r θ φ

C-1

x = rsinθ cosφ

y = rsinθsinφ (C.1)

z = rcosθ

r 0 ∞ θ 0

π φ 0 2π

C-1

dV = dxdydz = r2sinθdrdθdφ (C.2)

f(r)

f (r)V

∫ dV = dr dθ0

π

∫ dφ0

2π

∫ f (r)0

∞

∫ r2sinθ

= 4π0

∞

∫ r2 f (r)dr (C.3)

∇2 =∂2

∂x 2+∂2

∂y 2+∂2

∂z2

=1

r2∂∂r

r2∂∂r

⎛

⎝ ⎜

⎞

⎠ ⎟ +

1

r2sinθ∂∂θ

sinθ∂∂θ

⎛

⎝ ⎜

⎞

⎠ ⎟ +

1

r2sin2θ∂2

∂φ 2 (C.4)

r

© K. Saito, 06/05/2010