2024 Spring BS102 Engineering Mathematics I
https://github.com/HyosangKang/em2024-hkang
Differential equations are mathematical tools for modeling natural and social phenomena. The solution to the differential equation is a differentiable function, which gives simulated results as well as estimated results. We can find such solutions both in mathematical formulas and numerical data. To interpret the solution, we need to understand the definition of properties of derivatives and integrals. In this course, we learn basic concepts of differentiation and interaction on single variable functions and various techniques for solving differential equations. We also emphasize modeling real-world problems using differential equations and how to formulate proper research questions and answer those questions mathematically. We will learn how to use Matlab to solve those problems.
This course is designed for the first-year university students. The fundamental goal of the university is to educate students morally and intellectually so that they can resolve important problems in society.
In this course, students will learn how to find and model problems. Societies have many problems and most of them are very complicated. Good scientists are those who unentangle the problems and make them as simple as possible. This is what modeling means.
Second, students will learn how to formulate problems. It is easy to state a problem, whose solution is not transparent. To solve a complicated problem, we should reformulate the problem into an answerable question. This is done by splitting the problem into smaller (easier) sub-problems. We then transform each small sub-problems into concrete questions. In science, most of those questions are stated in terms of mathematical formulas and equations. This is called mathematical modeling.
Lastly, students will learn how to solve specific mathematical questions. Differential equations are the tools for mathematical modeling. They are used for modeling physical systems, engineering structures, social behavior, and analyzing information. Engineering Mathematics is a subject of study that concerns modeling problems into differential equations and solving them. This is why we learn Engineering Mathematics in the first year of the university.
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This course will run in PBL(project-based learning) style.
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The goal of this course is to learn how to set and solve problems using mathematical tools. We will do this by mathematical discussions and programming. In the discussion, we will start with the basic concepts of functions such as limits, continuity, and differentiability. And then we will discuss more on how to think and solve mathematical problems.
Programming is an important tool for all scientific areas. We will learn how to use GO programming language to simulate real-life data with mathematical models. The following is a list of mathematical concepts that we will learn from this course.
Albert Einstein(1879-1955) once said:
<aside> ☝ If you can't explain it simply, you don't understand it well enough.
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Also, Stan Gudder(1937-), an American mathematician, said:
<aside> ☝ The essence of math is not to make simple things complicated but to make complicated things simple.
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We often say that mathematics is the language of science. It is true in some sense. But it is not the entire reason why we need mathematics. Mathematics is a unique scientific area that studies proof-based statements. From mathematics, one can learn basic intellectual skills such as logical thinking, concrete expositions, and simplifying problems.
Mathematicians have worked with various ideas for thousands of years. Those ideas originated from intuitively clear questions. For example, the mathematical concept of limit is based on the question: “What does it mean by moving continuously?” The concept of integration is based on the question: “How can we find the area inside of a curved boundary?” These questions are based on our common sense. Mathematical definitions help to reformulate the questions into answerable problems. Also, it helps to understand the intuitive concept concretely.
A trained mathematician can do a complicated job. Having concrete mathematical concepts, she/he can clarify the problems. Then she can communicate and collaborate with others efficiently to solve problems. Therefore, every student should aim to become (at least) a semi-professional mathematician. Unprofessional mathematicians are easy to make logical errors and miscommunications, which often lead to devastating results. That is why it is dangerous to train students with only technical math skills.
The learning process of a person is very complicated. However, we must admit that learning begins when the person is fully motivated. Our motivation comes from our personal experiences. Such experiences are often related to encountering problematic situations. We get joyful experiences by solving those problems with our intellectual reasoning.
To learn something effectively, we need to set a related problem and have a firm reason why the problem is important to us. The problem has to be stated clearly so that anyone can understand it. Also, we should break down the problem into several smaller ones and lay out what kind of ideas and tools are available to solve them.
Now let’s talk about learning mathematics. Learning mathematics is very similar to learning a language. There is a set of rules in the mathematical expressions and the usage of symbols. There is a tradition that mathematicians have developed over thousands of years that anyone who tries to enter the realm of mathematics should follow. These rules and traditions are passed from teachers to students through lectures, seminars, and texts.
Babies are very good at learning languages. They listen to words that fit the best in each situation. Then they say those words in similar situations. They see the feedback from their parents and start to understand the meaning of those words. Later they put the words into sentences: we can talk. When babies become young, they can write, and increase their thoughts and verbal skills. This is the whole process of learning a language. The process of learning mathematics is almost the same as this.
If a person has a good vocabulary, she/he can think about broader subjects and deliver her/his ideas effectively. The same is true for mathematics. As we have more knowledge and understanding of mathematics, we can expand scientific ideas and communicate them with others more effectively.
Academic integrity is fundamental to this course. It requires all students to maintain honesty, fairness, and responsibility in their academic work. Upholding academic integrity entails submitting original work, properly crediting sources, and adhering to ethical standards in all assignments and assessments. Any form of academic dishonesty, including but not limited to plagiarism and cheating, is strictly prohibited and will result in appropriate disciplinary action.
Midterm and final are assessed individually, while homework and projects are assessed via groups. The final score is rounded from the first decimal point.
If you miss more than 7 classes, you will fail the class.
If you miss either midterm or final, you will fail the class.
| Score | Grade |
|---|---|
| 90 or above | A+ |
| 80 or above | A0 |
| 75 or above | A- |
| 65 or above | B+ |
| 55 or above | B0 |
| 50 or above | B- |
| 40 or above | C+ |
| 30 or above | C0 |
| 25 or above | C- |
| 15 or above | D+ |
| 5 or above | D0 |
| below 5 | D- |
| 0 | F |