by Eq.(A.1).Expanding the triple
product according to Eq.(A.1) while arranging an ordering of terms where the∇operates
only onCgives

A×



∇

︸︷︷︸

middle

×︸︷︷︸C

outer



=



∇

︸︷︷︸

middle

︸︷︷︸C

outer



·A−



A·∇

︸︷︷︸

middle



 C

︸︷︷︸

outer

. (A.8)

Thus, the first term on the right hand side has its vector direction determined by∇with the
other two terms dotted together, and the parenthesis indicates that the∇operates only on
C. The second term on the right hand side has its vector direction determined byCwith
the other two terms dotted together and again the∇operates only onC.
This can be rearranged as
(∇C)·A=A×(∇×C)+A·∇C. (A.9)

InterchangingAandCgives

(∇A)·C=C×(∇×A)+C·∇A. (A.10)

Adding these last two expressions gives

(∇C)·A+(∇A)·C=A×(∇×C)+A·∇C+C×(∇×A)+C·∇A. (A.11)