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Modeling and Applications


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 application-oriented-problems.


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


Tools for Model Building


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 extra-numerical 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 pre-calculus 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 high-level 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.



Fundamental to calculus are derivatives, integrals, and limits. Calculus is useful first and foremost as the foundations for most high-level 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.


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