Mathematics 2

Enrolment into the first year of the program.

 

Vector spaces: linear dependence, basis, dimension, subspaces.

Inner product spaces: orthogonality, orthonormal basis, orthogonal complements, Gram-Schmidt orthogonalization




Matrices: operations with matrices, transpose matrix, square matrices, invertible matrices, Gauss elimination, systems of linear equations, the row canonical form, rang, inverse.


Determinants: properties, computation by expansion over rows or columns, determinant of the product of matrices, Cramer’s rule.

Linear operators: matrix of a linear operator,
eigenvalues and eigenvectors, characteristic and minimal polynomial, diagonalizability,
Cayley-Hamilton theorem, invariant subspaces.

Operators on inner product spaces: the adjoint,
unitary, self-adjoint and positive operators,
diagonalization of selfadjoint operators.


Bilinear and quadratic forms: canonical form for linear and for orthogonal transformations, quadratic surfaces.

Metric spaces: open and closed sets, interior,
closure and boundary, limit points, compactness, compactness in euclidean spaces, continuous maps.

Vector functions of several variables:
differentiability and the Jacobian, the chain rule, extrema, Hessian, the inverse and the implicit function theorem, constrained extrema and Lagrange multipliers.

S. I. Grossman, Elementary linear algebra. Saunders College Publishing, Orlando, 1991.
P. R. Halmos, Finite-dimensional vector spaces. Springer-Verlag, New York-Heidelberg, 1974.
F. Križanič, Temelji realne matematične analize. Državna založba Slovenije, Ljubljana, 1990.
F. Križanič, Linearna algebra in linearna analiza. Državna založba Slovenije, Ljubljana, 1993.
D. C. Lay, Linear algebra and its applications. Addison-Wesley, Reading, 1997.
B. Magajna, Linearna algebra, metrični prostor in funkcije več spremenljivk, DMFA, Ljubljana, 2011.
M. Dobovišek, B. Magajna in D. Kobal, Naloge iz algebre I, DMFA, Ljubljana, 2011.
M. H. Protter, C. B. Morrey, Intermediate Calculus. Springer-Verlag, New York-Heidelberg, 1985.
I. Vidav, Višja matematika I. Društvo matematikov, fizikov in astronomov Slovenije, Ljubljana, 1994.

Objectives: to familiarize students with basic concepts of linear algebra, metric spaces and with the derivative of a vector function in several variables.

Competences: students should be able to apply the methods of linear algebra and calculus in several varibles to problems in physics.

Knowledge and understanding: understanding of basic concepts and definitions.

Application: there are numerous applications of linear algebra and several variable calculus to physics (and other disciplines)
Reflection: this part of mathematics is basic for understanding classical and modern physics.

Transferable skills: ability to apply mathematical methods. Identification of problems and critical analysis.

Lectures and exercise sessions, homework.

exercise-based exam or 2 midterm exercise-based exams instead of the exercise-based exam, homework (optional)
theoretical knowledge exam / oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Janez Mrčun:
1. I. Moerdijk, J. Mrčun, On the developability of Lie subalgebroid,
Adv. Math. 210 (2007), 1-21.
2. J. Mrčun, On isomorphisms of algebras of smooth functions, Proc.
Amer. Math. Soc. 133 (2005), 3109-3113.
3. I. Moerdijk, J. Mrčun, On integrability of infinitesimal actions,
Amer. J. Math. 124 (2002), 567-593.

Oliver Dragičević

-Dragičević, Oliver: Weighted estimates for powers and the Ahlfors-Beurling operator. Proc. Amer. Math. Soc. 139 (2011), no. 6, 2113–2120.

-Dragičević, Oliver; Volberg, Alexander: Bilinear embedding for real elliptic differential operators in divergence form with potentials. J. Funct. Anal. 261 (2011), no. 10, 2816–2828.

-Carbonaro, Andrea; Dragičević, Oliver: Bilinear embedding for divergence-form operators with complex coefficients on irregular domains. Calc. Var. Partial Differential Equations 59 (2020), no. 3, Paper No. 104, 36 pp.