## Lecture Mathematics B

#### Thursday at 8 am in C12

Basic information

Topics for oral exam

__Slides from lecture__

__Syllabus __ – here

- Vectors and matrices, matrix arithmetic, dot product. Linear independence of vectors and rank of a matrix.
- Systems of linear algebraic equations. Determinant of a matrix, cross product.
- Inverse matrices. Eigenvalues of a matrix. Geometry in the plane and three-dimensional space.
- Euclidean space, metric, norm, properties of subsets of the Eucidean space.
- Functions of several variables. Partial derivatives, partial derivatives of compositions of functions. Directional derivatives, gradient of a function. Total differential, tangent plane.
- Taylor polynomial of functions of two variables. Newton’s method for a system of two non-linear equations of two variables.
- Extrema of functions of two variables. Least square method.
- Implicitly defined functions of a single and several variables, derivatives of implicitly defined functions.
- Parametric curves, tangent vector to a curve, smooth curve, orientation and a sum of curves.
- Vector field in the plane and space. Curvilinear integral of a vector field and its physical meaning.
- Path independence of the curvilinear integral of a vector field. Scalar potential of a vector field. Differential forms and their integrals.
- Double integral and its geometrical meaning. Fubini theorem. Substitution for double integral. Polar coordinates.
- Laplace integral. Revision and discussion.
- Systems of two first order differential equations. Solving autonomous systems of differential equations with constant coefficients. Predator-prey model.

__Consultations__ – by agreement personal or email