2020 Vector Analysis

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Academic unit or major
Undergraduate major in Mechanical Engineering
Hasegawa Jun  Sasabe Takashi  Tanahashi Mamoru 
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Course description and aims

For students who have advanced to the field of mechanical engineering, the following topics on vector analysis, which is essential for learning specialized subjects (mechanics, elastic body mechanics, fluid mechanics, electromagnetics, etc.), are covered:
1) Algebraic operation of vector (scalar product, vector product, etc.)
2) Differentiation of vector (motion of mass point, derivative of vector, etc.)
3) Metrics of curves and surfaces with vectors (Frenet-Seret's law, quadratic surface, etc.)
4) Gradient of scalar field and divergence and rotation of vector field
5) Vector field integral theorem (Gauss theorem, Stokes theorem, etc.)

Student learning outcomes

At the end of this course, students will be able to:
1) Perform the basic algebra operation of vectors.
2) Understand the differentiation of vectors and apply it to the calculation of particle motion.
3) Understand the metrics of curves and surfaces with vectors, and calculate the values.
4) Understand the gradient of scalar field and the divergence and rotation of vector field, and calculate those values.
5) Solve practical problems using the integral theorems of vector field (Gauss theorem, Stokes theorem, etc.).


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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

In the beginning of each class, solutions to exercise problems given in the previous class are reviewed. In the end of the class, students are given exercise problems related to the contents of the lecture. Students should check the course schedule and what topics will be covered beforehand, and it is strongly recommended for students to prepare and review those topics.

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 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 3 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 4 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 5 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 6 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 7 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.

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.


None required.

Reference books, course materials, etc.

Toda, Morikazu, Vector Analysis, Tokyo: Iwanami-shoten; ISBN4-00-007773-2.
H.P. Hsu, Vector Analysis, Tokyo: Morikita-shuppan; ISBN978-4-627-93020-9. (Japanese)
Shimizu, Yuji, Fundamentals and Applications of Vector Analysis, Tokyo: Science-sha; ISBN4-7819-1133-1. (Japanese)

Assessment criteria and methods

Students' knowledge on vector algebra, vector calculus, and vector applications to physics will be assessed by final exam (50%) and exercise problems (50%).

Related courses

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

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

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

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