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.
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.
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)
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
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 | |
---|---|---|
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. |
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.
Kreyszig, E., 2011, Advanced Engineering Mathematics, 10th edition, John Wiley, New York.
Material will be distributed where necessary (Fujii, Utsumi)
Hildebrand, F. B., 1976, Advanced Calculus for Applications, 2nd edition, Prentice-Hall, New Jersey. (Fujii, Utsumi)
Material will be distributed where necessary (Fujii, Utsumi)
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)
not specially.
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.