Advanced book on Mathematics Olympiad

580 Real Analysis

=

∫

D×{b}

G(x,y,t)

−→

k ·d−→n +

∫

D×{a}

G(x,y,t)

−→

k ·d−→n

+

∫b

a

∫b 1

a 1

F 1 (x, a 2 )dx−

∫b

a

∫a 1

b 1

F 1 (x, b 2 )dx

+

∫b

a

∫b 2

a 2

F 2 (b 1 , y)dy−

∫b

a

∫a 2

b 2

F 2 (a 1 ,y)dy.

If we introduce the vector field

−→

H =F 2

−→

i +F 1

−→

j +G

−→

k, this equation can be written
simply as
∫∫

∂V

−→

H·−→ndS= 0.

By the divergence theorem,
∫∫∫

V

div

−→

HdV=

∫∫

∂V

−→

H·−→ndS= 0.

Since this happens in every parallelepiped, div

−→

H must be identically equal to 0. There-
fore,

div

−→

H =

∂F 2

∂x

+

∂F 1

∂y

+

∂G

∂t

= 0 ,

and the relation is proved.

Remark.The interesting case occurs when

−→

FandGdepend on spatial variables (spatial
dimensions). ThenGbecomes a vector fieldB, or better a 2-form, called the magnetic
flux, whileFbecomes the electric field strengthE. The relation

d

dt

∫

S

B=−

∫

∂S

E

is Faraday’s law of induction. Introducing a fourth dimension (the time), and redo-
ing mutatis mutandis the above computation gives rise to the first group of Maxwell’s
equations

divB= 0 ,

∂B

∂t

=curlE.

534.In the solution we ignore the factor 41 π, which is there only to make the linking
number an integer. We will use the more general form of Green’s theorem applied to the
curveC=C 1 ∪C 1 ′and surfaceS,