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A30449

2MVA   06 25667   Level I

Multivariable & Vector Analysis

Summer examinations 2018-19

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 (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 ﬁeld = 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

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.