Physical Chemistry Third Edition

1240 B Some Useful Mathematics

dz

dx

dy

x

y

z

Volume element of

dimensionsdx by dy by dz

Figure B.3 An Infinitesimal Volume Element in Cartesian Coordinates.

Volume element of

volumer^2 sin(
)dd
dr

dr

r

rsin(
)d

rsin(
)d

r d

d



rsin(
)

z

y

x

Figure B.4 An Infinitesimal Volume Element in Spherical Polar Coordinates.

of the volume element in therdirection is equal todr. The length of the box in

theθdirection (the direction in which an infinitesimal change inθcarries a point

in space) is equal tordθifθis measured in radians, since the measure of an angle

in radians is the ratio of the arc length to the radius. The length of the volume element in

theφdirection isrsin(θ)dφ, which comes from the fact that the projection ofrin the

xyplane has lengthrsin(θ), as shown in the figure. The volume of the element of

volume is thus

d^3 rr^2 sin(θ)dφdθdr (B-26)

whered^3 ris a general abbreviation for a volume element in any coordinate system. An

integral of an integrand functionfover all of space using spherical polar coordinates is