Physical Chemistry Third Edition

1244 B Some Useful Mathematics

or aCartesian tensor. Each of its components is multiplied by a product of two unit

vectors.

Thedivergenceof a vector functionFis denoted by∇·Fand is defined by

∇·F

(

∂Fx

∂x

)

+

(

∂Fy

∂y

)

+

(

∂Fz

∂z

)

(B-44)

The divergence is a scalar quantity. If the vector function represents the flow velocity of

a fluid, the divergence is a measure of the spreading out of the streaming curves along

which small elements of the fluid flow. A positive value of the divergence corresponds

to a decrease in density along a curve following the flow. See the discussion of the

equation of continuity in Section 11.2.

The divergence of the gradient of a scalar function is called theLaplacian. The

Laplacian of a scalar functionfis given in Cartesian coordinates by

∇^2 f

∂^2 f

∂x^2

+

∂^2 f

∂y^2

+

∂^2 f

∂z^2

(B-45)

The Laplacian is sometimes called “del squared.”

The vector derivative operators can be expressed in other coordinate systems. In

spherical polar coordinates, the gradient of the scalar functionfis

∇fer

∂f

∂r

+eθ

1

r

∂f

∂θ

+eφ

1

rsin(θ)

∂f

∂φ

(B-46)

whereeris the unit vector in therdirection (the direction of motion ifris increased

by a small amount, keepingθandφfixed),eθis the unit vector in theθdirection, and

eφis the unit vector in theφdirection. In spherical polar coordinates, the Laplacian is

∇^2 f

1

r^2

[

∂

∂r

[

r^2

∂f

∂r

]

+

1

sin(θ)

∂

∂θ

[

sin(θ)

∂f

∂θ

]

+

1

sin^2 (θ)

∂^2 f

∂φ^2

]

(B-47)

In cylindrical polar coordinates the three coordinates arez(same as in Cartesian coor-

dinates),φ(same as in spherical polar coordinates), andρ, equal to

√

x^2 +y^2. The

gradient of a scalar functionfis given by

∇feρ

∂f

∂ρ

+eφ

1

ρ

∂f

∂φ

+k

∂f

∂z

(B-48)

whereeρis the unit vector in theρdirection,eφis the unit vector in theφdirection,

andkis the unit vector in thezdirection.

For example, the liquid in a pipe with radiusRhas a velocity that depends onρ, the

distance from the center of the pipe such that

ukuz(ρ)kA(ρ^2 −R^2 ) (B-49)

whereAis a constant. All of the nine components of the gradient of this velocity will

vanish except for∂uz/∂ρ:

∇uzeρ

∂uz

∂ρ

eρ 2 Aρ (B-50)

The gradient of the flow velocity points at right angles to the velocity.