# MAT-04007 Engineering Mathematics 123, 10 cr

#### Additional information

The course cannot be included in a bachelor's degree.

Jani Hirvonen

#### Lessons

 Implementation Period Person responsible Requirements MAT-04007 2019-01 1 - 2 Hari Nortunen Exam(s) and compulsory exercises. Detailed information is published in Moodle.

#### Learning Outcomes

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

 Content Core content Complementary knowledge Specialist knowledge 1. Basics of Matlab and revision. Mathematics basic skill test. The MathWorks online course Matlab Fundamentals. 2. Complex numbers and their sum, remainder, product and quotient. Absolute value and complex conjugate. Root of complex number. Converting between cartesian, polar and exponential forms. Roots and factorization of a polynomial with real coefficients. Roots of a polynomial with complex coefficients. 3. Vectors. Vectors and analytic geometry: Linear combination, dot and cross product, equations of a line and a plane. Linear independence and orthogonality of vectors. Angle and distance between vectors. Sections of lines and planes. Axioms of vector space. Subspace. Metric requisites. 4. System of linear equations. Solving system of linear equations by Gaussian elimination. Applications of systems of linear equations. The MathWorks online course Introduction to Linear Algebra with Matlab. 5. Matrices. Simple calculations for matrices, inverse of matrix, determinants, scalar triple product. Eigenvalues and vectors. Eigendecomposition of matrices. Similarity and diagonalization. 6. Sets and operations on sets. Logical consequence and equivalence. Existential and universal quantification. Direct proof, proof by contradiction and proof by mathematical induction. Axioms of probability. Propositional logic and truth tables. Probability distributions. Joint probability distributions. Functions of random variables. 7. Statistics, statistical hypothesis testing. Central limit theorem. Distribution of the sample mean. Confidence intervals. Proof of the central limit theorem. 8. Functions, limits and continuity. Composite function. Inverse function. Hyperbolic functions and their inverse functions. Epsilon-delta proof of limits. 9. Sequences and series. Limit of a sequence, increasing and decreasing sequences. Geometric series, series with nonnegative terms, harmonic series, alternating series. Convergence of series. Series representation and polynomial approximation of a function. The most common convergence tests. Other convergence tests. Computing limits and integrals using series. 10. Derivative. Derivative by limit of a difference quotient. Chain rule. Differentiation of elementary functions. Differentiation of an inverse function. Mean value theorem. L'Hospital's rule. 11. Integration. Simple integration techniques (such as integration by parts and integration by substitution). Riemann integral. Applications of integrals such as area, arc length, volume and surface area of a solid of revolution. Numerical integration. 12. Ordinary differential equations. Linear and separable first order differential equations. Applications of differential equations. Numeric methods for solving differential equations. 13. Second order differential equations. Higher order differential equations with constant coefficients. Systems of differential equations.

#### Study material

 Type Name Author ISBN URL Additional information Examination material Book Modern Engineering Mathematics Glyn James 987-1-292-08073-4 Yes

Additional information about prerequisites
At least 10 credits university or applied university mathematics.

#### Correspondence of content

 Course Corresponds course Description MAT-04007 Engineering Mathematics 123, 10 cr MAT-04006 Engineering Mathematics 123, 7 cr

 Updated by: Kunnari Jaana, 05.03.2019