2026 (Current Year) Faculty Courses School of Environment and Society Undergraduate major in Civil and Environmental Engineering
Basic Mathematics for Physical Science
- Academic unit or major
- Undergraduate major in Civil and Environmental Engineering
- Instructor(s)
- Taizo Maruyama / Nobuyuki Utsumi
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 1-2 Mon (W5-106) / 1-2 Thu (W5-106)
- Class
- -
- Course Code
- CVE.M201
- Number of credits
- 110
- Course offered
- 2026
- Offered quarter
- 1Q
- Syllabus updated
- Mar 5, 2026
- Language
- Japanese
Syllabus
Course overview and goals
This course has two parts. The first part focuses on vector calculus. This topic includes derivative of a vector function, parametric representation of a curve, tangent to a curve and arc length of a curve, gradient of a scalar field, directional derivative, divergence and curl of a vector field, line integrals, complex function, Green’s theorem in the plane, surface integrals, divergence theorem of Gauss and Stokes’s theorem. Vector calculus is important and is essential for the study of engineering. Students learn the basics of vector differential calculus and vector integral calculus and will be able to solve some practical problems in engineering (e.g., hydrodynamics). In the second part, Fourier transform and partial differential equation are dealt with. These topics are important to understand dynamic problems in the fields of civil engineering. The following topics are discussed: Complex analysis, Fourier series, Fourier integral, formulation of partial differential equation, its general solutions, and method of separation of variables.
Course description and aims
By completing this course, students will be able to:
1) Understand the concepts of scalar fields and vector fields.
2) Understand and formulate line integrals, surface integrals and complex function.
3) Understand the surface integrals, divergence theorem of Gauss and Stokes’s theorem.
4) Understand the basic theory of Fourier transform.
5) Understand the relationships between frequency and time domain.
6) Formulate and solve some basic partial differential equations.
Keywords
vector functions, vector fields, gradients of scalar fields, divergence of vector fields, rotations, line and area integrals, complex function, Green's theorem, Gaussian divergence theorem and Stokes' theorem, Fourier series, Fourier integral, frequency domain, partial differential equation, strings, method of separation of variables
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Part of each class is devoted to fundamentals and the rest to advanced content or applications. To allow students to get a good understanding of the course contents and practical applications, problems related to the contents of this course are given in homework assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Vector Analysis Basics: Vectors, scalars, vector spaces, inner and outer products, vector-valued functions, complex functions (Utsumi) |
Understand vectors, scalars, vector spaces, inner and outer products, complex numbers and complex planes, etc. |
| Class 2 | Scalar and vector fields (1): Examples of scalar and vector fields, gradient vector (grad) of scalar fields, differentiation of complex functions (Utsumi) |
Understand examples of scalar and vector fields, gradient vectors (grad) of scalar fields, differentiation of complex functions, and regular functions, etc. |
| Class 3 | Scalar and vector fields (2): Gradient of scalar fields, divergence of vector fields, rotation of vector fields, differentiation of complex functions (Utsumi) |
Understand the gradient vector (grad) of a scalar field, the divergence (div) and rotation (rot f) of a vector field, and the Cauchy-Riemann equations. |
| Class 4 | Line integrals and area integrals (1): Line integrals of scalar and vector fields, line integrals of complex functions (Utsumi) |
Understand line integrals of scalar and vector fields, and line integrals of complex functions. |
| Class 5 | Line integrals and area integrals (2): Area integrals of scalar and vector fields (Utsumi) |
Understand the area of scalar and vector fields. |
| Class 6 | Line integrals and area integrals (3): Gauss' divergence theorem (Utsumi) |
Understand Gauss's divergence theorem. |
| Class 7 | Line integrals and area integrals (4): Stokes' theorem, Green's theorem, integration of complex functions (Utsumi) |
Understand Stokes' theorem, Green's theorem, and analysis using the integral of complex functions. |
| Class 8 | Checking understanding of vector analysis (lessons 1-7) |
Checking understanding of vector analysis (lessons 1-7) |
| Class 9 | Application of complex analysis (Maruyama) |
Understand basic theorems of complex analysis and infinite integral using complex analysis. |
| Class 10 | Fourier integral and its properties (Maruyama) |
Understand definition of Fourier series and its mathematical properties. |
| Class 11 | Mathematical properties of Fourier series (Maruyama) |
Understand mathematical properties and applications of Fourier series |
| Class 12 | Formulation of partial differential equation (Maruyama) |
Understand examples of partial differential equation and its physical background. |
| Class 13 | Wave equation and its solution method (Maruyama) |
Understand formulation and solution of wave equation. |
| Class 14 | Diffusion equation and method of separation of variables (Maruyama) |
Understand diffusion equation and its solution using method of separation of variables. |
| Class 15 | Examination on Fourier transform and partial differential equation (lessons 9-14) |
Examination on Fourier transform and partial differential equation |
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)
Kreyszig, E., 2011, Advanced Engineering Mathematics, 10th edition, John Wiley, New York.
Material will be distributed where necessary (Utsumi, Maruyama)
Reference books, course materials, etc.
Material will be distributed where necessary (Utsumi)
Evaluation methods and criteria
Students' knowledge of the topics on this course, and their ability to apply them to problems will be assessed.
exercises (final exam) 35%, homework 15% (Maruyama)
exercises (final exam) 35%, homework 15%. (Utsumi)
Related courses
- CVE.A210 ： Structural Dynamics in Civil Engineering
- CVE.M202 ： Basic Mathematics for System Science
- CVE.B201 ： Hydraulics I
- CVE.B202 ： Hydraulics II
Prerequisites
not specially.
Other
Depending on the progress of the lectures and exercises, the schedule may be changed and make-up lectures may be given.