Questions in Mathematics II

 1. Linear space, linearly dependent vectors, basis, dimension, 
    subspace, examples of linear spaces, R^n, C^k(I).
 2. Linear map, kernel. Linear map from R^n to R^m, its matrix 
    representation.  Invertible map, one-to-one map. Inverse matrix.
 3. Linear differential equation of 2. order. Wronskian. 
    Space of solutions.  Variation of constants. Initial condition.
 4. Linear differential equation with constant coefficients and with 
    a special right-hand-side, method of unknown coefficients. 
    Boundary condition.
 5. System of 2 differential equations of 1. order, autonomous and 
    non-autonomous equations, stationary solution.  
    Predator-prey model. Euler method.
 6. System of homogeneous linear differential equations with constant 
    coefficients. Eigenvalues and eigenvectors of a matrix.
 7. Complex function of a real argument, its application for solution 
    of differential equations.
 8. Euclidean space R^n, distance, neighborhood, open sets, 
    closed sets, boundary, domain, convex sets.
 9. Continuity and limit of function of more variables.
10. Partial derivative, directional derivative, total differential.
    Tangent plane. Taylor polynomial of a function of 2 arguments.
11. Derivative of composed functions of more arguments. Jacobi matrix.
12. Newton's method for solution of a system of 2 nonlinear algebraic
    equations for 2 unknowns. Its geometric meaning.
13. Local extrema of functions of 2 arguments, stationary point, 
    saddle point. Least squares method.
14. Implicit function theorem.
15. Riemann definite integral. Geometrical and physical applications 
    of definite integral. Mean value of a function.
16. Double integral, its geometrical meaning, Fubini theorem.
17. Substitution for double integral, polar coordinates.
18. Curves in R^3, tangent vector. Curve integral of a scalar field,
    its physical meaning.
19. Curve integral of a vector field, its properties. Work.
20. Path-independent curve integral of a vector field, potential,
    its existence and meaning. Integration of total differential.
    
Prague, May 19, 2006                          Pavel Pokorny