Study Guide

Objective

Topics of the course are basic knowledge of all engineer trades and most of them belong to SEFI's (The European Society for Engineering Education) Core Level 1. In that level most of the topics will be covered in the first year of engineering studies. Topics in Core Level 1 are essential for all engineers, because they are a base for the upcoming special know-how of each engineer trade. After completing the course, the students are able to continue to the master's mathematics course, where the topics belong to SEFI's Level 2.

Content

Basics of Matlab and revision. Mathematics basic skill test. Completing Mathematics Remedial Instruction. The extended Matlab course in Moodle.
2. Sets and operations on sets. Logical consequence and equivalence. Existential and universal quantification. Direct proof, proof by contraposition and proof by mathematical induction. Propositional logic and truth table. Boolean algebra and logical operations / binary operations.
3. Complex numbers and their sum, remainder, product and quotient. Absolut value and complex conjugate. Root of complex number. Converting between polar and exponential forms. Roots of polynomial with real coefficients and representing polynomial with roots. Roots of polynomial with complex coefficients.
4. Derivative. Derivative by limit of difference quotient, chain rule, differentiation of elementary functions. L¿Hospital¿s rule Differentiation of inverse function. Mean value theorem. Epsilon-delta proof of limits.
5. Integration. Simple integration techniques as integration by parts and integration by substitution. Riemann¿s integral. Applications of integrals, as area, arc length, item's volume and surface area. Numerical integration.
6. Vectors. Vectors and analytic geometry: Linear combination, dot and cross product, equation of line and plane, metric requisites. Vectors' linear independence. Vectors' orthogonality. Angle and distance between vectors. Lines and planes sections. Axioms of vector space. Subspace.
7. System of linear equations. Solving system of linear equations by Gaussian elimination. Applications of systems of linear equations Finding the LU decomposition
8. Matrices. Simple calculations for matrices, inverse of matrix, determinants, scalar triple product. Eigenvalues and vectors. Similarity ja diagonalization. Eigenvalue decomposition of matrices. Singular value decomposition of matrices.
9. Sequences. Limit of a sequence, increasing and decreasing sequences. An application of the epsilon-delta-method to limit of the sequence
10. Series (geometric, with positive terms, harmonic, alternating) and their convergence. Approximating a function with a polynomial. Testing the convergence. Computing limits and integrals using the series.
11. Ordinary linear and separable first order differential equations. Applications of differential equations Numeric methods for solving differential equations
12. Differential equations. Second order differential equations. High order differential equations with constant coefficient. Systems of differential equations.
13. Project work of topic which are linked to the training programme. Writing mathematical text Solving real life problems with Matlab

Assessment criteria, satisfactory (1)

Grading done by TUT

Assessment criteria, good (3)

Grading done by TUT

Assessment criteria, excellent (5)

Grading done by TUT