The calculus of variations concerns problems in which one wishes to find the minima or extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics and physics. They range from the problem in geometry of finding the shape of a soap bubble, a surface that minimizes its surface area, to finding the configuration of a piece of elastic that minimises its energy Perhaps most importantly, the principle of least action is now the standard way to formulate the laws of mechanics and basic physics. In this course it is shown that such variational problems give rise to a system of differential equations, the Euler-Lagrange equations. Furthermore, the minimizing principle that underlies these equations leads to direct methods for analysing the solutions to these equations. These methods have far reaching applications and will help develop students? mathematical techniques.

### Calculus of Variations (MATH 479)

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Programs\Type | Required | Core Elective | Area Elective |

BA- Political Science | |||

BA-Cultural Studies | |||

BA-Cultural Studies | |||

BA-Economics | |||

BA-Economics | |||

BA-International Studies | |||

BA-International Studies | |||

BA-Management | |||

BA-Management | |||

BA-Political Sci.&Inter.Relat. | |||

BA-Social & Political Sciences | |||

BA-Visual Arts&Visual Com.Des. | |||

BA-Visual Arts&Visual Com.Des. | |||

BS-Biological Sci.&Bioeng. | |||

BS-Computer Science & Eng. | |||

BS-Computer Science & Eng. | |||

BS-Electronics Engineering | * | ||

BS-Electronics Engineering | * | ||

BS-Industrial Engineering | * | ||

BS-Manufacturing Systems Eng. | * | ||

BS-Materials Sci. & Nano Eng. | * | ||

BS-Materials Science & Eng. | * | ||

BS-Mechatronics | * | ||

BS-Mechatronics | * | ||

BS-Microelectronics | |||

BS-Molecular Bio.Gen.&Bioeng | |||

BS-Telecommunications | * | ||

Mathematics | |||

Physics |

### CONTENT

### OBJECTIVE

Students will learn rigorous results in the classical and modern one-dimensional calculus of variations and see possible behaviour and application of these results in examples. Also, they will be able to formulate variational problems and analyse them.

### LEARNING OUTCOME

Learn rigorous results in the classical and modern one-dimensional calculus of variations

See possible behaviour and application of these results in examples.

Formulate variational problems and analyse them.

### PROGRAMME OUTCOMES

**1.** Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. 3

**2.** Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 5

**3.** Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects. 4

**4.** Communicate effectively in Turkish and English by oral, written, graphical and technological means. 5

**5.** Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. 2

**1.** Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 5

**2.** Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 5

**3.** Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. 5

**4.** Ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose. 5

**5.** Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 5

**6.** Knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 5

**7.** Knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; understanding of professional and ethical responsibility. 5

**1.** Use mathematics (including derivative and integral calculations, probability and statistics, differential equations, linear algebra, complex variables and discrete mathematics), basic sciences, computer and programming, and electronics engineering knowledge to
(a) Design and analyze complex electronic circuits, instruments, software and electronics systems with hardware/software.
or
(b) Design and analyze communication networks and systems, signal processing algorithms or software 5

**1.** Applying fundamental and advanced knowledge of natural sciences as well as engineering principles to develop and design new materials and establish the relation between internal structure and physical properties using experimental, computational and theoretical tools. 5

**2.** Merging the existing knowledge on physical properties, design limits and fabrication methods in materials selection for a particular application or to resolve material performance related problems. 4

**3.** Predicting and understanding the behavior of a material under use in a specific environment knowing the internal structure or vice versa. 3

**1.** Familiarity with concepts in statistics and optimization, knowledge in basic differential and integral calculus, linear algebra, differential equations, complex variables, multi-variable calculus, as well as physics and computer science, and ability to use this knowledge in modeling, design and analysis of complex dynamical systems containing hardware and software components. 5

**2.** Ability to work in design, implementation and integration of engineering applications, such as electronic, mechanical, electromechanical, control and computer systems that contain software and hardware components, including sensors, actuators and controllers. 3

**1.** Formulate and analyze problems in complex manufacturing and service systems by comprehending and applying the basic tools of industrial engineering such as modeling and optimization, stochastics, statistics. 5

**2.** Design and develop appropriate analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy. 5

**3.** Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 3

### Update Date:

### ASSESSMENT METHODS and CRITERIA

Percentage (%) | |

Midterm | 70 |

Assignment | 30 |

### RECOMENDED or REQUIRED READINGS

Readings |
1. U. Brechtken-Manderscheid, Introduction to the Calculus of Variations, Chapman & Hall, 1991. |