MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

TEST #3. COURSE 2005-2006

UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

TEST #3. MARCH 23, 2006

INSTRUCTIONS:

Write your name and your student number on the front page of each of your examination booklets.

Show and explain your work in all questions.

Use both sides of the papers, if necessary. Do not tear out any pages.

Do not use pencils. Only pen written answers will be considered for remarking.

No calculators or any other aids are permitted. Duration: 110 minutes.

1. (15 marks) Evaluate   where  E  is the solid tetrahedron bounded by the four planes

x = 0 ,  y = 0 ,  z = 0  and  6 x + 3 y + 2 z = 6 .

2. (15 marks) Evaluate    where  R  is the parallelogram

enclosed by the lines  y = xy = x – 2 ,  x + 2 y = 1  and  x + 2 y = 4 .

3. (10 marks) Let  f  and  g  be any two functions of three variables such that both  f  and  g  are

differentiable and have continuous second-order partial derivatives. Show that  div ( Ñ f  ´ Ñ g ) = 0 .

4. (10 marks) Let  F ( x , y , z ) = ( y + ez sin x ) i + x j + ( 1 + ez cos x ) k  and  C  be the

curve given by the vector function  r ( t ) = ( t 2t + p ) i + t 2 j  + ( t 3 – 1 ) k ,  0 ≤ t ≤ 1 .

Evaluate the line integral   .

5. (10 marks) Evaluate the line integral    where  C

is the positively oriented triangle with vertices  ( 0 , 0 ) ,  ( 2 , 1 )  and  ( – 2 , 1 ) .

6. (15 marks) Compute the surface area of the torus represented parametrically by the coordinate

functions  x = ( 3 + cos s ) cos ty = ( 3 + cos s ) sin t  and  z = sin s , where  0 ≤ s ≤ 2 p  and

0 ≤ t ≤ 2 p  .

7. (15 marks) Evaluate the surface integral   where  F ( x , y , z ) = 3 x i + y j + z k

and  S  is the hemisphere  x 2 + y 2 + z 2 = 4 ,  z ³ 0 .

8. (10 marks) Show that for each pair of points  G  and  H  on the xy-plane, there exists a pair of

real constants  p  and  q  such that  = 0 , where  C  is the line segment

that joins  G  and  H .

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