This course focuses on vector analysis, which is essential to learn advanced courses of science and engineering, and covers its fundamentals and practical applications. Topics include vector algebra, differential calculus of vector (differential calculus of multivariable function), gauging of curved line and surfaces with vector, gradient of scalar field, divergence and rotation of vector field, relationship between coordinate transformation and tensor, integral theorem of vector and its practical applications.
Nowadays, vector analysis is regarded as not only a part of applied mathematics, but also the most important part of mathematical analysis. It is a generalization of single variable calculus to arbitrary dimensions, and provides powerful analytical methods based on differential and integral calculus and linear algebra to scientists and engineers. Therefore, understanding of vector analysis is one of the goals of undergraduate curriculum for analytical mathematics. In this course, we focus on the physical meaning of vector analysis formulae by learning practical applications to flow fields and electromagnetic fields as well as the mathematical proofs of the formulae. This course offers students both an opportunity of systematic learning of vector and its applications and an introduction to advanced courses such as continuum mechanics, fluid dynamics, and electromagnetics.
At the end of this course, students will be able to:
1) Perform the basic algebra operation of vectors.
2) Understand the differential operation of vectors and apply it to the calculation of particle motion and rotation of body.
3) Understand the gauging of curved line and surface 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, tensor, Gauss' theorem, Stokes' theorem, orthogonal curvilinear coordinates
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | ✔ |
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 | |
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Class 1 | Basic properties of vector -- unit vector, vector components, scalar product, vector product, scalar triple product, vector triple product, coordinate transformation. | Perform vector operation using formulae of vector algebra. |
Class 2 | Differential calculus of vector -- velocity vector, acceleration vector, equation of motion, differential operation, rotating operation | Understand differential calculus of vector, and compute motion of particle and rotation of body. |
Class 3 | Curved lines -- horizontal curve, space curve, tangent vector, normal vector, curvature, torsion | Understand gauging of curved line by vector, and compute curvature and torsion. |
Class 4 | 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 5 | 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 6 | Vector field II -- rotation of vector field, coordinate transformation and scalar, vector, and tensor | Understand rotation of vector field, the relationship between coordinate transformation and scalar and vector, and the role of tensor. |
Class 7 | 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 8 | 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. |
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)
Students' knowledge on vector algebra, vector calculus, and vector applications to physics will be assessed by final exam (50%) and exercise problems (50%).
Students must have successfully completed Calculus I & II and Linear algebra I & II, or have equivalent knowledge.