UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

FALL-WINTER 2005-06

ASSIGNMENT #3. DUE ON DECEMBER 1.

PROBLEMS

DO NOT SUBMIT YOUR SOLUTIONS WITHOUT THE COVER PAGE.

READ THE INSTRUCTIONS WRITTEN ON THAT PAGE.

1.  a)  Let  u ( x ₁ , x ₂ , … , x _n ) = , where  a  is a constant, and  n  is given.

Find all the values of the constant  a , if any, for which the function  u  satisfies the partial differential equation  .

b)  Let  F : R ² ® R  be a function whose partial derivatives are continuous and suppose that the equation  F ( r cos q , z + r sin q  ) = 0  implicitly defines  z  as a function of the variables  r  and  q  . Find all the values of the constant  b , if any, for which  z  satisfies the partial differential equation  .

2. a)  Suppose that  E  is the ellipsoid   , where  a > 0 ,  b > 0  and  a ¹ b .

Let  d ( x , y , z )  denote the distance from the point  ( 0 , 0 , 0 )  to the line that is normal to the ellipsoid  E  at the point  ( x , y , z ) . Find the maximum value of the function  d  and determine all points  ( x , y , z )  where this maximum value is reached.

b)  A small insect is seating at the point  ( 4 , 3 , 1 )  inside a cage in the shape of the hollow rectangular region  0 £ x £ 5 ,  0 £ y £ 5 ,  1 £ z £ 4 . Suppose that the temperature at any point  ( x , y , z )  in the cage is given by the function  T ( x , y , z ) = 30 + x ² + 2 y ² – z ² . In order to cool off as soon as possible, the insect flies off from its position at the bottom of the cage in such a way that it experiences the maximum possible rate of cooling at each of the points of its trajectory. What are the coordinates of the point at which the insect reaches the top of the cage?

3.  a)  Question 2-5 from Spivak:

Let  f : R ² ® R  be defined by   .

Show that  f  is not differentiable at  ( 0 , 0 ) .          

b)  Define  f , g : R ³ ® R  by    and

 , where  u , v , w : R ³ ® R  are continuous functions, with continuous partial derivatives, and such that for all  ( x , y , z ) ,

u _x ( x , y , z ) + v _y ( x , y , z ) + w _z ( x , y , z ) = 0 .

Show that  f ,  g  and  w  satisfy the partial differential equation  f _x + g _y = w .

4. a)  Question 2-24 from Spivak:

Define  f : R ² ® R  by  .

a1) Show that  f _y ( x , 0 ) = x  for all  x  and  f _x ( 0 , y ) = – y  for all  y .

a2) Show that  f _{x y} ( 0 , 0 ) ¹ f _{y x} ( 0 , 0 ) .

b)    Question 2-32 from Spivak:

b1) Let  f : R ® R  be defined by   .

Show that  f  is differentiable at  0  but  f ¢  is not continuous at  0 .

b2) Let  f : R ² ® R  be defined by   .

Show that  f  is differentiable at  ( 0 , 0 )  but  f _x  and  f _y  are both discontinuous at  ( 0 , 0 ) .