A: K. Rektorys: Survey of Applicable Mathemaics, Springer 2nd edition (March 31, 1994)

1. Vectors and matrices, matrix algebra, scalar product. Linear independence of vectors and rank of a matrix.

2. Systems of linear algebraic equations. Determinant of a matrix, vector product.

3. Inverse matrix. Eigenvalues of matrices. Plane and space geometry.

4. Euclidean space, metric, norm, properties of subsets.

5. Functions of more real variables. Partial derivatives, partial derivatives of composite functions. Directional derivative, gradient. Total differential, tangent plane.

6. Taylor polynomial of functions of 2 variables. Newton's method for a system of 2 nonlinear equations of 2 unknowns.

7. Extremes of functions of two variables. Least squares method.

8. Implicit functions of one and more variables and their derivatives.

9. Curves given parametrically, tangent vector to a curve, smooth curve, orientation and sum of curves.

10. Vector fields in plane and space. The line integral of a vector field and its physical meaning.

11. Independence of the curve integral on the integration path. Potential vector field. Differential forms and their integration.

12. Double integral and its geometric meaning. Calculation of double integral by iterated integral - Fubini’s theorem.

13. Substitution for the double integral. Polar coordinates. Laplace integral.

14. Systems of two first-order differential equations. Solution of autonomous systems of linear differential equations with constant coefficients. "Predator-prey" model.

General skills:

1. basic mathematical terms

2. knowledge and understanding of basic algorithms

3. individual problem solving

4. basic mathematical background for formulation and solving of natural and engineering problems

5. numerical algorithms (systems of differential equations).

Mathematics I