The term
“modeling and applications” has been increasingly used to denote
all kinds of relationships whatsoever between the real world
applications and mathematics. Using mathematics to solve real
world problems is often called applying mathematics, and
a real world situation which can be tackled by means of
mathematics is called an ‘application’ of mathematics.
The term modeling, on the other hand, is the process of representing
the behavior of a real system by a collection of mathematical
equations. That means it focuses on the direction from reality
to mathematics while application focuses on the opposite
direction from mathematics to reality. Modeling
aims at providing students with a better understanding of
mathematical concepts and teaching them to formulate and solve
applicationorientedproblems.
Mathematical models are developed to help in the understanding
of physical systems. Engineers use models to represent the
elements of any system. Models are generated for manufactured
elements and devices in order to facilitate understanding and
establish the operating characteristics of the elements and
devices.
Modeling of systems, such as manufacturing
systems, can be achieved using a number of tools and techniques
one of which is simulation. Simulation is the technique
of building a model of a real or proposed system to study the
behavior of the system under various conditions as time
progresses.
Lesson on Modeling
The
development of models needs mathematical tools of major
components. Engineers use mathematical expressions, such as
constraint equations, to describe, analyze, and communicate
models of physical devices and their behavior. The quantities
represented in these expressions are different from purely
numerical values, and the algebra for operating over them must
account for extranumerical considerations such as dimensional
consistency and units of measure. The
mathematical tools students need are developed to a more
advanced level, enabling students to model physical problems
such as human walk, fluid motion, thermal profiles, and electric
circuits.
Algebra and Geometry
There is one thing we
would like to focus on before all else: algebra. If the student
can go into university knowing algebra, he/she will be ready to
take on everything else. The student should not only focus on
how it is done but why each technique works and when it is
allowed. Taking precalculus or trigonometry in high school may
sound very exciting, but a strong algebra base is going to be
much more helpful. Another important subject
is geometry, which gives the student the opportunity to learn
how to do proofs, which is a vital talent in all highlevel
mathematics. Trigonometry will be very useful, however, students
take a lot of that along the way in most high school mathematics
chain even if they do not take a specific course in it.
Functions and Relations
In
science and engineering there are numerous problems which can be
described and solved with the help of
functions. A physical quantity is either a constant
quantity or a function quantity. The mass in our example model
is a constant quantity, like 5 kg. The altitude of a particle
over time is a function quantity, mapping quantities of time to
quantities of length.
In general, functions of
engineering importance include
transcendental and
algebraic functions, and their properties and inverses. For
example, elementary functions (sine, cosine, tan, exponentials,
and logarithms) are the most commonly used mathematical
functions in science and engineering. Complex
functions are useful tools in
engineering especially when used
with computer algebra systems. Main fields of applications
include: steady state plane vector fields in mechanical and
electrical engineering such as fluid dynamics, heat flow,
electromagnetic fields; control systems, and signal processing.
Scalar and Vectors
Physical
quantities include not only scalars but also vectors, which are
distinct from scalars, in that complete specification of a
vector constant requires a statement of direction or
orientation. The students should be able to differentiate scalar
quantities from vector quantities. We provide a list of scalar
and vector quantities which the student will encounter
throughout his study. For instance, force as a physical quantity
in engineering could be a good example of a vector. One can
consider the problem of calculating the force relative to two
charges in a space. This problem may be easily visualized and
students can recall their own experience in electricity.
Matrices and Linear
Algebraic
Matrices and linear algebra, in particular, help
computers to solve algebraic problems. Quite often, students struggle
when confronted with different notations used in algebraic
equations. Often, they had memorized algorithms that relied on
one form of notation.
Calculus
Fundamental
to calculus are derivatives, integrals, and limits.
Calculus
is useful first and foremost as the foundations for most
highlevel calculus such as multivariate calculus which is, as
the name implies, the same calculus but applied to three or more
dimensions.
In a more practical setting, this technique helps to
solve problems that deal in multiple dimensions. For example,
those dimensions could be temperatures, port speeds, processing
speeds, resistance, voltage, and so on.
Based on our experience, students
are not confident in the concept of derivative or integration. Calculus
is the highest mathematics offered in high school curriculum. In
university, it is one of the lowest mathematics classes offered.
This results in a totally diverse instruction approach.
Differential Equations
Differential
equations
seem to be a stumbling block for many students. Differential
equations are used to construct mathematical models of physical
phenomena such as control systems, fluid dynamics or celestial
mechanics. Physicists and engineers are interested in how to
compute solutions to differential equations. These solutions are
usually used to solve various engineering problems.
Reverse Engineering
