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(MATH102)[2005](f)midterm~1659^_10419.pdf
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HKUST MATH 102
Midterm Examination

Multivariable and Vector Calculus
21 Dec 2005
Answer ALL 8 questions
Time allowed C 180 minutes

Directions C This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit.
Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet.
Student Name:
Student Number:
Tutorial Session: Problem 1
Question No. Marks
1 /20
2 /20
3 /20
4 /20
5 /20
6 /20
7 /20
8 /20
Total /160


(a)
Assume a, b and c are three dimensional vectors and if (a b) c = .a + b + .c.

Use su.x notation to .nd ., and . in terms of the vectors a, b and c. Can you say something about the direction of the vector (a b) c.


(b)
Let a be a constant vector and r =(x, y, z), use su.x notation to evaluate


(i) . r, (ii) . (a r), (iii) . (a r).
Problem 2

(a) Sketch and describe the parametric curve C
r = t cos t i + t sin t j + (2. . t) k, 0 . t . 2..
Show the direction of increasing t. Find the project curve C onto the yz-plane.

(b) Find a change of parameter t = g( ) for the semicircle
r(t) = cos t i + sin t j, 0 . t . .
such that (i) the semicircle is traced counterclockwise as varies over the interval [0, 1],
(ii) the semicircle is traced clockwise as varies over the interval [0, 0.5].
Problem 3
If a wheel with radius a rolls along a .at surface without slipping, a point P on the rim of the wheel traces a curve C , .nd the parametric equation of the point P . Suppose that the point P on the wheel is initially at the origin. Find also the arc length of the curve C if the wheel makes one complete turn (no need to carry out the integration).
Problem 4

(a) Verify the formula for the arc length element is cylindrical coordinates,
.
..2 ..2 ..2
dr ddz
ds = +(r(t))2 + dt.
dt dtdt

(b)
Find a similar formula as in (a) for the arc length element in spherical coordinates.

(c)
Use part (b) or otherwise, .nd the arc length of the curve in spherical coordinates: =2t, = ln t, = ./6; 1 . t . 5.


C1C
Problem 5
2
2xy(x. y2 )
if (x, y) .= (0, 0),Let f (x, y)= x2 + y2
0 if(x, y) = (0, 0).

(a)
Is the function continuous at (0, 0)?

(b)
Calculate fx (x, y), fy (x, y), fxy (x, y) and fyx (x, y) at point (x, y) = (0., 0). Also calculate these derivatives at (0, 0).

(c)
Is fyx (x, y) continuous at (0, 0)?

(d)
Explain why fyx (0, 0) .(0, 0).


= fxy
Problem 6

Find the distance from the origin to the plane x +2y +2z = 3,
(a)
using a geometric argument (no calculus),

(b)
by reducing the problem to an unconstrained problem in two variables, and

(c)
using the method of Lagrange multipliers.


Prob