Notes on Laplace Transforms
Laplace Transforms
Laplace transforms are invaluable for any engineer's mathematical toolbox as
they make solving linear ODEs and related IVP, as well as systems of linear
ODEs, much easier. Applications abound: electrical netw

Notes on Determinants of Matrices
The Determinant of a Matrix
The Determinant of 2 2 Matrices
We rst introduce the concept of determinant of a 2 2 matrix.
.
Note that only
square matrices have determinants
Denition. The determinant of the 2 2 matrix
a
A =

Notes on ERO operations and Eigenvalues and
Eigenvectors
Elementary Row Operations
Denition. An elementary row operation (ERO) on an augmented matrix pro-
duces a new augmented matrix corresponding to a new (but equivalent) system
of linear equations. Two

MEC702 Wk 13 Lect 1
Partial Dierential Equations (PDEs)
Denition.
A partial dierential equation (PDE) is an equation involving one or more
partial derivatives of an (unknown) function, call it u, that depends on
two or more variables, often time t and on

MEC702 Wk 12 Lect 2
The Fourier Transform of the Convolution
The convolution
f g
of function
f
g
and
is dened by
f (x p)g(p) dp.
f (p)g(x p) dp =
h(x) = (f g)(x) =
Theorem. (Convolution Theorem) Suppose that f (x) and g(x) are piecewise
continuous, bounde

MEC702 Wk 13 Tutorial
Fourier Cosine Transform. Fourier Sine Transform.
1. Find the Fourier cosine transform
fc (w)
and
fs (w)
of
0<x<1
1,
f (x) = 1, 1 < x < 2
0,
x > 2.
2. Find
fc (w)
and
fs (w)
of
(
f (x) =
3. Find
fc (w)
and
fs (w)
x, 0 < x < 2
.
0, x

MEC702 Wk 13 Lect 2
D'Alembert's Solution of the Wave Equation.
First, we aim to transform the two partial derivatives in the wave equation, uxx
and utt . Let
w = x ct .
v = x + ct,
Then
vx = 1,
wx = 1 .
Since u(v, w) is a function of both v and w, then
u

MEC702 Wk 12 Tutorial
Evaluation of Integrals
Show that the integrals represents the indicated function. Hint: Use the Fourier
integral and the Fourier integral coecients formulae; the integral tells you
which one, and its value tells you what function to

MEC702 Wk 11 Lect 2
Fourier Integrals
Fourier series are powerful tools for problems involving functions that are periodic or are of interest on a nite interval only. Since, of course, many problems
involve functions that are nonperiodic and are of intere

MEC702 Wk 11 Lect 1
Fourier Series - Generalizing from Period p = 2
to Any Period p = 2L
Let f (x) have period p = 2L.
We introduce new variable v such that f (x), converted to a function of v , f (v)
has period 2 . Then
v
x
=
p
2
= v =
2x
= x = x = v
p
L

MEC702 Wk 11 Tutorial
Fourier Series of Period p = 2
1. Find the Fourier series of the given function which is assumed to have the
period
(a)
(b)
(c)
(d)
2 .
Show details of your work.
(
x, < x < 0
f (x) = |x| =
, period p = 2 .
+x, 0 < x <
(
x + , < x <

MEC702 Wk 10 Lect 2
Fourier Analysis
Fourier series are innite series that represent periodic functions in terms of
cosines and sines. As such, Fourier series are of grestest importance to the
engineer and applied mathematician. To dene Fourier series, we

MEC702 Wk 10 Lect 1
Background on Partial Fractions
There is a theorem in advanced algebra which states that every proper rational
fraction can be expressed as a sum
P (x)
= F1 (x) + F2 (x) + . . . + Fn (x)
Q(x)
where
F1 (x), F2 (x), . . . , Fn (x)
are ra

MEC702 Wk 9 Lect 2
Laplace Transforms
Laplace transforms are invaluable for any engineer's mathematical toolbox as
they make solving linear ODEs and related IVP, as well as systems of linear
ODEs, much easier. Applications abound: electrical networks, spr

MEC702 Wk 9 Lect 1
Criteria for Critical Points. Stability.
Critical Points of a System of ODE
For the system of ODEs
y10
=
a11 y1 + a12 y2
y20
=
a21 y1 + a22 y2 ,
we have the y1 y2 -plane (phase plane) and can obtain the derivative
y 0 dt
dy2
= 20
dy1
y

MEC702 Wk 9 Lect 3
Transforms of Derivatives and Integrals. ODEs.
Laplace Transform of Derivatives
Theorem.
The transforms of the rst and second derivatives of f (t) satisfy
1. L (f 0 ) = sL (f ) f (0)
2. L (f 00 ) = s2 L (f ) sf (0) f 0 (0).
Laplace Tran

MEC702 Wk 8 Tutorial
Matrices
1. Find a.) A + B , b.) A B , c.) 2A, d.) 2A B and e.) B + 21 A.
1
2 1
,B=
;
1
1 8
1
4
6 1
4 , B = 1 5 ;
(b) A = 2
3 5
1 10
(a) A =
1
2
2. Find a.) AB and b.) BA (if they are dened).
1 2
2 1
,B=
;
2 4
1 8
1 1 2
1 1 7
A = 2 1

MEC702 Wk 8 Lect 2
Basic Theory of Systems of ODEs
First-Order System of
ODEs
n
The system of n rst-order of ODEs is given by
y10
= f1 (t, y1 , . . . , yn )
y20
= f2 (t, y1 , . . . , yn )
yn0
= fn (t, y1 , . . . , yn )
.
.
.
.
.
.
This can be written as a

MEC702 Wk 8 Lect 1
Background on Matrices
Given are the constant c and the matrices of size 2 2
a
A = 11
a21
Matrix Addition
A+B =
a11
a21
a12
a22
b
B = 11
b21
a12
b
+ 11
a22
b21
b12
b22
b12
a + b11
= 11
b22
a21 + b21
,
a12 + b12
.
a22 + b22
Note tha

MEC702 Wk 7 Lect 2
Modeling: Electrical Circuits.
Designing good models is a task the computer cannot do.
Hence setting up
models has become an important task in modern applied mathematics.
The
best way to gain experience in successful modeling is to care

MEC702 Wk 6 Lect 1
Modeling: First Order ODE.
Given the RL-circuit
solve the resulting ODE for the current I(t) Ampheres, where t is time.
Assume that EMF is 48 volts, which has a resistor R = 11 and an inductor of
L = 0.1 H, and that the current is initi

MEC702 Wk 7 Lect 1
Non-homogeneous Linear ODEs
We now turn from homogeneous to non-homogeneous linear ODEs of nth order.
We write them in standard form as
y (n) + pn1 (x)y (n1) + + p1 (x)y 0 + p0 (x)y = r(x) .
Just like in Chapter 2, the solution of non-h

MEC702 Wk 7 Tutorial
Modeling
1. If a massspring system with an iron ball of weight 98 nt (about 22 lb) can be regarded as
undamped, and the spring is such that the ball stretches it 1.09 m (about 43 in.), how many
cycles per minute will the system execut

MEC702 Wk 6 Lect 2
Higher Order Linear ODEs
Homogneneous Linear ODEs
The ODE is of nth
order
if it includes the nth derivative
y (n) =
d4 y
dx4
of the unknown functiony(x) in the Ordinary Dierential Equation. Thus
the ODE is of the form
F (x, y, y 0 , ,