UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

FALL-WINTER 2005-06

ASSIGNMENT #1. SOLUTIONS

1. Compute the area of the surface generated when the parametric curve

is rotated about the line  x = 1 .

Solution:

Setting up the integral corresponding to this surface area we obtain:

A =  , where

1  x ( t ) =  ,  x ( t ) =  ,  y ( t ) =  , and

[ x ( t ) ] 2 + [ y ( t ) ] 2 =  .

So, the area is  A =  =  =  .

2. Determine the maximum and minimum values of the curvature at points of the polar curve  r = 3 + sin q  .

Solution.

The given curve can be parametrized as:

x(q ) = rcosq  = 3cosq  + sinq cosq  = 3cosq  + sin(2q ) , and

y(q ) = rsinq  = 3sinq  + sin 2q  = 3sinq  + (1  cos(2q )) , 0≤q ≤2p .

Now,  x(q ) =  3sinq  + cos(2q ) ,  y (q ) = 3 cosq  + sin(2q ) ,

x²(q ) =  3cosq   2sin(2q ) ,  y ²(q ) =  3sinq  + 2cos(2q ) ,

x(q ) y ²(qx²(q ) y (q )=11+9(sin(2q )cosq sinq cos(2q ))=11+9sinq , and

[x(q )] 2 + [y (q )] 2 =10+6(sin(2q )cosq sinq cos(2q ))=10+6sinq .

Therefore,  K (q ) =

and  K (q ) = .

The possible extrema of  K (q )  are at  q = 0, p /2, 3p /2, arcsin(1/3) .

Testing, we find  K(0)=11/103/2 » 0.3479 , K(p /2)=20/163/2 = 0.3125 ,

K(3p /2)=2/43/2 = and K(arcsin(1/3))=8/83/2=» 0.3536 .

So, the minimum and maximum curvatures are  and  , respectively.

3. Let  p  denote a real number such that  0 < p < 1  and let  v 1 , v 2 , v 3 ,   be vectors in  R3  such that    and    for all  k = 1 , 2 , 3 ,   Find the values of  p , if any, for which  .

Solution.

Notice that

=

=  , and

=  .

Now,

=  .

We want to find the values of  0 < p < 1  such that  4 p 3  3 p 2 + 4 p  1 = 0 .

The polynomial function  f ( p ) = 4 p 3  3 p 2 + 4 p  1  is continuous and differentiable

everywhere. The function  f  has at least one zero over  ( 0 , 1 )  because  f ( 0 ) =  1  and

f ( 1 ) = 4 . That is the only real zero of  f  because  f ( p ) = 12 p 2  6 p + 4 > 0  for all  p .

So, there is a unique value of  p . An approximated value is  p » 0.2884 .

4. Prove or disprove each of the following propositions:

a) If  C 1 , C 2 , C 3 ,   are all closed sets in  R n , then the set    is also a closed set in  R n .

Solution:

This proposition is false. We will give a counterexample.

Consider the sets  C 1 = { 1 } , C 2 = { 1 / 2 } ,    , C k = { 1 / k } ,

All of these sets are obviously closed but their union

is not a closed set because  implies that  0  is a boundary

point of  C  and  0 Ο C .

b) If  A 1 , A 2 , A 3 ,   are all non empty compact sets in  R n  such that  A k + 1 Μ A k  for all  k = 1 , 2 , 3 ,   , then the set    is also non empty.

Solution:

This proposition is true.

We will prove that the assumption A  is an empty set leads to a contradiction.

Notice that all the sets  A k  are closed because they are compact. Therefore, the sets    are all open. Assume now that the set  A  is empty. If the set  A  is empty then

. It means that    is an open cover of  R n  and

therefore an open cover of  A 1 Μ R n . Consider now any finite choice of some of these open sets, say   where  k 1 < k 2 < k 3 <  < k m . Notice that   because  therefore  but    because  Ζ Ή Μ A 1 .

So    is an open cover of  A 1  that does not contain a finite subcollection of open sets which also covers  A 1 . This is a contradiction because  A 1  is a compact set.

  HOME