Begin2.DVI

7-37. Evaluate the surface integral I=

∫∫

S

f(x, y, z )dS, where f(x, y, z ) = 2(x+ 1)y

and Sis the surface of the cylinder x^2 +y^2 = 1, 0 ≤z≤ 3 ,in the first octant.

7-38. Evaluate the integral I=

∫∫

S

F
·dS,
where F
= (x+z)ˆe 1 + (y+z)ˆe 2 −(x+y)ˆe 3

and Sis the surface of the sphere x^2 +y^2 +z^2 = 9 where z≥ 0.

7-39. Evaluate the surface integral I=

∫∫

S

F
·dS ,
where F
= 4yˆe 1 + 4(x+z)ˆe 2 and

Sis the surface of the plane x+y+z= 1 which lies in the first octant.

7-40. (Lagrange multipliers)

Lagrange multipliers are used to help find the maximum or minimum values as-

sociated with functions of several variables when the variables are subject to certain

constraint conditions. The following is a two-dimensional example of finding the

minimum value of a function when the variables in the problem are subject to con-

straints. Let Ddenote the distance from the origin (0 ,0) to a point (x, y )which lies

on the line x+y+ 2 = 0.Let F(x, y ) = D^2 =x^2 +y^2 denote the square of this distance.

The mathematical problem is to find values for (x, y )which minimize F(x, y ) when

(x, y )is constrained to move along the given line. Mathematically one writes

Minimize F(x, y ) = x^2 +y^2

subject to the constraint condition G(x, y ) = x+y−2 = 0.

The point (x, y ),where F has a minimum value, is called a critical point.

(a) Show that at a critical point ∇F is normal to the curve F=constant and ∇Gis

normal to the line G= 0.

(b) Show that at a critical point the vectors ∇Fand ∇Gare colinear. Consequently,

one can write

∇F+λ∇G=
 0

where λis a scalar called a Lagrange multiplier.

(c) Show that at a critical point which minimizes F, the function H=F+λG, satisfies

the equations

∂H

∂λ = 0,

∂H

∂x = 0,

∂H

∂y = 0.

Calculate these equations and find the point (x, y )which minimizes F.