Page 1 of 5 Continued overleaf

UNIVERSITY OF GLASGOW

Degrees of MEng, BEng, MSc and BSc in Engineering

MATHEMATICS AE2X (ENG2042)

Sample Paper


Attempt ALL Questions
Attachments:

Table of some standard derivatives and integrals

The numbers in square brackets in the right-hand margin indicate the marks allotted to the
part of the question against which the mark is shown. These marks are for guidance only.
An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.


Page 2 of 5 Continued overleaf
Q1. (a) Let C be the boundary of the finite region, A , enclosed by the curve 2y x=
and the line y = 2x.
(i) Sketch the region A , indicating co-ordinates of limit points. [3]
(ii) Evaluate the line integral 3 2 2
C
xy dx x y dy⎡ ⎤+⎣ ⎦∫ directly, where C is
the anticlockwise path around the boundary of A. [7]
(iii) Evaluate the integral in (b) using Green’s Theorem. [6]
(b) A circular helix is represented as a function of a parameter, t, by the position
vector r(t) = [ ]2cos 2 , 2sin 2 , 6t t t .
(i) Calculate the unit tangent vector at the point for which 4t
π
= . [4]
(ii) Calculate the length of the curve between points (2,0,0)A and
(2,0,12 )B π . Sketch the curve between these points. [5]
Q2. (a) A function on the x-y plane is defined by 2 31( , )
3
f x y x y y= − .
(i) Find the direction of maximum increase of the function at the point
P(2,3), and determine the rate of change of f in that direction. [4]
(ii) Using the result from (i) or otherwise find the equation of the tangent to
the level curve of f at the point P. [4]
(b) A vector field is defined as
2
2[2 , , 2 3 ]
2
zxy x z y xzα β δ= + + +v .
(i) Find constants α, β, δ such that the vector field is irrotational. [6]
(ii) For the above values of α, β, δ determine a potential function f (x, y, z)
for v [7]
(iii) Hence evaluate the line integral
C∫ v dri where C is any curve joining
point (2,1, 4) to (3, 2,6)− . [4]


Page 3 of 5 Continued overleaf
Q3. (a) Let a scalar field f and vector fields v and w be defined as follows on 3\ :
( , , )f x y z zy yx= + ; [ ]( , , ) , , 4x y z y z z x= −v ; 2 2 2( , , ) , ,x y z y z x⎡ ⎤= ⎣ ⎦w
(i) State whether the following operations are valid or invalid, giving
reasons in each case
grad( ) , curl( ) , div( ) , curl( )f• • × •v w v w v v w [8]
(ii) Evaluate all valid expressions in part (i) [10]
(b) Let r be the position vector [x, y, z] , with r = |r| , and let 3∈a \ be a constant
vector. Show that:
( )3 3 5grad 3r r r
⎛ ⎞
= −⎜ ⎟⎝ ⎠
a ra r a r
ii [7]

Page 4 of 5 Continued overleaf
Q4. (a) Let S be the surface of the paraboloid, ( )2 2( , ) 2z x y x y= − + , above the x-y
plane as indicated in Fig. Q4a.
(i) Show that, given a function G(x,y,z) defined on S ,
( ) 2 2( , , ) , , ( , ) 1 4 4
S A
G x y z dS G x y z x y x y dxdy= + +∫∫ ∫∫
where A is the circular region obtained by projecting S onto the x-y
plane. [7]
(ii) Hence, by employing polar coordinates or otherwise, calculate the
surface area of the paraboloid. [10]

Fig. Q4a
(b) Let S be the surface of the circular cylinder with enclosed region
{ }2 2: 1 , 0 2V x y z+ ≤ ≤ ≤ , illustrated in Fig. Q4b
Using the Divergence Theorem of Gauss, show that the surface integral
S
dS∫∫ F ni can be reduced to the following form:
28
S A
dS x dxdy=∫∫ ∫∫F ni
where 2 2 3 23 , , 3xy yx y zx⎡ ⎤= −⎣ ⎦F , n is the outward pointing normal of S
and A is the projection of V into the x-y plane. [8]




Fig. Q4b
y
x
z y
x
y
x 0 1
z
2
x
0 y
1
Page 5 of 5 End of question paper
Table of some standard derivatives and integrals






































































































































































































































学霸联盟