College of Engineering and Computer Science
Mechanical Engineering Department
Mechanical Engineering 501A
Seminar in Engineering Analysis
Fall 2004 Number 17472 Instructor: Larry Caretto
Goals and Assignments for the First Week
First week goals
This week will provide a review of topics that you should have seen in your undergraduate
courses, but may have forgotten because you have not used them in a while.
The Tuesday lecture will show the derivation of some differential equations to define engineering
problems, which we will seek to solve by the engineering analysis and numerical analysis
techniques to be learned in this course. We will return to these problems as we learn the tools
that allow us to solve them for systems of interest.
We will begin the course material with a basic review of vector concepts that you have seen in
courses from high school physics to your junior and senior courses in mechanics. This includes
the basic idea of representing a vector, which we think of has having a magnitude and a direction,
as a set of components. These components are not actually vectors, but are scalars that multiply
unit vectors in the two or three coordinate directions that describe our problem. We will also
discuss the dot product of two vectors, a•b = |a||b|cos(), where  is the angle between the two
vectors and |a| and |b| represent the magnitude of the vectors. You should be able to solve
problems using simple vector operations and dot products.
Next, we will discuss the basic properties of matrices and show the use of row and column
matrices to represent vectors. We will discuss the addition and multiplication of matrices, the
multiplication of a matrix by a scalar, and the transposition of matrices. You should be able to
work problems with all these concepts.
On Thursday, we will build on the introductory material on matrices to will discuss determinants;
the determinant is related to, but not the same as a matrix. Determinants play an important role
in theoretical concepts of linear systems, but are generally not used in computations. You should
be able to compute the determinant of a square array of numbers or symbols.
The final topic on Thursday will be the inverse of a matrix. Just as we can solve the scalar
equation ax = y to say y = a-1x, we can formally solve the matrix-vector equation, Ax = y by the
statement y = A-1x., where A-1.is the inverse of the matrix A. Although we will often use the
concept of a matrix inverse, we will learn that we will generally not compute the inverse to solve
the equation Ax = y. You should understand the use of matrix inverses and the formulae for
computing the components of the inverse matrix.
Reading for August 24, 2004
The reading assignments that ideally you would have read before this week’s class are the
following pages from Kreyszig; these should be a review for most students, although many
students may not have used these concepts recently.
Pages 83-87, 585-587, and 616–618 review modeling of engineering systems with which you
should be familiar from your undergraduate classes: oscillations in a spring-mass system and the
vibrating sting. This illustrates the derivation of equations for engineering problems, whose
mathematical solutions we will be considering in this course.
Engineering Building Room 1333
E-mail: lcaretto@csun.edu
Mail Code
8348
Phone: 818.677.6448
Fax: 818.677.7062
Assignments for week one
ME501A, L. S. Caretto, Fall 2004
Page 2
Pages 400–413 review the basic concepts of vectors and dot products as used in courses in
mechanics. This provides the background for the general concept of vector spaces we will be
using in the course.
Pages 303–321 cover basic matrix concepts, including matrix multiplication and the use of
matrices in network problems. We will be using matrices for analysis of engineering systems that
have several linked components.
The corresponding material in the lecture notes on vectors and matrices is on pages 1-8
(introduction and matrix multiplication).
Reading for August 26, 2004
Pages 341–346 discuss the basic properties of determinants and the evaluation of determinants.
These results are important in mathematical analysis, but computations with linear systems
should generally not be done with determinants.
Pages 350–358 cover he basic concept of a matrix inverse and formulae for computing the
inverse.
The corresponding material in the lecture notes on vectors and matrices is on pages 15-20.
Homework due August 31, 2004
Page 407, problems 11, 18, and 20. These illustrate basic relationships among vectors and
vector components.
Page 413, problems 20, 23, and 31. These illustrate the use of the dot product of vectors, and its
abilities to determine the angle between two vectors and the component of one vector in the
direction of the other.
Pages 309-310, problems 5 and 11. These require the use of the basic rules for matrix
multiplication.
Page 319, problem 4. This illustrates matrix multiplication and the need for compatibility of the
two matrices to be multiplied.
Page 349, problem 13. This problem shows that you can choose any row or column to apply the
general formula for evaluating a determinant.
Page 357, problem 12. This problem requires a theoretical proof that the inverse of the transpose
is the same as the transpose of the inverse. This problem is approached by using the basic
relations that AA-1 = A-1A = I, and (AB)T = BTAT.