Objectives and competences
Objectives:
To know thoroughly basics of scalar and vector field theory, integration of functions of several variables and vector and scalar functions, basics of complex analysis and their implementation in engineering problems.
Competences:
The ability to use the acquired knowledge for computing some quantities in engineering and technics which appear in different basic and applied sciences.
Content (Syllabus outline)
Integrals of functions of several variables (parametric integrals, double integral, triple integral, substitutions. center of gravity, moment of inertia).
2. Differential geometry (curves and surfacec in the space and their parametrization, connection with physics).
3. Scalar and vector fields (gradient, rotor, divergence, operator nabla, directional derivative, curve and flat integrals of scalar and vector functions, connection with the integrals of different types, Green, Stokes, and Gauss theorems, applications of fomulas – length of the arc, surface area, work, potential, flow).
4. Complex analysis (complex function, analytical function, elementary complex functions, integrals of complex functions, infinite series – Taylor and Lauren series, theory of residues and connection with the inverse Laplace transformation).
Learning and teaching methods
Lectures, tutorial, homework assignments.
Intended learning outcomes - knowledge and understanding
Knowledge and understanding:
At the end of the course students are going to:
1. Define double integrals and interpret the geometric meaning of the integral sum and use it in the calculation of some physical quantities.
2. Define the triple integral and interpret the fundamental theorem on triple integrals.
3. Differ between a scalar function and a vector function and be able to use the derivative and integral of those functions.
4. Differ and use a curve and a surface in the space and understand the curve length and curvature.
5. Differ between integrals in the space (in particular, triple-, curve, and surface integrals) and be able to interpret the geometric meaning of the integral sum; use those integrals for evaluation of masses, moments of inertia, deviational and static moments; be able to use the properties of the integrals in the space; describe the physical interpretation of curve and surface integrals of the vector function.
6. Show the understanding of the theorem of substituting new variables in multiple integrals and be able to apply the theorem to transform complex integrals into simpler ones. Show the understanding of polar, spherical, and cylindrical coordinates and be able to use them sensibly.
7. Explain the Green, Gauss and Stokes theorems, and apply them to curve and surface integrals.
8. Consider the basics of complex functions, criterion for analyticity of those functions, and their Taylor and Laurent power series expansion. Formulate Cauchy integral formula and Residue Theorem and apply them to complex integrals.
Intended learning outcomes - transferable/key skills and other attributes
Transferable/key competences and other abilities:
• Communication skills: unequivocal and precise expression.
• Computational skills: application of formulas and methods for computing real integrals of functions of several variables, vector functions and complex integrals.
• Problem solving: identifying and solving mathematical problems that occur in energy and that are described by the theory of scalar and vector fields and integrals of several variables and by the theory of complex functions.
Readings
Osnovna/Basic:
- M. Mencinger, P. Šparl, B. Zalar, Zbirka rešenih nalog iz matematike II, FG UM, Maribor, 2007.
- G. Tomšič, Matematika III, Založba FE in FRI, Ljubljana, 2005.
Dodatna/Additional:
- E. Kreyszig, Advanced Engineering Mathematics, J. Wiley and Sons, 2011.
Prerequisits
A knowledge of the (partial) derivative and integral (indefinite and definite) and basics of linear algebra is recommended.
Additional information on implementation and assessment Method (written or oral exam, coursework, project):
• Written exam
• Oral exam
• coursework
Notes:The student must finish each part of the exam (written exam, oral exam, coursework) with at least 50%. In the case that a student is offered the opportunity to take a written exam in the form of two midterm exams, each must be graded with at least 30% and both with an average of at least 50% to pass the written part of the exam.
