Objectives and competences
Objectives:
To know the basics of mathematical analysis and linear algebra and their applications in engineering mathematics.
Competences:
The ability to apply the acquired knowledge for solving of mathematical problems in the area of mathematical analysis and linear algebra.
The ability to transfer and to use the acquired knowledge for computing some quantities in engineering and technics which appear in different basic and applied sciences, for example Physics, Thermodynamics, Electrical engineering.
Content (Syllabus outline)
1. Numerical sets (predicate logic, integer, rational, real, irational, complex numbers).
2. Vectors in the plane and in the space (definition, geometrical interpretation, basic operations, scalar product, vector product, mixed product, linear dependence and independence of vectors, analitical geometry in space – straightlines and planes, connection with physics).
3. Sequences (definition of a sequence, accumulation point, limit, upper and lower bound).
4. Functions (basic elementary functions, zeros, continuity of function, domain, codomain, inverse function, limit of function).
5. Derivative (definition, problems of extrems, curvature, higher derivations, L’Hospital rule, connection with physics).
6. Integral (definition of indefinite integrals and basic properties, integrals of basic elementary functions, methods of integration, definition of definite integral and its connection to indefinite integral, applications – surface, rotary body, length of the arc).
7. Matrices (operations, different kinds of matrices, determinant and its properties, inverse matrix, systems of linear equations, Gauss elimination, Cramer's rule, matrix rank, linear transformations in the plane -, rotations, mirroring, projections; eigenvalues and eigenvectors of matrix, diagonalization of the 2x2 matrix).
8. Numerical methods (numerical solutions to equations – simple iteration, Newton's method, secant method, Jacobi and Gauss – Seidel iteration; numerical integration – midpoint, trapezoidal and Simpson's rules).
Learning and teaching methods
Lectures: students learn about the theory.
Exercises: avditorial exercisee present practical examples and discuss in more detail selected content. These hours of exercises include also 5 hours of field visiti and work where students consolidate their theoretical knowledge during visti to a radioactive waste managment facility. Demonstration of practical work with radiaoctive waste is includeed in the field visit.
Intended learning outcomes - knowledge and understanding
Knowledge and understanding:
At the end of the course students are going to:
1. Recognize basic mathematical notation and use basic notions of predicate logic and set theory.
2. Define basic notions concerning real sequences.
3. Apply standard methods for plotting graphs of functions.
4. State different types of matrices and perform basic arithmetic operations with matrices and vectors (addition, subtraction, product of the vector or matrix by scalar, transposition, multiplication of matrices).
5. Recognize scalar, vector, and mixed product of vectors and use them correctly in calculating the angle between vectors, vector length, figure area, body volume, checking the perpendicularity, parallelism, and linear dependence of vectors.
6. Write the equations of basic objects (straightline, plane) of analytical geometry in space and calculate the distances between these objects.
7. Calculate the determinant of a smaller dimension matrix (2x2, 3x3) using a chain rule and show an understanding of the calculation of a higher dimension matrix determinant using row or column development.
8. List the properties of the determinant that can help us in the calculation.
9. Calculate the inverse of a 2x2 or 3x3 matrix using an extended matrix or an adjunctive matrix and use it to solve matrix equations.
10. Solve smaller (2x2, 3x3) systems of linear equations using Gaussian elimination and Cramer's rule.
11. Use the basic transformations of a plane (rotation, mirroring, projection) and their composite to map a vector or point.
12. Find the eigenvalues and the corresponding eigenvectors and the diagonalization of the 2x2 matrix.
13. Apply standard methods for solving basic differential calculus problems (extremes of functions, the tangent and normal lines, incresing and decreasing of functions, curvature, inflection points).
14. Solve the indefinite integral using table of integrals, substitution of new variable and ''per partes'' methods, and the method of partial fractions .
15. Use integral for computing the area of the shape, the area and volume of the rotating body, and the length of the arc.
16. Numerically approximate the definite integral, use iteration methods for solving f(x)=0 and distinguish between Jacobi and Gauss-Seidel iteration
Intended learning outcomes - transferable/key skills and other attributes
Transferable/key competences and other abilities:
1. Communication skills: unequivocal and precise expression.
2. Computational skills: performing a variety of computational operations.
3. Problem solving: identifying and solving mathematical problems that occur in energy and that are described by the functions of one variable and the theory of linear algebra.
Readings
Osnovna/Basic:
- R. Jamnik, Matematika, DMFA Slovenije, Ljubljana, 2008.
- P. Žigert Pleteršek, Matematika za študente VS programa, FKKT UM, 2009
- M. Žulj, Naloge iz predmeta Matematične metode I, FE UM, Krško, 2011.
- T. Sovič, S. Špacapan, Matematika 1, FGPA UM 2019.
Dodatna/Additional:
- I. Vidav, Višja Matematika I, DMFA Slovenije, Ljubljana, 2008.
- J. Žerovnik, I. Banič, I. Hrastnik, S. Špacapan, Zbirka rešenih nalog iz tehniške matematike, Fakulteta za strojništvo Maribor, 2005.
Additional information on implementation and assessment Type (examination, oral, coursework, project):
Oral exam,
seminar project
