Systems of linear algebraic equations. Inverse matrix. Eigenvalues and eigenvectors of matrices, generalized eigenvectors.
Singular values, singular value decomposition. Least squares solution of a system of linear algebraic equations. Normal equations.
Linear and nonlinear regression.
Solving systems of nonlinear equations, Newton method. Newton method for systems of nonlinear equations.
Implicit function of one or more variables, general theorem for implicit functions.
Numerical solution of ordinary differential equations, initial value problem: Euler's method. Runge-Kutta methods, multistep methods.
Numerical solution of ordinary differential equations, boundary value problem, method of shooting.
Dynamical systems, the trajectory of the system, equilibrium conditions, phase portrait. Invariant set, ω-limit sets of trajectories.
Systems of linear ODEs with constant coefficients: Solving linear systems using eigenvalues, eigenvectors and generalized eigenvectors.
Phase portraits of linear systems in R^1,R^2.
Systems of nonlinear differential equations: classification of equilibrium states of nonlinear systems. Principles of construction of phase portraits in the plane. Homoclinics and heteroclinics.
Line integrals of scalar and vector fields. Potential vector field.
Surface integrals of scalar and vector fields. Gauss's and Stokes's theorems.
Basics of vector and tensor calculus. Nabla operator algebra. Green's theorem.