Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 183

We apply this to

F(z)=

f(z)

z−a

=

(u(x, y)+iv(x, y))(x−iy+α−iβ)

(x+α)^2 +(y+β)^2

,

wherea=α+iβ. It is routine to check thatFis holomorphic. We thus have
∮



f(z)

z−a

dz=

∮



f(z)

z−a

dz.

The change of variablez=a+eiton the right-hand side yields
∮



f(z)

z−a

dz=

∫π

−π

f(a+eit)

eit

ieitdt=i

∫π

−π

f(a+eit)dt.

When→0 this tends to 2πif(a), and we obtain
∮



f(z)

z−a

dz= 2 πif(a).

Hence the desired formula. 

526.Assume that a curve(x(t), y(t))runs counterclockwise around a regionD. Prove
that the area ofDis given by the formula

A=

1

2

∮

∂D

(xy′−yx′)dt.

527.Compute the flux of the vector field
−→
F (x, y, z)=x(exy−ezx)

−→

i +y(eyz−exy)

−→

j +z(ezx−eyz)

−→

k

across the upper hemisphere of the unit sphere.

528.Compute
∮

C

y^2 dx+z^2 dy+x^2 dz,

whereCis the Viviani curve, defined as the intersection of the spherex^2 +y^2 +z^2 =

a^2 with the cylinderx^2 +y^2 =ax.

529.Letφ(x, y, z)andψ(x, y, z)be twice continuously differentiable functions in the
region{(x,y,z)|^12 <

√

x^2 +y^2 +z^2 < 2 }. Prove that

∫∫

S

(∇φ×∇ψ)·−→ndS= 0 ,

whereSis the unit sphere centered at the origin,−→nis the normal unit vector to this

sphere, and∇φdenotes the gradient∂φ∂xi+∂φ∂yj+∂φ∂zk.