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Chapter Introductions


“I hear, and I forget. I see, and I remember. I do, and I understand.”
– Confucius

Chapter 1: DIMENSIONAL ANALYSIS–A SHORTCUT TO OBTAIN A MATHEMATICAL MODEL FOR THE LAWS OF NATURE
Everything and its state in our world can be described by five fundamental quantities: Mass (M), length (L), time (T), temperature (Θ), and electric current I. These five are called fundamental dimensions. The dimension of a given quantity, such as energy, can be expressed in terms ofthese five fundamental dimensions. For example, the dimension of energy is ML2T−2. This chapter shows the dimensions of commonly used quantities, such as force, pressure, electric field, and magnetic field. Each physics phenomenon must have a constraint of dimensions, which helps us discover a mathematical model for the phenomenon using dimensional analysis. The approach is that the two sides of an equation must have the same dimension or units.

Chapter 2: BASICS OF R PROGRAMMING
R is free and ranked seventh among the IEEE's top programming languages.
R is the easiest to learn among these top languages for mathematical modeling students.
It is popular in today's mathematical modeling learning process to use computers and smartphones to deal with complex and tedious algebras so that students can focus on efficient and correct usage of the mathematical tools for accurate statement of the problem, precise description of assumptions, clear outline of the modeling method, comprehensive interpretation of the results. Among many software packages used in applied mathematics, engineering, and the big data community, R's popularity has dramatically increased in recent years due to its simplicity and enormous power of handling big data. The freely available R programs can be used to do statistics, perform both numerical and symbolic calculations, plot graphics, and generate animations. We thus choose to include the basics of R for this book. A student who has mastered the R examples used in this book should have sufficient skills to develop R projects independently. A companion Python code for this book is available from the future book website.

Chapter 3: BASICS OF PYTHON PROGRAMMING
Python is free and ranked 1st among the IEEE’s top 10 programming languages in 2022. Python skill is a shortcut for a mathematical modeling students to find a good job. It has many libraries for data science and machine learning. Both ChatGPT and Copilot can be your AI programming assistants, in addition to conventional help from Google search.

It is popular in today’s mathematical modeling learning process to use computers and smartphones to deal with complex and tedious algebras so that students can focus on efficient and correct usage of the mathematical tools for accurate statement of the problem, precise description of assumptions, clear outline of the modeling method, comprehensive interpretation of the results. Among many software packages and programming languages used in applied mathematics, engineering, and the big data community, Python and its libraries appear to be relatively new in mathematical learning, but clearly are on their way to dominance due to their popularity in all kinds of applications and job market. The freely available Python programs can be used to do statistics, perform both numerical and symbolic calculations, plot graphics, and generate animations. We thus choose to include the basics of Python for this book. When you have mastered the Python examples used in this book, you should have sufficient skills to develop Python projects independently.

Chapter 4: LINEAR MODELS USING REGRESSION AND DATA
Linear models mean that the mathematical models are in linear relationships, such as when the data of and are given and form the points on a scatter plot, the best straight line that best fits the points is a linear model for the data. This chapter discusses the geometric meaning and its derivation of a linear model using the concept of anomaly data. Computer code and real-world data are used to show the model development.

Chapter 5: PRINCIPLES OF MATHEMATICAL MODELING
You may become a consultant using your mathematical modeling skills. Then what steps and principles you should follow so that you can deliver an excellent consulting report to your client and hence do a good job? The steps may be summarized from the previous examples of this book and from our common sense from the point of view of a client.
Usually, a mathematical modeling process has five steps, denoted by DAESI: Description (D), abstraction (A), equations (E), solution (S), and interpretation (I). Real-world examples are used to describe the five-step method.

Chapter 6: MATHEMATICAL MODELING BY LINEAR ALGEBRA
Mathematics is a logic, precise and scientific language to describe almost any quantitative problems in our real life, including the best route to commute to work, financial planning, climate change, insurance, aircraft design, and rocket science. The real life mathematical models almost surely use data and can encounter all kinds of areas of mathematics: trigonometry, linear algebra, calculus, complex analysis, differential equations, probability, and statistics. This chapter discusses examples of math modeling using linear algebra.

Chapter 7: MATHEMATICAL MODELING BY CALCULUS
Using calculus for mathematical models has been common in engineering and science. In modern mechanical technology, calculus modeling is almost inevitable, such as calculating the thrust of a rocket, the orbit of a spacecraft, and heat dissipation in our cellphone. This chapter will use the DAESI five-step method and a few examples to help you learn how to develop a mathematical model using calculus.

Chapter 8: PROBABILISTIC MODELS
Many events around us are random and non-deterministic. The measure of the randomness is probability, a percentage chance at which a particular event can occur. For example, what is the probability of getting a winning lottery ticket? What is the chance to rain in my city on April 5th? Although we cannot be sure what something can definitely happen due to the randomness nature, it is possible in many cases to develop models to find out the probability of the occurrence of a specific event. This chapter presents a few examples of mathematical models for probability calculations. Please be advised that probabilistic models have vast varieties and there are numerous books devoted to the subject. This chapter only presents a few simple examples with computer simulations. For some problems, analytic mathematical modeling might be very difficult or impossible, while probabilistic computer simulations, often known as Monte .

Chapter 9: STOCHASTIC MODELS
This chapter starts with a nowhere differentiable but everywhere continuous model. From here, we introduce white noise, random walk, Ito calculus, and fractal dimensions.

Chapter 10: VISUALIZE MATHEMATICAL MODELS BY R and Python
Chapter 3 already showed how to plot a simple time series using plot(xtime, ydata). Climate science often requires one to plot two different quantities, such as two time series, on the same plot so that direct comparisons can be made. For example, to see whether a hot year is also a dry year, one may plot the temperature data on the same figure as the precipitation data. The left side of the y-axis shows temperature and the right side shows precipitation. This chapter provides the R and Python code for plotting figures commonly used in physics and engineering.