Calculus & Linear Algebra II Problem Set (Empty 13/9/2022)

The a-s-s-e-s-s-m-e-n-t will be 7 calculation questions. Please see the questions shown in the screenshot. I will send you all info after being hired, eg PPTs, student access etc. Please send a draft in 12hrs -1 day time, day 2, and day 3 as well. + Will need to draft some questions to ask the teacher and revise base on feedback (Send bk ard in 1 day max)

MATH7000 PROBLEM SET
SEMESTER 2 2022
Due at 4:00pm 19 October. Marks for each question are shown. Total marks: 30
Submit online via the MATH7000 Problem Set submission link in Blackboard.
(1) (5 marks.) Consider the non-homogeneous ODE
√
d2 y
dy
−
y
=
1 + x2 .
+
x
dx2
dx
The corresponding homogeneous ODE is
(1 + x2 )
d2 y
dy
− y = 0.
+
x
dx2
dx
(i) Show that by the variable change x = sinh t, the homogeneous ODE is transformed to
d2 y
− y = 0.
dt2
(ii) A fundamental set of solutions of this equation can be chosen to be y1 =
sinh t and y2 = cosh t. Deduce that a fundamental
set of solutions of the
√
homogeneous ODE are y1 = x and y2 = 1 + x2 .
(iii) Find the general solution of the non-homogeneous ODE.
(1 + x2 )
(2) (5 marks.) Determine for what real numbers a and b, the map f : R2 × R2 given
by
f ((x, y), (x0 , y 0 )) = axx0 + byy 0
is an inner product.




x1
y1
(3) (5 marks.) Suppose that the orthogonal transformation
= P
x2
y2
2
2
2
2
transforms f = x1 − 4×1 x2 + 4×2 into f = 4y1 + 4y1 y2 + y2 . Find the orthogonal
matrix P .
(4) (4 marks.) Suppose that f (x, y) is continuous on D = {(x, y)|x2 + y 2 ≤ y, x ≥ 0}
and satisfies
p
8
f (x, y) = 1 − x2 − y 2 −
f (x, y) dxdy.
π
x
D
Find f (x, y).
y
y
(5) (4 marks.) You are given that
V
f (x, y, z) dxdydz =
f (x(u, v, w), y(u, v, w), z(u, v, w)) |J| dudvdw,
R
1
under the variable transformation x = x(u, v, w), y = y(u, v, w) and z = z(u, v, w),
which maps the region R (in the uvw-space) one-to-one into the region V (in the
xyz-space). Here J is the Jacobian of the variable transformation defined by
 ∂x ∂x ∂x 
∂u
∂v
∂w
∂(x, y, z)
∂y
∂y
∂y 
.
J=
= det  ∂u
∂v
∂w
∂(u, v, w)
∂z
∂z
∂z
∂u
∂v
∂w
Use an appropriate variable transformation to evaluate the triple integral
s
 2
3/2
x
y2 z2
1−
dV,
+ 2 + 2
a2
b
c
y
V
2
2
2
where V is the region enclosed by xa2 + yb2 + zc2 = 1.
(6) (3 marks.) Suppose that z = f (x, y) is continuous and differentiable and satisfies
the relation
∂z
∂z
2
2
+y
= e(x−1) +y .
(x − 1)
∂x
∂y
Evaluate
Z
∂z
∂z
dy −
dx,
∂y
c ∂x
√
where C is the curve from (2, 0) to (0, 0) along y = 2x − x2 .
R
R
Hint: Note that c F(r) · dr = c F(x, y) · T(x, y) dS, where T(x, y) is a unit
tangent vector to C at point (x, y) on C.
(7) (4 marks.) Compute the line integral
I
(4x − y)dx + (x + y)dy
,
4×2 + y 2
C
where C is the circle x2 + y 2 = 2 traversed in an anti-clockwise direction.
2