CALCULUS OF VARIATIONS

Course Code:

MATH 454

Course Type:

Area Elective

P:

3

Lab:

0

Laboratuvar Saati:

0

Credits:

3

ECTS:

7

Prerequisite Courses:

PARTIAL DIFFERENTIAL EQUATIONS

Course Language:

English

Course Objectives:

To understand the problems of Physics and Engineering better and find their solutions

Course Content:

Euler-Lagrange equations and generalizations.Hamiltonian functions. Invariant integrals. Noether theorem. Second variation and Jacobi fields. Constraint variational problems. Isoperimetric problems. Non-holonomic systems.

Course Methodology:

1: Lecture, 2: Problem Solving, 3:Question-answer, 4: Homework

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Program Learning Outcomes	Teaching Methods	Assessment Methods
1) Knows variations and their properties	3,4,5,7,8,9	1,2	A
2) Derives Euler equation	3,4,5,7,8,9	1,2	A
3) Can perform generalizations	3,4,5,7,8,9	1,2	A
4) Investigates moving boundary value problems	3,4,5,7,8,9	1,2	A
5) Knows the direct method and Ritz method	3,4,5,7,8,9	1,2	A
6) Can investigate the multiple independent variables case	3,4,5,7,8,9	1,2	A

Course Flow

Week	Topics	Study Materials
1	Minima and maxima of differentiable functions
2	Variations and its properties
3	Euler equation
4	Some generalizations
5	Parametric representations of variation problems
6	Variation problems with moving boundaries
7	Variation problems with moving boundaries (continued)
8	Sufficiency for an extremum
9	Problems with constrained extrema
10	Problems with constrained extrema (continued)
11	Direct methods
12	Ritz’s method
13	Generalizations to more than one independent variables
14	Generalizations to more than one independent variables

Recommended Sources

Textbook	L. E. Elsgolc; Calculus of Variations
Additional Resources	F. B. Hildebrand; Methods of Applied Mathematics

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	100
Quizzes	-
Assignments	-
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		60
Total		100

COURSE CATEGORY

Expertise/Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)				X
2	Acquiring fundamental knowledge on fundamental research fields in mathematics			X
3	Ability form and interpret the relations between research topics in mathematics					X
4	Ability to define, formulate and solve mathematical problems					X
5	Consciousness of professional ethics and responsibilty			X
6	Ability to communicate actively			X
7	Ability of self-development in fields of interest					X
8	Ability to learn, choose and use necessary information technologies			X
9	Lifelong education			X

ECTS

Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	3	42
Hours for off-the-classroom study (Pre-study, practice)	14	5	70
Mid-terms (Including self study)	2	15	30
Quizzes	0	0	0
Assignments	0	0	0
Final examination (Including self study)	1	30	30
Total Work Load			172
Total Work Load / 25 (h)			6.88
ECTS Credit of the Course			7

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