COURSE GOALS: Aquire technics of differential and integral calculus in higher dimensions and provide a foundation in differential geometry of surfaces.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon successful completion of the course, the student is able to:
* apply the concept of diagonalization of a linear operator in the context of differential and difference equations and, as a special case, to interpret the spectral theorem;
* graphically and analytically interpret the directional derivatives and differential of functions of several variables, using the correct notation and mathematical precision;
* apply the basic theorems of differential calculus (a chain rule, the mean value theorems, theorems on inverse and implicit function);
* determine the constrained extrema of a functions;
* interpret and apply Green's, Stokes' and the divergence theorem.
[1-3] Translation space of d-dimensional Euclidean space, inner product space. Tensor product of vectors, invariants of operators, axial vector of antisymmetric operator in three dimensions, spectrum of symmetric operator.
[4-7] Differential calculus in Rd: derivative, differential, gradient, Jacobi matrix, derivatives of higher order. Mean value theorems, theorem on implicit and inverse function. Extrema of functions with several arguments and applications.
[8-10] Integral calculus in Rd: Riemann integral on parallelepiped, change of variables and Fubini theorem.
[11-14] Vector fields, divergence and rotation. Derivative of determinant and inverse matrix. Space curves, tangent line and length. Surfaces in space, tangential space, constrained extrema. Line and surface integrals, Stokes and Gauss theorem. Differential forms and some applications.
REQUIREMENTS FOR STUDENTS:
Students should attend 70% of all lectures and tutorials and pass all mid-term exams.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Final exam is in written or oral form. Final grade is a combination of grades obtained in mid-term exams and the final exam.
- Theodore Shifrin: Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds, Wiley, 2004
- Paul Bamberg, Shlomo Sternberg: A course of mathematics for students of physics, Cambridge, 1991.
- Šime Ungar: Matematička analiza 3, PMF-Matematički odjel, Zagreb, 2002.
- Morton E. Gurtin: An introduction to continuum mechanics, Academic Press, 1981.