Elementary results from vector analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 515

and then the gradient of the product of scalars,

∇ ( φψ ) = φ ∇ ψ + ψ ∇ φ.

(A.6)

The divergence of the sum of two vectors becomes

∇· ( a + b ) =∇· a +∇· b . (A.7)

The curl of the sum of two vectors is

∇× ( a + b ) =∇× a +∇× b . (A.8)

The divergence of the product of a scalar and a vector yields

∇· (φ a ) = a ·∇ φ + φ ∇· a , (A.9)

while the curl of the product of a scalar and a vector gives

∇× (φ a ) =∇ φ × a + φ ∇× a . (A.10)

The gradient of the scalar product of two vectors develops as

∇ ( a · b ) = ( a ·∇ ) b + ( b ·∇ ) a + a × ( ∇× b ) + b × ( ∇× a ). (A.11)

The divergence of the cross product of two vectors can be expanded as

∇· ( a × b ) = b ·∇× a − a ·∇× b , (A.12)

while the curl of the cross product of two vectors becomes

∇× ( a × b ) = a ∇· b − b ∇· a + ( b ·∇ ) a − ( a ·∇ ) b . (A.13)

In Cartesian co-ordinates, the curl of a vector, taken twice, can be expressed as

2 a .

∇× ( ∇× a ) =∇ ( ∇· a ) −∇

(A.14)

The curl of the gradient of a scalar vanishes identically,

∇×∇ φ ≡ 0, (A.15)

and the divergence of the curl of a vector also vanishes identically,

∇· ( ∇× a ) ≡ 0. (A.16)