Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 179

523.Show that fora, b >0,
∫∞

0

e−ax−e−bx

x

dx=ln

b

a

524.Let|x|<1. Prove that

∑∞

n= 1

xn

n^2

=−

∫x

0

1

t

ln( 1 −t)dt.

525.LetF(x)=

∑∞

n= 1

1

x^2 +n^4 ,x∈R. Compute

∫∞

0 F(t)dt.

3.3.3 The Many Versions of Stokes’ Theorem......................

We advise you that this is probably the most difficult section of the book. Yet Stokes’
theorem plays such an important role in mathematics that it deserves an extensive treat-
ment. As an encouragement, we offer you a quote by Marie Curie: “Nothing in life is to
be feared. It is only to be understood.’’
In its general form, Stokes’ theorem is known as
∫

M

dω=

∫

∂M

ω,

whereωis a “form,’’dωits differential, andMa domain with boundary∂M. The
one-dimensional case is the most familiar; it is the Leibniz–Newton formula
∫b

a

f′(t)dt=f(b)−f(a).

Three versions of this result are of interest to us.

Green’s theorem.LetDbe a domain in the plane with boundaryCoriented such that
Dis to the left. If the vector field

−→

F (x, y)=P(x, y)

−→

i +Q(x, y)

−→

j is continuously
differentiable onD, then
∮

C

Pdx+Qdy=

∫∫

D

(

∂Q

∂x

−

∂P

∂y

)

dxdy.

Stokes’ Theorem.LetSbe an oriented surface with normal vector−→n, bounded by a
closed, piecewise smooth curveCthat is oriented such that if one travels onCwith the
upward direction−→n, the surface is on the left. If

−→

Fis a vector field that is continuously
differentiable onS, then
∮

C

−→

F·d

−→

R=

∫∫

S

(curl

−→

F ·−→n)dS,

wheredSis the area element on the surface.