1000 Solved Problems in Modern Physics

(Romina) #1
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
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