8-37. Let F=xˆe 1 +yˆe 2 +zˆe 3 and evaluate the surface integral
I=
∫∫
S
F·dS ,
where Sis the surface enclosing the volume bounded by the planes x= 0 , y = 0, z = 0
and 2 x+ 3 y+ 4 z= 12 Hint: The volume of a tetrahedron having sides a, b and cis
given by V=^16 abc.
8-38. Use Stokes theorem to evaluate the integral
I=
∫
C
©F ·dr, where F=yˆe 1 + 2 zˆe 2 + (4 y+ 2 x)ˆe 3
and Cis the simple closed curve consisting of the line segments
P 1 P 2 +P 2 P 3 +P 3 P 1
connecting the points P 1 (0 , 0 ,0), P 2 (1 , 1 ,0),and P 3 (0 , 0 , 2
√
2).
8-39. Let v =v (x, y, z, t)denote the velocity of a fluid having density ρ=ρ(x, y, z, t).
Construct an imaginary volume of fluid V enclosed by a surface Slying within the
fluid.
(a) Show the mass of the fluid inside V is given by M=
∫∫∫
ρ(x, y, z, t)dV
(b) Show the time rate of change of mass is ∂M
∂t
=
∫∫
∂ρ
∂t
dV
(c) Show the mass of fluid leaving V per unit of time is given by
∂M
∂t =−
∫∫
S
ρv ·n dS
(d) Use the divergence theorem to show
∫∫∫
∂ρ
∂t dV =−
∫∫
ρv ·n dS =−
∫∫∫
V
∇(ρv)dV
(e) Since V is an arbitrary volume show that ∇J+∂ρ
∂t
= 0, where J =ρv. This
equation is known as the continuity equation of fluid dynamics.
8-40. In parabolic cylindrical coordinates (ξ, η, z),find
(a) The unit vectors ˆeξ, eˆη, eˆz
(b) The metric components gij
8-41. In the paraboloidal coordinates (ξ, η, φ ),find
(a) The unit vectors eˆξ,ˆeη,eˆφ
(b) The metric components gij
