Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 175

There are three special situations worth mentioning:

• The change in two dimensions from Cartesian to polar coordinatesx =rcosθ,
  y=rsinθ, with the Jacobian∂(x,y)∂(r,θ)=r.


• The change in three dimensions from Cartesian to cylindrical coordinatesx=rcosθ,
  y=rsinθ,z=z, with the Jacobian∂(x,y,z)∂(r,θ,z)=r.


• The change in three dimensions from Cartesian to spherical coordinates x =
  ρsinφcosθ,y=ρsinφsinθ,z=ρcosφ, with the Jacobian∂(ρ,θ,φ)∂(x,y,z)=ρ^2 sinφ.

As an illustration, we show how multivariable integrals can be used for calculating
the Fresnel integrals. These integrals arise in the theory of diffraction of light.

Example.Compute the Fresnel integrals

I=

∫∞

0

cosx^2 dx and J=

∫∞

0

sinx^2 dx.

Solution.For the computation of the first integral, we consider the surfacez=e−y
2
cosx^2
and determine the volume of the solid that lies below this surface in the octantx, y, z≥0.
This will be done in both Cartesian and polar coordinates. We will also make use of the
Gaussian integral

∫∞

0

e−t

2

dt=

√

π

2

,

which is the subject of one of the exercises that follow.
In Cartesian coordinates,

V=

∫∞

0

∫∞

0

e−y

2

cosx^2 dydx=

∫∞

0

(∫∞

0

e−y

2

dy

)

cosx^2 dx

=

∫∞

0

√

π

2

cosx^2 dx=

√

π

2

I.

In polar coordinates,

V=

∫ π 2

0

∫∞

0

e−ρ

(^2) sin (^2) θ
cos(ρ^2 cos^2 θ)ρdρdθ

=

∫ π 2

0

1

cos^2 θ

∫∞

0

e−utan

(^2) θ
cosududθ=
∫ π 2
0

1

cos^2 θ

·

tan^2 θ

1 +tan^4 θ

dθ,

where we made the substitutionu=u(ρ)=ρ^2 cos^2 θ. If in this last integral we substitute
tanθ=t, we obtain