Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

In addition to these, we note that the gradient operation also obeys the chain

rule as in ordinary differential calculus, i.e. ifφandψare scalar fields in some

regionRthen

∇[φ(ψ)]=

∂φ

∂ψ

∇ψ.

10.7.2 Divergence of a vector field

Thedivergenceof a vector fielda(x, y, z) is defined by

diva=∇·a=

∂ax

∂x

+

∂ay

∂y

+

∂az

∂z

, (10.33)

whereax,ayandazare thex-,y-andz- components ofa. Clearly,∇·ais a scalar

field. Any vector fieldafor which∇·a= 0 is said to besolenoidal.

Find the divergence of the vector fielda=x^2 y^2 i+y^2 z^2 j+x^2 z^2 k.

From (10.33) the divergence ofais given by

∇·a=2xy^2 +2yz^2 +2x^2 z=2(xy^2 +yz^2 +x^2 z).

We will discuss fully the geometric definition of divergence and its physical

meaning in the next chapter. For the moment, we merely note that the divergence

can be considered as a quantitative measure of how much a vector field diverges

(spreads out) or converges at any given point. For example, if we consider the

vector fieldv(x, y, z) describing the local velocity at any point in a fluid then∇·v

is equal to the net rate of outflow of fluid per unit volume, evaluated at a point

(by letting a small volume at that point tend to zero).

Now if some vector fieldais itself derived from a scalar field viaa=∇φthen

∇·ahas the form∇·∇φor, as it is usually written,∇^2 φ,where∇^2 (del squared)

is the scalar differential operator

∇^2 ≡

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

. (10.34)

∇^2 φis called theLaplacianofφand appears in several important partial differ-

ential equations of mathematical physics, discussed in chapters 20 and 21.

Find the Laplacian of the scalar fieldφ=xy^2 z^3.

From (10.34) the Laplacian ofφis given by

∇^2 φ=

∂^2 φ

∂x^2

+

∂^2 φ

∂y^2

+

∂^2 φ

∂z^2

=2xz^3 +6xy^2 z.