ME 5331 Analytical Methods in Engineering
Lecture Notes
Vector Calculus (Part I)
Differentiation of Vectors:
Continuity: A vector function
is said to be continuous at t=t0 if it is defined in some
neighborhood of t0 and
If we introduce a Cartesian coordi

ME 5331 Analytical Methods in Engineering
Fall 2015
DEPARTMENT OF MECHANICAL
AND
AEROSPACE ENGINEERING
UNIVERSITY OF TEXAS AT ARLINGTON
Mid-Tenn Exam #1
CLOSED BOOK AND NO NOTES
(No Electronic Devices)
Show all your work
Illegible writings or incomplete e

Fall 2014
ME5331-003 Homework 4
Assignment given on September 24, 2014
Assignment due on October 1, 2014
Please solve the following problems, showing all steps of your analysis.
1. Find the solution of the IVP using Laplace transforms
2. Solve the followi

Fall 2014
ME5331-003 Homework 3
Assignment given on September 17, 2014
Assignment due on September 24, 2014
Please solve the following problems, showing all steps of your analysis.
1. Solve the following non-homogeneous differential equations using the me

Fall 2014
ME5331-003 Homework 7
Assignment given on October 22, 2014
Assignment due on October 29, 2014
Please solve the following seven problems, showing all steps of your analysis.
1. Compute the following vector operations:
with the vectors
and
,
2. Co

Fall 2014
ME5331-003 Homework 9
Assignment given on November 26, 2014
_
Please solve the following seven problems, showing all steps of your analysis.
1. Derive the Fourier series of the periodic functions below. Evaluate their
convergence: At which value

Fall 2014
ME5331-003 Homework 1
Assignment given on September 3, 2014
Assignment due on September 10, 2014
Please solve the following problems, showing all steps of your analysis.
1. Find the general solution of the following differential equation
2. Find

Fall 2014
ME5331-003 Homework 6
Assignment given on October 15, 2014
Assignment due on October 22, 2014
Please solve the following problems, showing all steps of your analysis.
1. Find and classify all local maxima, local minima, and saddles for the funct

ME5331-003 Homework 8
Fall 2014
Assignment given on November 12, 2014
Assignment due on November 19, 2014
Please solve the following seven problems, showing all steps of your analysis.
1. Verify the divergence theorem by working out both sides of the Equa

ME 5331 - Handout 1
9 September 2015
The following are examples worked out in the textbook:
Integrating Method
Problems to Practice
Note: the only problems with solutions in the book are (b), (e), and (f).

ME 5331 Homework #2
Due: 9/18/2013
1 (10pts) Find the general solution of xy '+ 2 y ' =
0
2 (10pts) Find the general solution of 16 y ' 8 y '+ 5 y =
0
3 (10pts) Find the general solution of 4 y '+ 16 y '+ 17 y =
0
4 (20pts) Solve the initial value problem

ME5331 Homework#5
Due:11/6/2013
(20 pts each problem)
(1). Find the directional derivative of given functions at the given point in the
given direction.
(a). (, ) = 2 + + 2 ; (1,2) ; =
3
(b). (, ) = 2 3 in the direction of + 2 at [2,-1, 4].
(2). Find a ve

ME 5331
Analytical Methods in Engineering
Prof. Albert Y. Tong
Ordinary Differential Equations: Part III
Solution of Non-homogeneous Equations
Beginning with the
order linear O.D.E.
The Existence and Representation theorem is used to guide our approach.

ME 5331
Analytical Methods in Engineering
Prof. Albert Y. Tong
Laplace Transform
Definition of Laplace Transform
Let f (t ) be a given function which is defined for all t > 0 . The Laplace
transform of f (t ) , denoted by f (s ) or Lcfw_ f (t ), is given

ME 5331 Analytical Methods in Engineering
Fall 2015
Mondays and Wednesdays 4:00 5:20 p.m.
Woolf Hall 404
Instructor:
Albert Y. Tong; Ph.D.
Office:
Room 206A Woolf Hall
Office Hours:
2:00 4:00 p.m. MW or by appointment.
Phone:
817-272-2297
Mailbox:
Box 190

ME5331 Homework 6
Due: 11/27/13
1. Let (, ) = ,and denote (), () = (). Evaluate /, where = + 1, = .
2. If = 2 () + 2 () + 2 () = () is the distance of a particle from the origin of a Cartesian x, y, z
coordinate system and t is the time, use chain rule to

ME5331 Homework#4
Due:10/21/2013
(20 pts each problem)
Find the Fourier series of f, given over the period as follows. At which values of x, if any, does the series
fail to converge to f(x)? To what values does it converge at those points?
1. () = (, ]
y

Fall 2014
ME5331-003 Homework 2
Assignment given on September 10, 2014
Assignment due on September 17, 2014
Please solve the following problems, showing all steps of your analysis.
1. Solve the following differential equations
(a)
(b)
(c)
(d)
(e)
(f)
+ 2

ME5331 Lecture 5
Solution of Non-Homogeneous Differential Equations with
Constant Coefficients
(1)
In order to solve a non-homogeneous linear equation, we must first know how to
solve the associated homogeneous equation.
The general solution of (1) will b

ME5331 Lecture 4
Higher Order Differential Equations
Introductory Example: Consider a long slender vertical column of length and
uniform cross section. The column is hinged at both ends. Find and analyze the
deflection if the column is subjected to a vert

ME5331 Lecture 14
Space Curves and Surfaces
1. Space Curves
In a Cartesian coordinate system, the position vector is given by
(1)
in which
and
over a closed interval
are continuous functions of a real parameter
.
defined
If the tail of the position vector

ME5331 Lecture 13
Review of Vectors in 3-D
A vector is a quantity consisting of a non-negative magnitude and a direction. Many
quantities behave as vectors, e.g., displacement, velocity, acceleration, force. Thus,
vectors in three-dimensional space are th

Second-Order Differential Equations:
Applications
y' ' ay' by f ( x)
d 2x
m 2 kx 0
dt
Free Vibration
d 2x
dx
m 2 c kx 0
dt
dt
Free Vibration with Viscous Damping
d 2x
m 2 kx N
dt
d 2x
m 2 kx F0 cos t
dt
d 2x
dx
m 2 c kx F0 cos t
dt
dt
Free Vibration with

ME5331 Lecture 12
Theory of Maxima and Minima: Extrema of Functions
Introduction - As we learned in calculus, for a function
critical) point is a point at which
absolute maximum at a point
absolute minimum at a point
domain of definition of ).
, a station

ME5331 Lecture 11
Taylors Series of functions of n-variables
Taylors theorem (discovered first by Gregory) states that any function satisfying
certain conditions can be expressed as a Taylor series.
Taylor series: a representation of a function as an infi

ME5331 Lecture 10
Differential Calculus of Functions of Several Variables
We begin analysis with real-valued functions of more than one variable,
.
The study of these multi-variable functions will include concepts such as derivatives, chain
differentiatio

ME5331 Lecture 8
Laplace Transform
Suppose we need to solve the following IVP where
Solution: To solve it with Laplace transform first we note that
with Laplace transform we get (as we shall see later)
The given IVP is then equivalent to
We deduce
Now we