Applied Mathematics Ⅳ

Course Information

College	Toyama College	Year	2019
Course Title	Applied Mathematics Ⅳ
Course Code	0211	Course Category	Specialized / Elective
Class Format	Lecture	Credits	Academic Credit: 1
Department	Department of Mechanical Engineering	Student Grade	5th
Term	First Semester	Classes per Week	前期:2
Textbook and/or Teaching Materials	Text book; ISBN978-4-87525-215-3, in Japanese
Instructor	Toshima Takeshi

Course Objectives

Vector analytics is an important basic knowlege for mathematically solving spatial correlation of physical quantity as well as position, motion and trajectory in multidimensional space, so we aim to understand the following vector rule .
· Vector's basic rule (inner product, outer product, figure notation method)
· Vector calculation (operator, calculus)
· Vector analysis (Green's theorem, Stokes 'theorem, Gauss' divergence theorem)
Eigenvalues, eigenvectors
Specifically, each item of the following rubric will be the target.

Rubric

	Ideal Level of Achievement	High Level of Achievement	Standard Level of Achievement	Unacceptable Level of Achievement)
1: Understand vector inner product / cross product.	Master the inner/cross product of vectors, recognize it as a spatial expression method and explain the correlation	Master the inner/cross product of vectors and recognize it as a space expression method.	Calculate simple inner/cross product of a vector.	It is impossible to calculate simple vector inner/cross product.
2: A graphic can be expressed using a vectors.	Arbitrary figures in three-dimensional space can be represented by vector product, linear combination, etc.	Arbitrary figures in two-dimensional space can be represented by vector product, linear combination, etc.	Derive vector expressions representing representative figures.	It is impossible to derive vector expressions representing representative figures.
3: Vector operator can be calculated.	Being able to calculate the vector operator, explain the mathematical meaning indicated by the obtained results.	Understand about vector operators and It is possible to calculate vector operators.	Calculate vector operators.	It is impossible to calculate vector operators.
4: Understand differentiate and integrate vectors.	Explain that the result of vector differentiation / integration is linked to physical phenomenon.	It is possible to calculate differential/integral of vector and linked to physical phenomenon.	Calculate differential/integral of a vector.	It is impossible to calculate differential/integral of vector.
5: Understand the vector integral formulas.	Explain the phenomenon by combining Green's theorem, Stokes 'theorem, and Gauss' divergence theorem with fluid dynamics and electromagnetism.	Explain Green's theorem, Stokes 'theorem, and Gauss' divergence theorem.	Calculate Green's theorem, Stokes' theorem, and Gauss' divergence theorem.	It is impossible to calculate Green's theorem, Stokes 'theorem, and Gauss' divergence theorem.
6: Understand the correspondence between matrix and coordinate transformation.	Understand matrices and linear transformations. Use coordinate transformation Explain the direction cosine	Understand matrices and linear transformations. Use coordinate transformation.	Calculate coordinate transformation with matrix calculations.	It is impossible to calculate coordinate transformation by matrix calculation.
7: Understand the relationship between eigenvalues and eigenvectors.	Understand the solution of the eigen equation and derive eigenvalues and eigenvectors.	It is possible to derive and solve the eigen equations and calculate the eigen values.	Solve the Eigenvalue equations and calculate the eigenvalues.	It is impossible to solve the eigen equations and calculate the eigen values.

Assigned Department Objectives

Learning and Educational Objectives of the “General Engineering” A-5 （工学的な諸問題に対処する際に必要な数学，自然科学及び情報の基礎的知識を理解し，それらを適切に活用することができる。）

JABEE 1(2)(c) （数学及び自然科学に関する知識とそれらを応用する能力）

Diploma policy 3 （豊かな教養と倫理観を身につけ，社会に貢献できる。）

Teaching Method

Outline:

Vector analysis is a mathematical method that can formulate complex spaces by continuously handling multiple amounts of information at the same time.
This is basic knowledge that is essential for engineers and researchers, since it forms the basis of fluid dynamics and electromagnetism.
In this lecture, we aim to understand the vector analysis method as a powerful mathematical tool and to
cultivate the fundamental powers that can be applied.

Style:

Evaluate the student’s degree of understanding according to the form of lecture and exercise, and in principle proceed according to the lesson plans.
(There may be changes to the lesson plan according to the student’s degree of understanding.)

Notice:

Advance classes on vector analysis on the premise that basic differentiation / integration of functions is acquired.
Although it shows detailed explanations and reviews on esoteric things, it sets challenges and difficulty of examination as preparations / review are done.

Course Plan

			Theme	Goals
1st Semester
	1st Quarter
		1st	Guidance: Review of vector Perform computational exercises on inner/cross product of vectors and check comprehension level	Understand the definition of vectors, calculate inner/cross product of vectors.
		2nd	Geometry and Vector I Learn how to represent figures using vector equations in a two-dimensional space.	A point, a straight line, a curve, a circle in a two-dimensional system can be described by a vector equation
		3rd	Geometry and Vector II Learn how to represent figures using vector equations in a three-dimensional space.	A straight line, a curve, a plane, a curved surface, and a spherical surface in a three-dimensional system can be described by a vector equation
		4th	Differentiation of vector Understand that vector functions can express curves and curved surfaces in space	It is possible to illustrate tangent vectors and normal vectors.
		5th	Vector operator I: Scalar field and gradient Learn concepts about scalar fields and gradients and understand grad operations using differential operators	It is possible to calculate the grad operation.
		6th	Vector operator Ⅱ: Vector field and divergence Learn the concept of vector field and divergence and understand div operation using differential operator	It is possible to calculate the div operations.
		7th	Vector operator Ⅲ: Vector field and rotation Learn the concepts of vector fields and rotations and understand rot (curl) operation using differential operators	It is possible to calculated the rot (curl) operation.
		8th	Vector operator IV: Laplace operator Learn about combinations of vector operations and derivation of basic formulas for Laplace's operation	It is possible to derive the basic formulas by combining each operations.
	2nd Quarter
		9th	Integration of vector Understand ordinary integration, line integral, area integral, and volume integral of vectors	It is possible to link the physical phenomena with differentiation and integration of vectors.
		10th	Vector integral formula I Understand Green's theorem, Stokes' theorem	It is possible to explain Green's theorem and Stoke's theorem
		11th	Vector integral formula Ⅱ Understand Gauss' divergence theorem	It is possible to explain Gauss' divergence theorem
		12th	Matrix and coordinate transformation Learn about matrix and linear transformation, understand coordinate transformation and direction cosine	It is possible to do the coordinate transformation by using matrix.
		13th	Eigenvalues and eigenvectors Understand eigenvalues and eigenvectors and learn about solving eigenvalue equations	It is possible to calculate eigenvalue equations.
		14th	Application of vector analysis Mathematically evaluate the fundamental law in hydrodynamics and electromagnetism using vector fields	It is possible to explain the relationship between formulas learned in hydrodynamics and electromagnetics and vector formulas.
		15th	Final exam
		16th	Return exam papers Explanation of exam Class questionnaire

Evaluation Method and Weight (%)

	Examination	Presentation	Mutual Evaluations between students	Behavior	Portfolio	Other	Total
Subtotal	60	0	0	0	40	0	100
Basic Ability	20	0	0	0	40	0	60
Technical Ability	40	0	0	0	0	0	40
Interdisciplinary Ability	0	0	0	0	0	0	0