Begin2.DVI

9-5. Consider the vector field E
=r^12 eˆrin polar coordinates. (a) Show this vector

field is irrotational. (b) Find a potential function φ=φ(r)satisfying φ(r 0 ) = 0, where

r 0 > 0.

9-6. True or false, if both A
and B
are irrotational, then the vector F
=A
×B
is

solenoidal.

9-7. Show that if φ=φ(x, y, z )is a solution of the Laplace equation ∇^2 φ= 0, then

(a) Show the vector
V =∇φis irrotational. (b) Show the vector V
=∇φis solenoidal.

9-8.

(a) Show the velocity field V
= kx

x^2 +y^2

ˆe 1 + ky

x^2 +y^2

ˆe 2 is both irrotational and

solenoidal and has the potential function Φ = k

2

ln(x^2 +y^2 )and the stream function

Ψ = ktan −^1 yx

(b) Show that

∂Φ

∂x =

∂Ψ

∂y and

∂Φ

∂y =−

∂Ψ

∂x

(c) Express the potential function and stream function in polar coordinates and

sketch the equipotential curves and streamlines. This type of velocity field is

said to correspond to a source at the origin if k > 0 or a sink at the origin if k < 0.

9-9. Verify that the velocity field
V = −ky

x^2 +y^2

ˆe 1 + kx

x^2 +y^2

ˆe 2 is both irrotational

and solenoidal. Find the potential and streamlines for this velocity field. This type

of flow is termed a circulation about the origin of strength k.

9-10. Sketch the field lines and analyze the vector fields defined by:

(a) F
=yˆe 1 +xˆe 2

(b) F
=yˆe 1 −xˆe 2

(c) F
=aˆe 1 +beˆ 2

(d) F
= 2xy ˆe 1 + (x^2 −y^2 )eˆ 2

(e) F
 = (x^2 +y^2 )ˆe 1 + 2xy ˆe 2

(f) F
=aˆe 1 +xˆe 2

9-11. Show in polar coordinates that the Cauchy-Riemann equations can be ex-

pressed as

∂φ

∂r

=^1

r

∂ψ

∂θ

and ∂ψ

∂r

=−^1

r

∂φ

∂θ