# 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