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2MVA   06 25667   Level I

Multivariable & Vector Analysis

Summer examinations 2018-19



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


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.



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 (uv), where

u = tan! 1  ╱   . v = ln 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 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 x2 +y2 +z2dz

! 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   +  + dz

9.     (a)  Consider the vector eld  = ay2 + 2y(x + z)jˆ + (by2 + z2 ).  For which values of the constants a and b will  be conservative? Show your work.

(b)  For the values of a and b determined in part (a), find a function f (x.y.z) such that  = φf .


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

the property that


 」d = 0.


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



by reducing it to an appropriate surface integral over S.