Mathematical Physics Vol 1

7.8 Examples

431

it follows

f ′ r h

r 2 i

∂ 2 f ∂ x i ∂ x j

∂ f ∂ x i

∂ r ∂ x i

x i x j r 2

x i x j

x i r

= f ′ ( r )

= f ′ ( r )

= f ′′

δ i j −

,

+

,

d 2 f d r 2 ⇒

d f d r

f ′ ( r )=

f ′′ ( r )=

;

d f ′ f ′

n − 1 r

n + 1 r

= −

f ′ ( r )= 0 ⇒

∆ f = f ′′ ( r )+

d r ⇒

f ′ ( r )= Cr 1 − n , C = const. ⇒ d f = C r n − 1 d r .

Analysis

We shall distinguish two cases 1) for n = 2

d r

d f = C

f = C ln r + D ,

r ⇒

i

2) for n > 2

C 2 − n

r 2 − n + D .

f =

R Note that in this appendix the convention on summation by repeated indices was used.