Begin2.DVI

7-41. (Lagrange multipliers)

Use Lagrange multipliers to

Minimize ω=ω(x, y, z ) = x^2 +y^2 +z^2 ,

subject to the constraint conditions: g(x, y, z ) = x+y+z−6 = 0

h(x, y, z ) = 3 x+ 5 y+ 7 z−34 = 0

7-42. Given the vector field F
= (x^2 +y−4) ˆe 1 + 3xy ˆe 2 + (2xz +z^2 )ˆe 3 .Evaluate the

surface integral

I=

∫∫

S

(∇× F
)·dS

over the upper half of the unit sphere centered at the origin.

7-43. Evaluate the surface integral

∫∫

S

F
·dS ,
where F
=x^2 ˆe 1 + (y+ 6) ˆe 2 −zeˆ 3 and

S is the surface of the unit cube bounded by the planes x= 0 , y = 0 , z = 0 and

x= 1, y = 1, z = 1.

7-44.

(a) Show in the special case the surface is defined by
r =
r (x, y ) = xˆe 1 +yˆe 2 +z(x, y )ˆe 3

the element of surface area is given by

dS = dx dy

|ˆen·ˆe 3 |

=

√

1 +

(

∂z

∂x

) 2

+

(

∂z

∂y

) 2

dx dy

(b) Show in the special case the surface is defined by
r =
r (x, z) = xˆe 1 +y(x, z )ˆe 2 +zˆe 3

the element of surface area is given by

dS =

dx dz

|ˆen·ˆe 2 |=

√

1 +

(

∂y

∂x

) 2

+

(

∂y

∂z

) 2

dx dz

(c) Show in the special case the surface is defined by
r =
r (y, z ) = x(y, z)ˆe 1 +yˆe 2 +zˆe 3

the element of surface area is given by

dS =|ˆedy dz

n·ˆe 1 |

=

√

1 +

(

∂x

∂y

) 2

+

(

∂x

∂z

) 2

dy dz

7-45. Given n particles having masses m 1 , m 2 ,... , m n. Let
r i, i= 1 , 2 ,... , n de-

note the position vector describing the position of the ith particle. Find the vector

describing the center of mass of the system of particles.