### 2023　Basic Mathematics for Physical Science

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Academic unit or major
Undergraduate major in Civil and Environmental Engineering
Instructor(s)
Morikawa Hitoshi  Fujii Manabu  Utsumi Nobuyuki
Class Format
Lecture / Exercise    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Mon1-2(W9-325(W934))  Thr1-2(W9-325(W934))
Group
-
Course number
CVE.M201
Credits
2
2023
Offered quarter
1Q
Syllabus updated
2023/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course has two parts.

In the first 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: Fourier series, Fourier integral, formulation of partial differential equation, its general solutions, and method of separation of variables.

The second part focuses on vector calculus. Topics include 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, 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.

### Student learning outcomes

By completing this course, students will be able to:
1) Explain the basic theory of Fourier transform.
2) Explain the relationships between frequency and time domain.
3) Formulate and solve some basic partial differential equations.
4) Explain the concepts of scalar fields and vector fields.
5) Calculate line integrals and surface integrals.
6) Proof and use Green’s theorem in the plane, surface integrals, divergence theorem of Gauss and Stokes’s theorem.

### Keywords

Fourier series, Fourier integral, frequency domain, partial differential equation, strings, method of separation of variables (Morikawa)

vector functions, vector fields, gradients of scalar fields, divergence of vector fields, rotations, line integrals, area fractions, Green's theorem, Gaussian divergence theorem and Stokes' theorem (Fujii / Utsumi)

### Competencies that will be developed

 ✔ 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/Required learning

Course schedule Required learning
Class 1 Fourier integral and its properties (Morikawa) Definition of Fourier series and its mathematical properties
Class 2 Fourier series (Morikawa) definition of Fourier series, relationships between Fourier series and integral
Class 3 Mathematical properties of Fourier series (Morikawa) mathematical properties and applications of Fourier series
Class 4 Formulation of partial differential equation (Morikawa) examples of partial differential equation and its physical background
Class 5 Solving wave equation and diffusion equation (Morikawa) formulation and solution of wave equation and diffusion equation
Class 6 method of separation of variables/free vibration of simple beam without damping (Morikawa) solution of partial differential equation using method of separation of variables/formulation and solution for free vibration of simple beam without damping
Class 7 Examination on Fourier transform and partial differential equation (Morikawa) Examination on Fourier transform and partial differential equation
Class 8 Fundamentals of Vector calculus (1): Vectors, scalars, vector spaces, inner and outer products, vector-valued function (Fujii, Utsumi) To understand the vectors, scalars, vector spaces, inner and outer products, etc.
Class 9 Scalar and vector fields (1): Examples of scalar and vector fields, scalar fields, gradient vector (grad f) (Fujii, Utsumi) Understand examples of scalar and vector fields, scalar fields, gradient vector (grad f)
Class 10 Scalar and vector fields (2): gradient vector of a scalar field (grad) and divergence of a vector field (div) (Fujii, Utsumi) Understand gradient vector of a scalar field (grad) and divergence of a vector field (div) .
Class 11 Scalar and vector fields(3):Rotation of vector fields (rot f) (Fujii, Utsumi) Understand the rotation of a vector field (rot f).
Class 12 Line Integral and Surface integral (1): Line Integral for Scalar and Vector Fields (Fujii, Utsumi) Understand Line Integral for Scalar and Vector Fields
Class 13 Line Integral and Surface integral (2): Surface integral for Scalar and Vector Fields (Fujii, Utsumi) Understand Surface integral for Scalar and Vector Fields
Class 14 Line Integral and Surface integral (3): Gauss's Divergence Theorem (Fujii, Utsumi) Understand Gauss's divergence theorem.
Class 15 Line Integral and Surface integral (4): Divergence theorem of Gauss and Stokes’s theorem. (Fujii, Utsumi) Understand Stokes' theorem and Green's theorem.

### Out-of-Class Study Time (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 (Fujii, Utsumi)

### Reference books, course materials, etc.

Hildebrand, F. B., 1976, Advanced Calculus for Applications, 2nd edition, Prentice-Hall, New Jersey. (Fujii, Utsumi)
Material will be distributed where necessary (Fujii, Utsumi)

### Assessment criteria and methods

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% (Morikawa)
exercises (final exam) 35%, homework 15%. (Fujii)

### Related courses

• CVE.A210 ： Structural Dynamics in Civil Engineering
• CVE.M202 ： Basic Mathematics for System Science
• CVE.B201 ： Hydraulics I
• CVE.B202 ： Hydraulics II

not specially.

### Other

In general, Fujii and Utsumi will give lectures on 8 to 15 on Mondays, and Morikawa will give lectures on 1 to 7 on Thursdays. Depending on the progress of the lectures and exercises, the schedule may be changed and make-up lectures may be given.