At the end of this course the student should be able to:
- Have a good understanding of the knowledge of the basic applied mathematics for engineers, within the wide area of the differential and integral calculus of one variable, of the series of numbers and functions, as well as of the linear algebra, which is adequate to his/her science.
- Know the new notions in the form of definitions and theorems that concern the basic contents of the course "Single Variable Calculus and Linear Algebra", in order to be able to apply them.
- Combine and make worthy of the knowledge that he/she acquired to other fields of the theoretical and applied mathematics, in which certain notions and principles of the present course are necessary and useful.
At the end of the course the student will have further developed the following skills and competences:
- Ability to demonstrate knowledge and understanding of essential concepts, principles and applications that are related to the differential and integral calculus of one variable, to the series of numbers and functions, as well as to the linear algebra.
- Ability to apply such knowledge to the solution of problems in other fields of the wide conception of theoretical and applied mathematics, related to the science of Chemical Engineering, or to the solution of multidisciplinary problems.
- Study skills needed for continuing profession development.
There are no prerequisite courses. It is, however, recommended that students should have a basic knowledge of the differential and integral calculus of one variable, as well as of the principal theory of vectors from school.
Cartesian and polar coordinates on the plane, introduction to the calculus of one variable and the method of the mathematical induction. Functions of one variable, the conception of representation, limit and continuity, Boltzano’s theorem. Derivative of first or higher order of functions and geometrical meaning, derivation rules and total differential. Inverse and composite functions, parametric equations, complex forms and L’ Hospital’s rule. Analysis, monotony and extremities of functions, asymptotes. Fermat’s theorem and theorems of mean value. Sequences, number series and convergence criterions. Series of functions, uniform convergence criterions and power series. Generalized mean value theorem or Taylor’s formula and local approximation of function, binomial expansion. Taylor’s and Maclaurin’s series, binomial series and convergence. Fourier’s series and total approximation of function. Applications of derivatives with the use of method of extremities for functions of physical interest and finding the curvature of a plane curve. Indefinite integral of functions and analytic techniques of integration. Riemann’s integral and definite integral. Introduction to the numerical methods of integration and to the generalized integrals. Applications of integrals to the calculation of plane areas, curve’s length, surface areas and domain volumes by rotation. Introduction of plane vectors and the meaning of the third spatial dimension. Inner, exterior, mixed and double–exterior product, geometrical meaning. Matrix theory and square matrices, determinant and inverse matrix. Homogeneous and non homogeneous systems of linear equations, solution with Gauss’ deletion method. Spectral analysis of matrix, in spaces of finite dimensions, eigenvalues and eigenvectors or characteristic magnitudes and physical meaning, Cayley–Hamilton’s theorem. Algebraic and geometric multiplicity of eigenvalues, diagonalization of square matrix. Degenerate eigenvalues, degeneration degree and generalized eigenvectors, Jordan’s matrix. Generalization of inner product, the meaning of norm, distance and orthonormalization with Gram–Schmidt’s method.
Use of web pages and e–class.
- Teaching (4 hours/week): lectures using blackboard of the theory and its application to typical mathematical problems of Chemical Engineering.
- Recitation (2 hours/week): solving on the blackboard exercises concerning mainly mathematical applications of the science of Chemical Engineering.
Total Module Workload (ECTS Standards):
The instruction language is Greek. Written or / and oral examination (100% of final mark).
- V.V. Markellos, "Applied Mathematics", Gotsis Publications, Patras, 2013 (Eudoxos / code 32998565).
- K.E. Papadakis, "Applied Mathematics", Tziolas Publications, Thessaloniki, 2014 (Eudoxos / code 41954961).
- D. Georgiou, S. Iliadis and A. Megaritis, "Real Analysis", Tziolas Publications, Thessaloniki, 2018 (Eudoxos / code 77106793).
- N. Milonas, C. Schinas and G. Papaschinopoulos, "Single Variable Calculus & Linear Algebra", Tziolas Publications, Thessaloniki, 2017 (Eudoxos / code 68369901).
- T.M. Rassias, "Mathematics I", Tsotras Publications, Athens, 2017 (Eudoxos / code 68375438).