SECTION A

1.  For the function,

f (x.y) = ln │x3 +y2 │ .

find  and  at the point (1, 2).

2.  Find and classify the critical points of the function

f (x.y) = 5 +3x2 +3y2 +x3 +2y3．

3.  Find the area enclosed by the two curves x = y2 and x = 2y ! y2 using a double integral.

4.     (a)  Define divergence and curl of a vector field in 3-dimensional space. (b)  Consider the vector field  = !y +xjˆ +z .

(i)  Compute its divergence. (ii)  Compute its curl.

5.  Evaluate the line integral

(y ! x)dx +(y ! z)dz

C

over the line segment C from P = (1. 1. 1) to Q = (2. 4. 6).

6.  Determine whether each of the following statements is true or false. If it is true, explain why, and if it is false, provide a counterexample.

(a)  If  and  are conservative vector fields defined on the whole 3-dimensional space, then the sum  +  is also conservative.

(b)  The flux of  = (0.y. 0) across a sphere of radius 1 centred at the origin is strictly less than its flux across a sphere of radius 2 centred at the origin, where both point outwards.

SECTION B

7.     (a)  For the following partial differential equation for the function f (x.y) (x ! y) +(x +y) = 0.

change the variables (x.y) to (u. v), where

u = tan! 1  ╱  、 . v = ln 尸x2 +y2．

(b)  Using the method of the Lagrange multipliers, find the maximum and minimum values of the function

f (x.y.z) = x +z.

where (x.y.z) is on the following surface

x2 +y2 +z2 = 1．

8.     (a)  Calculate the following integral using cylindrical coordinates

1              ì1!x2                  ì 1!x2 !y2                                                 

I =      dx               dy                      z 尸x2 +y2 +z2dz．

0            0                      0

(b)  Calculate the integral I in question 8(a) using spherical coordinates.

(c)  Using the results obtained in 8(a), or otherwise, calculate

1               ì1!x2                   ì 1!x2 !y2                                                                                        

J =        dx                  dy                         ╱sin(x)z2 +z 尸x2 +y2 +z2、dz．

! 1           !ì1!x2               0

(d)  Using the results obtained in 8(a), or otherwise, calculate

K =  02 dx  0 ì 1!x2 /22 dy  0 ì 1!x2 /22 !y2 /32 z 尸  +  + dz．

(c)  For the values of a and b determined in part (a), give the equation of a surface S having

the property that

Q

 」d = 0.

P

for any two points P and Q on the surface S.

10.  Let (x.y.z) = (yz. !xz. !1). Let S be the portion of the paraboloid z = 4 ! x2 ! y2 which lies above the first octant x 珪 0, y 珪 0, z 珪 0. Let C be the closed curve C = C1 +C2 +C3, where the curves C1 . C2 . C3 are formed by intersecting S with the xy.yz. and xz planes, respectively, so that C is the boundary of S. Orient C so that it is transversed counterclockwise when seen from above in the first octant.

(a)  Compute curl().

(b)  State Stokes’ theorem. Please be careful to include all the hypotheses.

(c)  Use Stokes’ theorem to compute the integral

 」d.

C

by reducing it to an appropriate surface integral over S.