wykłady prof. Rovenskiego z Haify

Szanowni Państwo,
W maju będzie mieć u nas dwa cykle wykładów prof. V. Rovenski z Technion---Israel Institute of Technology Haifa w Izraelu. Wykłady mogą być interesujące dla szerokiego spektrum studentów - od zainteresowanych geometrią różniczkową po osoby interesujące się matematyką stosowaną i komputerową.


Tematy kursów to:

Course 1: Geometry and Visualizing with Maple. 30 hours

Course 2: Modeling in mathematics and physics with Maple. 30 hours

Opisy kursów poniżej.

Course 1: Geometry and Visualizing with Maple. 30 hours

Abstract.

The introductory course is devoted to using Maple program in symbolic/numeric calculations and visualizing of curves and surfaces.

It includes the following themes:

0. Introduction to Maple (menu, help, menu, basic symbolic commands, graphics and programming).

1. Functions and graphs (in Cartesian and other coordinates, using complex numbers and quaternions).

2. Geometric transformations in 2 and 3 dimensions (matrices, isometries, affine and projective transformations).

4. Mobius transformations and hyperbolic geometry (in the half-plane and in the half-space).

4. Geometry of curves (examples of curves, tangent lines, singular points, length, curvature and torsion, fractals, variation problems).

5. Geometry of surfaces (examples of surfaces and polyhedra, tangent plane, singular points, I and II forms, length of a curve,

Gaussian curvature, geodesics).

6.* Piecewise curves and surfaces (interpolation, Bezier curves, geometric splines).

Bibliography:

[1] Rovenski V. Geometry of curves and surfaces with Maple, Birkhauser, Boston, 2000.

[2] Rovenski V. Modeling of curves and surfaces with Matlab, SUMAT, Springer, 2010.

[3] Oprea J. Differential Geometry and Its Applications, Prentice Hall, 1997.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

Course 2: Modeling in mathematics and physics with Maple. 30 hours


Abstract.


The course is devoted to using Maple program in symbolic/numeric solutions of differential equations, geometrical structures in mechanics and physics, and their visualizing.

It includes the following traditional and advanced themes:



0. Introduction to Maple: menu, help, basic symbolic commands, graphics and programming, vector calculus.


1. Differential equations

a) Differential equations: Eigenvalue problems, Fourier series, Solving PDE's (Laplace, heat, wave, hyperbolic),

perturbation methods.

b) Dynamical systems: planar systems, limit cycles, Lyapunov functions, bifurcations, chaos, Poincare map, discrete dynamical systems.

c) Mathematical models in anisotropic elasticity and piezo-electricity:

Dirichlet/Neumann and biharmonic BVP's in cartesian and polar coordinates,

exact and approximate polynomial and Fourier series based solutions, coupled beam analysis.

d) Instability analysis in fluid mechanics:

Kelvin-Helmholtz instability, linearized Navier-Stokes equations, perturbation method and dispersion relation.


2. Geometry and Physics.

a) Tensors: metric connection, Euler-Lagrangue equation, geodesics, curvature tensor, Jacobi fields.

b) Examples form Relativity: Galilean and Lorentz transformations, Maxwell equations,

Einstein equation, the Schwarzschild metric solution, perturbation method and gravitational Wave solutions.


Bibliography:


[1] Rand O. and Rovenski V. Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Boston, 2005.

[2] Rovenski V., Gaissinski I. and Kelis O. Hydrodynamic Instability Analysis. Perturbation Methods. Verlag Dr. Muller, 2009.

[3] Gaissinski I. and Rovenski V. Non-Linear Models in Mechanics: Instabilities and Turbulence. Mathematical Methods and Applications, Lambert Academic Publishing, 2010.

[4] Lynch S. Dynamical systems with applications using Maple. 2-nd edition, Birkhauser, 2010.

[5] Articolo G.A. PDE's and BVP's with Maple. 2-nd edition, Elsevier, 2009.

[6] O'Neill B. Semi-Riemannian geometry with applications to relativity, Academic Press, 1983.

[7] Carroll S. Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, 2004.