Mathematics

(Course Code: 401Υ, Course outline)

Semester: 1 Teaching Credits: 4 ECTS Credits: 5 Type:

Compulsory

Prerequisite Courses:  – Course type: General background Instructor: Demetris F. Lekkas
Upon successful completion of the course students should have acquired the following:
  • Generalize key calculus concepts and develop deeper understanding.
  • Functions (Functions and Their Graphs, Combining Functions)
  • Limits and Continuity
  • Differentiation
  • Applications of Derivatives
  • Integration
  • Applications of Definite Integrals
  • Basic skill in solving environmental design applications.
  • Knowledge of quantitative and qualitative characteristics of environmental systems and ability to calculate pertinent parameters
  • An ability to calculate and/or estimate basic design parameters
Topics per Week:  PART A – Calculus

Lecture 1: Introduction to Calculus

– Numbers, functions, graphs, inverse of a function

Lecture 2: Limits and Derivatives

– Definition and properties of limits and derivatives, physical and geometric meaning of  derivatives, applications     

Lecture 3: Integrals

– Indefinite integral, integration rules, definite integral, properties 

Lecture 4: Ordinary Differential Equations I

– Introduction to differential equations, class and order of differential equations, solution methods  of 1st order differential equations, separation of variables, linear and homogenous equations

Lecture 5: Ordinary Differential Equations II

– Applications of differential equations on problems in environmental science and engineering

Lecture 6: Multivariable Functions

– Examples of multivatiable functions in modeling environmental processes, partial differentiation multiply integrals, applications of partial derivatives and multiple integrals 

PART B – Linear Algebra

Lecture 7: Introduction to Vectors and Matrices

 – Definition of vectors and matrices, operations with matrices and vectors (addition, multiplication)         

Lecture 8: Solutions of Systems of Linear Equations I

– Matrix formulation of systems of linear equations, the inverse of a matrix and the solution of  systems of linear equations  

Lecture 9: Solutions of Systems of Linear Equations II

– Gauss and Cramer’s method

Lecture 10: Eigenvalues and Eigenvectors (Definitions)

– Definition and estimation of the eigenvalues and the eigenvectors of a matrix

Lecture 11: Applications of Eigenvalue Analysis 

– Solution of systems of Differential Equations

Lecture 12: Applications of Eigenvalue Analysis

Theory – Lectures
(hours / week)
:
 3
Exercises – Laboratories
(hours / week)
:
 2
Other Activities:  –
Grading:  Final exam (100%)

Weekly tests and homeworks (optional, max. 30% of the total grade)

The final grade is estimated by the final exam (100%) increased by a factor proportional to the optional weekly tests and homeworks given throughout the course. The increase of the final exam mark cannot be higher than 30%, and cannot exceed the value of 10 (maximum grade).

Notes:
Basic Textbook:
  • Thomas Jr G.B., Finney, R.L., Weir, M.D. and Giordano, F.R., 2004, «Thomas’ Calculus», 11th Edition, Pearson, 2007

  • Strang G., «Linear Algebra and its Applications», Harcourt, 1988

Bibliography:

Language:

The course is taught in Greek. English literature, instructions, and examinations can be provided in English for exchange students.

Internet Links:

Thomas Calculus

http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/

MIT open coursework:

http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2006/CourseHome/index.htm

http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2006/

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/

Math archives, Calculus resources on-line

http://archives.math.utk.edu/calculus/crol.html