UNIVERSITY OF NEW SOUTH WALES
MATH2019
ENGINEERING MATHEMATICS 2E
Test 2
SAMPLE PAPER 2
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7 qu
LECTURE 21
FIRST ORDER DIFFERENTIAL EQUATIONS
SEPARABLE DIFFERENTIAL EQUATION
These dierential equations are of the form
dy
= f (x)g(y).
dx
and are solved by separating xs and ys on each side of the equation and integrating.
LINEAR DIFFERENTIAL EQUATION
T
LECTURE 22
APPLICATIONS OF FIRST ORDER DIFFERENTIAL EQUATIONS
When making substitutions into a D.E. remember to take care of
dy
as well as y
dx
SEPARABLE DIFFERENTIAL EQUATIONS
These are of the form
dy
= f (x)g(y).
dx
and are solved by separating and inte
LECTURE 25
NON-HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS
The non-homogeneous second order dierential equation with constant coecients
ay + by + cy = r(x)
is solved by rst solving the homogeneous problem
ay + by + cy = 0
to obtain a homogenous soluti
LECTURE 26
FORCED OSCILLATIONS AND RESONANCE
When simple periodic forcing is added to the mechanical or electrical system studied
earlier, we have to solve an equation like
my + cy + ky = F0 sin t
(1)
where, as before, m > 0, c > 0, k > 0.
This models a m
LECTURE 24
HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS
The homogeneous second order dierential equation with constant coecients
ay + by + cy = 0
is solved by rst forming the auxiliary equation (also called the characteristic
equation)
a2 + b + c = 0.
LECTURE 27
REVISION OF MATRIX THEORY
If A is an m n matrix and B is an p q matrix then AB exists i n = p (inner
dimensions match) and the product is an m q matrix (the outer dimensions).
The identity matrix I serves as the 1 of matrix theory.
The trans
LECTURE 29
EIGENVALUES AND EIGENVECTORS
Given a square matrix A, a non-zero vector v is said to be an eigenvector of A if Av = v
for some R. The number is referred to as the associated eigenvalue of A.
We rst nd eigenvalues through the characteristic equa
LECTURE 30
SPECIAL MATRICES
A matrix A is said to be symmetric in A = AT .
The eigenvectors from dierent eigenvalues of a symmetric matrix are mutually perpendicular.
A matrix Q is said to be orthogonal if QT Q = I or equivalently Q1 = QT .
The columns of
LECTURE 34
LAPLACE TRANSFORMS
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s+1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(s2 b2 )
sin bt bt cos bt
2b3
LECTURE 17
DOUBLE INTEGRALS IN POLAR COORDINATES
x = r cos()
y = r sin()
x2 + y 2
r=
tan() =
y
x
dA = dxdy = dydx = rdrd
Sometimes integrals become much simpler when expressed in terms of the polar coordinates (r, ) rather than the usual Cartesian coordin
LECTURE 16
CHANGING THE ORDER OF INTEGRATION AND AREAS
It is important to be able to convert
f (x, y) dxdy into
f (x, y) dydx and vice
versa. This should always be done by rst carefully sketching the region of integration .
First a revision example from t
LECTURE 12
VECTOR CALCULUS
By establishing a time dependent position vector of a particle
r(t) = x(t)i + y(t)j + z(t)k
it is possible to not only specify paths in R3 but to also to use calculus to analyse the
velocity v(t) = x(t)i + y(t)j + z(t)k and acce
UNIVERSITY OF NEW SOUTH WALES
MATH2019
ENGINEERING MATHEMATICS 2E
Test 2
SAMPLE PAPER 1
This sheet must be lled in and stapled to the front of your answers
Students Surname
Initials
Student Number
Tutorial Code
Tutors Surname
Time allowed: 50 minutes
7 qu
MATH 2019: ENGINEERING MATHEMATICS 2E
SEMESTER 2, 2015
Course information:
Course Authority: Prof Jeya Jeyakumar
Lecturer: Prof Jeya Jeyakumar
Oce: Red Centre 2073
Email: v.jeyakumar@unsw.edu.au
Lecturer (Problem Class): Milan Pahor, Red Centre 3091, m.pa
LECTURE 2
CHAIN RULE
The Chain Rule for functions of more than one variable:
If z = f (x, y) and x = x(t) and y = y(t) then
z x z y
dz
=
+
dt
x t
y t
If z = f (x, y) and x = x(u, v) and y = y(u, v) then
z
z x z y
=
+
u
x u y u
and
z
z x z y
=
+
v
x v y v
LECTURE 3
TAYLOR SERIES AND ERROR ESTIMATION
The Taylor Series for functions of two variables:
The Taylor Series of f (x, y) about the point (a, b) is
f (x, y) = f (a, b) + (x a)
+
1
2!
(x a)2
f
f
(a, b) + (y b) (a, b)
x
y
2f
2f
2f
(a, b) + (y b)2
(a,
LECTURE 11
APPLICATIONS OF GRAD
Given a scalar eld , the directional derivative of in the direction of the vector b
is given by (grad ) b where b is the unit vector in the direction of b.
Given a scalar eld at a point P the direction of maximum increase o
LECTURE 14
LINE INTEGRALS
Line integrals are used to calculate the work done in moving a particle P from A to B
along a path C in a force eld F.
F dr =
C
(F1 dx + F2 dy + F3 dz)
C
In general, this integral depends not only on F but also on the path C we t
LECTURE 39: THE HEAVISIDE FUNCTION
LAPLACE TRANSFORMS
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s+1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(s2 b2
LECTURE 40
DIFFERENTIAL EQUATIONS VIA LAPLACE TRANSFORMS 1
LAPLACE TRANSFORMS
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s+1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2 + b2 )
sinh bt
b/(s2 b2 )
cosh bt
s/(
LECTURE 41
DIFFERENTIAL EQUATIONS VIA LAPLACE TRANSFORMS 2
LAPLACE TRANSFORMS
Oliver Heaviside
u(t 1850) u(t 1925)
est f (t)dt = F (s)
Lcfw_f (t) =
0
f (t)
1
F (s)
1/s
t
1/s2
tm
m!/sm+1
t , ( > 1)
( + 1)/s+1
eat
1/(s + a)
sin bt
b/(s2 + b2 )
cos bt
s/(s2
LECTURE 1
PARTIAL DIFFERENTIATION
The deﬁnition of partial diﬁ'erentiation:
Suppose :, = f(.r.y). Deﬁne the partial derivatives of f with respect to .r and y as
_ Hm ft-I'+A-r-y}-f{-r-y) Q; “m flJ‘-!I*-'3.U)—fl-I'-!ll
air—m Ar ‘ é)” _Xy-n Ag;
Notation
ﬂ _
LECTURE 15
DOUBLE INTEGRALS
For a region S! in tlw .r —- 3; plane and a surface
/ fir. 3;}rl'yr.’.!'.
‘ - II
J
= ﬁr. y; in 33 11:0 rimlhlv integral
evaluates tht‘ vuimln- n? 111v solid nlmw E! and hv'um' : fixayi.
“'Ilvn r‘mlur-Uing {lullhh‘ inn-gr
LECTURE 2.!
FIRST ORDER DIFFERENTIAL EQUATIONS
SEPARABLE DIFFERENTIAL EQUATION
Tiit‘ﬁt’ tiiift'ri-ntinl (-quntitms :m- of tilt- form
rf
1:;mmn
(II
and are solwd l)_\' squirming .i":-: and y's ml mnh side of the equation and init‘grnting.
LINEAR DIFFEREN
LECTURE 54
HEAT EQUATION WITH INSULATED ENDS
The equation
is called the one-dimensionai heat equation. It governs the heat ﬂow across a homogenous bar
where c is determined by the thermal properties of the bar.
The adiabatic boundary conditions
@5303, t)
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JUNE 2014
MATH2019
ENGINEERING MATHEMATICS 2E
(1) TIME ALLOWED 2 hours
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUE
TOPIC 2 : SCALAR FIELDS
2.1 EXTREME VALUES
Second Derivative Test
If f and all its first and second partial derivatives are continuous
in the neighbourhood of (a, b) and fx (a, b) = fy (a, b) = 0 then
(i) f has a local maximum at (a, b) if fxx < 0 and D =
TOPIC 1 : FUNCTIONS OF SEVERAL VARIABLES
1.1 PARTIAL DIFFERENTIATION
The definition of partial dierentiation:
The partial derivative of z(x, y) with respect to x and y is defined as
@z
z(x +
= zx = lim
x !0
@x
x, y)
x
z(x, y)
@z
z(x, y +
= zy = lim
y !0
@