Tensors for Physics

7.6 Rules for Nabla and Laplace Operators 105

7.6 Rules for Nabla and Laplace Operators

7.6.1 Nabla

The application of the nabla operator∇μto a field, which is a tensor of rank, yields

a tensor of rank+1. The appropriate decomposition shall be discussed latter. Here

the obvious product and chain rules are listed, which have already been used above.

Letfandgbe components of tensors, which depend onr. Then one has

∇μ(fg)=g∇μf+f∇μg. (7.76)

Now, letfbe the component of a tensor, which is a function of the scalarg, which

in turn, depends onr. Then the chain rule applies:

∇μ(f(g))=

∂f

∂g

∇μg. (7.77)

The position vectorris equal to the product of its magnituderand of the unit vector
̂r, viz.:rμ=rr̂μ, withr̂μ=r−^1 rμ. In some applications, it may be convenient and
useful to decompose the spatial derivative into differentiations with respect torand
with respect tor̂μ. This is accomplished by observing

∇μ≡

∂

∂rμ

=

∂r

∂rμ

∂

∂r

+

∂r̂ν

∂rμ

∂

∂r̂ν

.

Due to

∂r

∂rμ

=r̂μ,

∂̂rν

∂rμ

=r−^1 (δμν−r̂μr̂ν),

the radial and the angular parts of the spatial derivative are

∇μ=r̂μ

∂

∂r

+r−^1 (δμν−r̂μ̂rν)

∂

∂r̂ν

. (7.78)

Multiplication of (7.78)byrμleads to

rμ∇μ=r

∂

∂r

. (7.79)

With the help of the anti-hermitian operator

Lμ=εμνλrν∇λ=εμνλ̂rν

∂

∂r̂λ

, (7.80)