Print

2024　Fundamentals of engineering mathematics

Font size  SML

Register update notification mail Add to favorite lecture list
Academic unit or major
Undergraduate major in Mechanical Engineering
Instructor(s)
Yoshida Kazuhiro  Yamazaki Takahisa 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Mon5-8(W2-401(W241))  
Group
-
Course number
MEC.A211
Credits
2
Academic year
2024
Offered quarter
1Q
Syllabus updated
2024/3/14
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

For students who have advanced to the field of mechanical engineering, this course focuses on the vector analysis used in the mechanics, elastic body mechanics, fluid mechanics, electromagnetics, etc., and the ordinary differential equations used in the analysis of linear systems and nonlinear systems, etc. Topics of the vector analysis include algebraic operation of vector, differentiation of vector, metrics of curves and surfaces with vectors, gradient of scalar field and divergence and rotation of vector field, and vector field integral theorem. Topics of the ordinary differential equations include 1st order ordinary differential equations, n-th order ordinary differential equations, Fourier series, etc. and also include the functions of elementary solutions of ordinary differential equation.

Student learning outcomes

By the end of this course, students will be able to:
1) Understand and perform the basic algebra operation and the differentiation of vectors.
2) Understand and calculate the metrics of curves and surfaces with vectors.
3) Understand and calculate the gradient of scalar field, the divergence and rotation of vector field, and the integral theorems of vector field.
4) Explain linear systems and nonlinear systems in ordinary differential equations.
5) Explain the fundamentals of the functions of general solutions.
6) Understand and perform solving method of n-th order ordinary differential equations using differential operators, etc.

Keywords

Vector algebra, differential calculus of vector, gauging of curved line and surface, integral calculus of vector, coordinate transformation, Gauss' theorem, Stokes' theorem, orthogonal curvilinear coordinates, ordinary differential equations, elementary solutions, differential operator, Fourier series, nonlinear ordinary differential equations, perturbation method.

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills
✔ Basic knowledge for mechanical engineering

Class flow

At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purpose.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Basic properties of vector -- unit vector, vector components, scalar product, vector product, scalar triple product, vector triple product Perform vector operation using formulae of vector algebra.
Class 2 Introduction of ordinary differential equations. 1st order ordinary differential equations Understand elementary solutions, particular solutions, and general solutions.
Class 3 Differential calculus of vector -- velocity vector, acceleration vector, equation of motion, differential operation Curved lines -- horizontal curve, space curve, tangent vector, normal vector, curvature, torsion Understand differential calculus of vector, and compute motion of particle. Understand gauging of curved line by vector, and compute curvature and torsion.
Class 4 n-th order ordinary differential equations Understand general solutions and Wronski determinant.
Class 5 Curved surface -- expression of curved surface, distance, area, and normal vector, curve line on curved surface, principal curvature Understand gauging of curved surface, and compute surface area and curvature.
Class 6 n-th order ordinary differential equations Understand differential operator and variation of parameters.
Class 7 Vector field I -- gradient of scalar field, divergence of vector field, equation of continuity, Laplacian Understand how to calculate gradient of scalar field and divergence of vector field, and physical meaning of these values.
Class 8 Simultaneous ordinary differential equations Understand eigenvalue.
Class 9 Vector field II -- rotation of vector field, coordinate transformation and scalar, vector Understand rotation of vector field, the relationship between coordinate transformation and scalar and vector, and the role of tensor.
Class 10 Solving by series Understand solving by series.
Class 11 Integral theorem of vector field I -- line integral, Gauss' theorem, electrostatic force and gravitational force, Poisson equation Understand linear integral of vector and Gauss' theorem, and apply them to problems of mechanics, electromagnetics, etc.
Class 12 Nonlinear ordinary differential equations, orthogonal functions Understand perturbation method and Bessel function.
Class 13 Integral theorem of vector field II and orthogonal curvilinear coordinate system -- Green's theorem, Stokes' theorem, cylindrical coordinate and spherical coordinate Understand Green's theorem and Stokes' theorem. Calculate gradients, divergences, and rotations in cylindrical coordinate and spherical coordinate.
Class 14 Fourier series Understand Fourier series.

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)

Kentaro Yano and Shigeru Ishihara: Basic Analysis, Shokabo Co., Ltd, .ISBN: 978-4-7853-1079-0. (Japanese)

Reference books, course materials, etc.

Fumiyuki Terada and Noriaki Kimura: Fundamentals of Vector Analysis, Saiensu-sha Co., Ltd. Publishers, ISBN: 4-7819-0876-4. (Japanese)
Osamu Takenouchi: Differential Equations and Their Application, Saiensu-sha Co., Ltd. Publishers, ISBN: 4-7819-1060-2. (Japanese)

Assessment criteria and methods

Students' knowledge of vector analysis, ordinary differential equations, and their ability to apply them to problems will be assessed by final examination (65%) and exercise problems (35%).

Related courses

  • LAS.M101 ： Calculus I / Recitation
  • LAS.M102 ： Linear Algebra I / Recitation
  • LAS.M105 ： Calculus II
  • LAS.M106 ： Linear Algebra II
  • LAS.M107 ： Calculus Recitation II
  • LAS.M108 ： Linear Algebra Recitation II
  • MCS.T301 ： Vector and Functional analysis
  • MEC.A212 ： Advanced engineering mathematics

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed Linear algebra I & II, and Calculus I & II, or have equivalent knowledge.

Page Top