Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES

∇Φ=

∂Φ

∂r

ˆer+

1

r

∂Φ

∂θ

eˆθ+

1

rsinθ

∂Φ

∂φ

ˆeφ

∇·a =

1

r^2

∂

∂r

(r^2 ar)+

1

rsinθ

∂

∂θ

(sinθaθ)+

1

rsinθ

∂aφ

∂φ

∇×a =

1

r^2 sinθ

∣∣

∣

∣∣

∣

∣∣

eˆr reˆθ rsinθeˆφ

∂

∂r

∂

∂θ

∂

∂φ

ar raθ rsinθaφ

∣∣

∣

∣∣

∣

∣∣

∇^2 Φ=

1

r^2

∂

∂r

(

r^2

∂Φ

∂r

)

+

1

r^2 sinθ

∂

∂θ

(

sinθ

∂Φ

∂θ

)

+

1

r^2 sin^2 θ

∂^2 Φ

∂φ^2

Table 10.3 Vector operators in spherical polar coordinates; Φ is a scalar field

andais a vector field.

x

y

z

r

rdθ

φ

dφ

dφ

dr

rsinθ

rsinθdφ

rsinθdφ

θ

dθ

Figure 10.10 The element of volume in spherical polar coordinates is given

byr^2 sinθdrdθdφ.

we can rewrite the first term on the RHS as follows:

1

r^2

∂

∂r

(

r^2

∂Φ

∂r

)

=

1

r

∂^2

∂r^2

(rΦ),

which can often be useful in shortening calculations.