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 
 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 1^{st} 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: 

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/1801Fall2006/CourseHome/index.htm http://ocw.mit.edu/courses/mathematics/1802multivariablecalculusfall2007/ http://ocw.mit.edu/courses/mathematics/1803differentialequationsspring2006/ http://ocw.mit.edu/courses/mathematics/1806linearalgebraspring2010/ Math archives, Calculus resources online 