The least upper bound property in R, equivalents and consequences. Metric spaces. Completeness, compactness, connectedness. Functions,continuity. Sequences and series of functions. Contraction mapping theorem and applications to calculus: Inverse and implicit function theorems.

### Introduction to Mathematical Analysis (MATH 301)

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

Computer Science and Engineering | * | ||

Computer Science and Engineering | * | ||

Electronics Engineering | * | ||

Electronics Engineering | * | ||

Industrial Engineering | * | ||

Industrial Engineering (Previous Name: Manufacturing Systems Engineering) | * | ||

Materials Science and Nano Engineering | * | ||

Materials Science and Nano Engineering (Previous Name: Materials Science and Engineering) | * | ||

Mathematics Minor | * | ||

Mechatronics Engineering | * | ||

Mechatronics Engineering | * | ||

Microelectronics | * | ||

Molecular Biology, Genetics and Bioengineering | * | ||

Molecular Biology, Genetics and Bioengineering (Pre. Name: Biological Sciences and Bioengineering) | * | ||

Telecommunications | * |

### CONTENT

### OBJECTIVE

Learning the basics of mathematical analysis; i.e. basic theorems and basic techniques in analysis.

### LEARNING OUTCOMES

- Comprehend the language of analysis
- Read and understand definitions, theorems, proofs.
- Produce their "own" proofs in some cases.
- Comprehend the structure of real numbers and the Euclidean space. Comprehend the basic theorems of Calculus.
- Comprehend the topology of the Euclidean space.
- Comprehend the notion and use the facts of continuity.
- Comprehend the notion and use the facts of differentiability.
- Comprehend the notion and use the inverse and implicit function theorems.
- Comprehend the notion and use the facts of Riemann integrals.

### 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. 1

**2.** Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 4

**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; have the ability to continue to educate him/herself. 2

**4.** Communicate effectively in Turkish and English by oral, written, graphical and technological means. 1

**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. 3

**1.** Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 4

**2.** Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 3

**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. 2

**4.** Have the 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. 2

**5.** Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 3

**6.** Possess knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 3

**7.** Possess 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; knowledge of behavior according to ethical principles, understanding of professional and ethical responsibility. 2

**8.** Have the ability to write effective reports and comprehend written reports, prepare design and production reports, make effective presentations, and give and receive clear and intelligible instructions. 2

**1.** Comprehend key concepts in biology and physiology, with emphasis on molecular genetics, biochemistry and molecular and cell biology as well as advanced mathematics and statistics. 4

**2.** Develop conceptual background for interfacing of biology with engineering for a professional awareness of contemporary biological research questions and the experimental and theoretical methods used to address them. 3

**1.** Design, implement, test, and evaluate a computer system, component, or algorithm to meet desired needs and to solve a computational problem. 2

**2.** Demonstrate knowledge of discrete mathematics and data structures. 2

**3.** Demonstrate knowledge of probability and statistics, including applications appropriate to computer science and engineering. 3

**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

**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. 4

**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. 3

**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. 4

**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. 2

**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. 2

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

**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 (%) | |

Final | 40 |

Midterm | 30 |

Homework | 30 |

### RECOMENDED or REQUIRED READINGS

Textbook |
Michael C. Reed, Fundamental Ideas of Analysis, John Wiley \& Sons, Inc., 1998. We will cover the first 6 chapters, we will skip some sections. |