Begin2.DVI

8-42. In cylindrical coordinates (r, θ, z ),show that

curl F
=∇× F
 =^1

r

∣∣

∣∣

∣∣

ˆer rˆeθ eˆz

∂

∂r

∂

∂θ

∂

∂z

Fr rF θ Fz

∣∣

∣∣

∣∣

8-43. In cylindrical coordinates (r, θ, z ),show that

div F
=∇·F
=

1

r

[

∂

∂r (rF r) +

∂

∂θ (Fθ) +

∂

∂z (rA z)

]

8-44. In cylindrical coordinates (r, θ, z ),show that

grad u=∇u=∂u

∂r

ˆer+^1

r

∂u

∂θ

ˆeθ+∂u

∂z

eˆz

8-45. In cylindrical coordinates (r, θ, z ),show that

∇^2 u=^1

r

∂

∂r

(

r∂u

∂r

)

+^1

r^2

∂^2 u

∂θ^2

+∂

(^2) u
∂z^2

8-46. In spherical coordinates (r, θ, φ ),show that

curl F
 =∇× F
 =^1

r^2 sin θ

∣∣

∣∣

∣∣

ˆer rˆeθ rsin θˆeφ

∂

∂r

∂

∂θ

∂

∂φ

Fr rF θ rsin θF φ

∣∣

∣∣

∣∣

8-47. In spherical coordinates (r, θ, φ ),show that

div F
=∇·F
=

1

r^2 sin θ

[

∂

∂r

(

r^2 sin θF r

)

+

∂

∂θ (rsin θF θ) +

∂

∂φ (rF φ)

]

8-48. In spherical coordinates (r, θ, φ ),show that

grad u=∂u

∂r

ˆer+^1

r

∂u

∂θ

ˆeθ+^1

rsin^2 θ

∂u

∂φ

ˆeφ

8-49. In spherical coordinates (r, θ, φ ),show that

∇^2 u=

1

r^2

∂

∂r

(

r^2

∂u

∂r

)

+

1

r^2 sin θ

∂

∂θ

(

sin θ

∂u

∂φ

)

+

1

r^2 sin^2 θ

∂^2 u

∂φ^2

8-50. Show that

(a) In cylindrical coordinates (r, θ, z ),the element of volume is dV =r dr dθdz.

(b) In spherical coordinates (r, θ, φ ),the element of volume is dV =r^2 sin θ dr dφ dθ.

(c) In a general orthogonal curvilinear coordinate system (u, v, w ),the element of

volume can be expressed as dV =huhvhwdu dv dw.