Tensors for Physics

140 8 Integration of Fields

8.3.6 Integration by Parts.

Letf=f(r)andg=g(r)be two field functions. Due to the product rule∇μ(gf)=
g∇μf+f∇μg, and with the help of the generalized Gauss theorem, the volume
integral

∫

Vg∇μfis equal to

∫

V

g∇μfd^3 r=−

∫

V

f∇μgd^3 r+

∫

∂V

fgnμd^2 s. (8.82)

In many applications, the surface integral

∫

∂Vfgnμd

(^2) sis taken over a surface, where
at least one of the two functionsfandgvanishes. Then, the integration by parts is
equivalent to ∫
V
g∇μfd^3 r=−

∫

V

f∇μgd^3 r. (8.83)

Withf=−∇μgandΔ=∇μ∇μ, the relation (8.83) implies

−

∫

V

gΔgd^3 r=

∫

V

(∇μg)(∇μg)d^3 r≥ 0. (8.84)

Thus, subject to the condition that the contribution of the surface integral, occurring
in connection with the integration by parts, is zero, the negative Laplace operator
−Δis a positive definite operator. This point is of importance for the kinetic energy
operator in wave mechanics, cf. Sect.7.6.4.

8.4 Further Applications of Volume Integrals.

8.4.1 Continuity Equation, Flow Through a Pipe

The mass density and the local velocity field of a fluid are denoted byρ=ρ(r)
andv=v(r), as in Sect.7.4.3. The vector fieldj(r)=ρvis theflux density.The
continuity equation, cf. (7.49),

∂ρ

∂t

+∇νjν= 0 , (8.85)

expresses thelocal conservationof mass. The integral ofρover a volumeVyields
the massMV=

∫

Vρd

(^3) rcontained withinV. Upon integration of the continuity
equation over a volumeVwhich does not change with time, the first term of the
equation is the time change of the massMV. The second term can be expressed as a
surface integral over∂V, due to the Gauss theorem. Thus (8.85) leads to