1000 Solved Problems in Modern Physics

2 1 Mathematical Physics

(b)

∫

CA.dr

(c)

∫

CA×dr

whereφis a scalar,Ais a vector andr=xiˆ+yˆj+zkˆ, is the positive vector.

Stoke’s theorem

∮

C

A.dr=

∫∫

S

(∇×A).nds=

∫∫

S

(∇×A).ds

The line integral of the tangential component of a vectorAtaken around a simple

closed curveCis equal to the surface integral of the normal component of the curl

ofAtaken over any surfaceShavingCas its boundary.

Divergence theorem (Gauss theorem)

∫∫∫

V

∇.Adv=

∫∫

S

A.nˆds

The volume integral is reduced to the surface integral.

Fourier series

Any single-valued periodic function whatever can be expressed as a summation of

simple harmonic terms having frequencies which are multiples of that of the given

function. Let f(x) be defined in the interval (−π,π) and assume that f(x) has

the period 2π, i.e. f(x+ 2 π)= f(x). The Fourier series or Fourier expansion

corresponding tof(x) is defined as

f(x)=

1

2

a 0 +

∑∞

n= 1

(a 0 cosnx+bnsinnx) (1.1)

where the Fourier coefficientanandbnare

an=

1

π

∫π

−π

f(x) cosnxdx (1.2)

bn=

1

π

∫π

−π

f(x)sinnxdx (1.3)

wheren= 0 , 1 , 2 ,...

Iff(x) is defined in the interval (−L,L), with the period 2L, the Fourier series

is defined as

f(x)=

1

2

a 0 +

∑∞

n= 1

(ancos(nπx/L)+bnsin(nπx/L)) (1.4)

where the Fourier coefficientsanandbnare