Begin2.DVI

8-18. For
r =xˆe 1 +yeˆ 2 +zˆe 3 and r=|
r |show

(i) curl rn
r =
0

(ii) div
r = 3

(iii) curl
r =
0

(iv) div rn
r = (n+ 3)rn

8-19. Show that the vector field ∇φis both solenoidal and irrotational if φis a

scalar function of position which satisfies the Laplace equation ∇^2 φ= 0.

8-20. Show that the following functions are solutions of the Laplace equation in

two dimensions.

(i) φ=x^2 −y^2

(ii) φ= 3x^2 y−y^3

(iii) φ= ln(x^2 +y^2 )

8-21. Verify the divergence theorem for F
=xy ˆe 1 +y^2 ˆe 2 +zˆe 3 over the region

bounded by the cylindrical surface x^2 +y^2 = 4 and the planes z = 0 and z = 4.

Whenever possible, integrate by using cylindrical or polar coordinates. Find the

sections of this surface which has a flux integral.

8-22.

(i) Verify Green’s theorem in the plane for

M(x, y ) = x^2 +y^2 and N(x, y ) = xy,

where Cis the closed curve illustrated in the figure.

(ii) Use line integration and appropriate values for Mand N in Green’s theorem to

determine the shaded area of the attached figure.

8-23. Verify Stokes theorem for F
=yˆe 3 over that portion of the unit sphere in

the first octant. Hint: Use spherical coordinates.

8-24. Verify the given differential equations are exact and then use line integrals

to find solutions.

(i) (2 xy +y^2 )dx + (x^2 + 2xy )dy = 0

(ii) (3 x^2 y+ 2xy^2 )dx + (x^3 + 2yx^2 + 2) dy = 0