Fundamentals of Plasma Physics

Appendix A 517

However, using the same arguments about maintaining the dot and satisfying theprod-
uct rule shows that

∇(A·C) = ∇(C·A)

= (∇A)·C+(∇C)·A. (A.12)

Here the∇operates once on theAand once on theCaccording to the product rule and
the dot is always between theAand theC.Combining Eqs.(A.11) and (A.12) gives the
standard vector identity

∇(A·C)=A×(∇×C)+A·∇C+C×(∇×A)+C·∇A. (A.13)

Finally, the triple product with two∇’s can be expressed as

∇×



∇

︸︷︷︸

middle

×︸︷︷︸A

outer



= ∇

︸︷︷︸

middle

       

∇·︸︷︷A︸

other two

dotted together

       

−∇︸︷︷︸^2

other two

dotted together

︸︷︷︸A

outer

(A.14)

or, without the labeling, as

∇×∇×A=∇∇·A−∇^2 A. (A.15)

The relationships∇·∇×A= 0and∇×∇ψ= 0can be proved by direct evaluation
using Cartesian coordinates.

Summary of vector identities

A×(B×C) = B(A·C)−C(A·B)

(A×B)×C = B(A·C)−A(B·C)

A·B×C = A×B·C,interchange dot and cross

A·B×C = B×C·A,cyclic permutation, cyclic order maintained

A·B×C = −A·C×B,cyclic permutation, cyclic order changed

∇·(ψA) = ψ∇·A+A·∇ψ

∇·(A×B) = B·∇×A−A·∇×B

(∇B)·A = A×(∇×B)+A·∇B

∇(A·B) = A×(∇×B)+A·∇B+B×(∇×A)+B·∇A

∇×∇×A = ∇(∇·A)−∇^2 A

∇·∇×A = 0

∇×∇ψ = 0