2025 (Current Year) Faculty Courses School of Engineering Undergraduate major in Mechanical Engineering
Advanced engineering mathematics
- Academic unit or major
- Undergraduate major in Mechanical Engineering
- Instructor(s)
- Yuko Aono / Takatoki Yamamoto
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-8 Fri (WL1-301(W531))
- Class
- -
- Course Code
- MEC.A212
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This lecture covers partial differential equations (including Laplace transform) and complex functions. Understanding partial differential equations is essential in mechanical engineering, which analyzes many phenomena in time and space. Complex numbers are a combination of real and imaginary numbers, and are likewise an indispensable mathematical tool for understanding various phenomena in mechanical engineering. In this lecture, objectives are to learn the fundamentals and applications of partial differential equations and complex function theory through a close combination of lectures and exercises.
In the first half, first- and second-order partial differential equations, Laplace transforms and their properties, and solving differential equations using Laplace transforms are explained, and the fundamentals of mathematical methods that can be widely applied to linear systems are acquired. In the latter half of the course, students will understand the basic concepts of differentiation of complex functions, and learn its relation to second-order partial differential equations and its application to the calculation of integrals of real functions, thereby acquiring mathematical skills that contribute to solving engineering problems.
Specifically, the lecture will focus on the following points:
1. Partial differentiation and partial differential equations
2. Partial differential equations and their basic solutions
3. Solution by Laplace transform
4. Integral equations
5. Calculus of functions of complex variables
6. Cauchy-Riemann equation
7. Applied topics such as series expansion and integration by use of divisors
Course description and aims
By taking Partial Differential Equations and Complex Function Theory, students will acquire the following abilities.
1) Understand partial differential equations, complex numbers and complex functions, and be able to perform basic calculations.
2) Understand the advantages of using partial differential equations, Laplace transforms, and complex functions, and be able to apply them to solve real engineering problems.
Keywords
Partial differential equation, Integral equation, Laplace transform, Complex derivative, Complex integral, Residues
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
After studying the basic content, students will study the more advanced and applied content. Exercises related to the lecture content will be conducted as necessary to develop a solid understanding and ability to apply the lecture content.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Fundamentals of partial differentiation and construction of partial differential equations |
Explanation and derivation of partial derivatives |
| Class 2 | Linear first-order partial differential equation |
Derivation of solutions to the Lagrangian equations |
| Class 3 | Linear second-order partial differential equation |
Classification of linear second-order partial differential equations |
| Class 4 | Solution of linear second-order partial differential equations |
Deriving solutions to linear second-order partial differential equations by separation of variables |
| Class 5 | Laplace transform and its properties |
Laplace transform of linear differential equations |
| Class 6 | Solution of ｄifferential equations by Laplace transform |
Derivation of solutions by inverse Laplace transform |
| Class 7 | Integral equations / 1st achievement assessment |
Series solution of integral equations |
| Class 8 | Differentiation of complex functions, Cauchy-Riemann equation |
Derivation of Cauchy-Lehmann's equation |
| Class 9 | Basics of linear second-order partial differential equations, Laplace equation |
Relational equations satisfied by elliptic-type second-order partial differential equations |
| Class 10 | Integration of complex functions, Cauchy's integral theorem |
Setting up an integral path in the integration of a complex function |
| Class 11 | Cauchy's integral formula |
Integral method using Cauchy's integral formula |
| Class 12 | Taylor series, Laurent series |
Derivation of series expansion |
| Class 13 | Residue, Use of distinctions for integral evaluation |
Example of integral calculation using residues |
| Class 14 | Application to real function integrals / 2nd assessment of achievement |
Calculation examples of real function integrals |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
スッキリわかる複素関数論, Minamoto, Kindai Kagaku-sha, Inc.
It is strongly recommended that each student purchase a textbook that is appropriate for him or her.
Reference books, course materials, etc.
複素解析, Miyaji, Nippon Hyoronsha
複素関数を学ぶ人のために, Ashida, Ohmsha
複素解析, Spiegel, translated by Soichi Ishihara, McGraw-Hill
A Complex Analysis Problem Book, 2nd Ed., Diniel Alpay, Birkhäuser
Evaluation methods and criteria
Assessment of achievement (80%)
Exercises (20%)
Related courses
- MEC.A211 ： Fundamentals of engineering mathematics
Prerequisites
It is desirable to have taken Fundamentals of engineering mathematics.
This course is the equivalent of the former MEC.B212.A "Complex Function Theory" and the former MEC.B213.A "Partial Differential Equations".
Students who have already earned credits for both "Complex Function Theory" and "Partial Differential Equations" cannot take this course.
Students enrolled before March 31, 2023 (~22B) who earn credits for this course will be
・If the student has already earned one of the credits, he/she will receive 1 credit for A (○) and 1 credit for the non-standard course.
・If both credits are not earned, two A (○) credits are earned.