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 (H. Morikawa)
vector functions, vector fields, curves, gradient of a scalar field, directional derivative, divergence of vector field, curl of vector field, line integrals, Green’s theorem in the plane, divergence theorem, Stokes’s theorem. (Anil C. W.)
|✔ 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||Vector differential calculus (1). Review of vector algebra. Vector and scalar functions and fields. Derivatives. (Anil C. W.)||Review sections 9.1-9.4 of textbook.|
|Class 9||Vector differential calculus (2). Curves and arc length. (Anil C. W.)||Review sections 9.5 of textbook.|
|Class 10||Vector differential calculus (3). Gradient of a scalar field. Directional derivative. (Anil C. W.)||Review section 9.7 of textbook.|
|Class 11||Vector differential calculus (4). Divergence and curl of a vector field.||Review sections 9.8-9.9 of textbook.|
|Class 12||Vector integral calculus (1). Line integrals. Path independence of line integrals. (Anil C. W.)||Review sections 10.1-10.2 of textbook.|
|Class 13||Vector integral calculus (2). Green’s theorem in the plane.||Review section 10.4 of textbook.|
|Class 14||Vector integral calculus (3). Surface integrals.||Review sections 10.5-10.6 of textbook.|
|Class 15||Vector integral calculus (4). Divergence theorem of Gauss and Stokes’s theorem. (Anil C. W.)||Review sections 10.7-10.9 of textbook.|
|Class 16||Examination on vector calculus (Anil C. W.)||Examination on vector calculus|
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.
Hildebrand, F. B., 1976, Advanced Calculus for Applications, 2nd edition, Prentice-Hall, New Jersey. (Anil C. W.)
Students' knowledge of the topics on this course, and their ability to apply them to problems will be assessed.
exercises 35%, homework 15% (H. Morikawa)
Final exam 30%, exercise problems 20%. (Anil C. W.)
Basic vector algebra. (Anil C. W.)
Anil and Morikawa provide their lecture on Monday and Thursday, respectively. If necessary, we offer makeup class and/or to change the schedule.
For 2022, assignments may be required every week instead of practice problems during class. The final exam may be performed on-line. For this, students are requested to prepare some devices and/or environments to upload answers in image on T2SCHOLA. (Morikawa)
For 2022, assignments every week - answers submitted by e-mail. The final exam will be a take-home exam (24 hrs) - answers submitted by e-mail. (Anil C. W.)