Single Variable Calculus and Linear Algebra

Cartesian and polar coordinates on the plane, introduction to the calculus of one variable and the method of the mathematical induction. Functions of one variable, the conception of representation, limit and continuity, Boltzano’s theorem. Derivative of first or higher order of functions and geometrical meaning, derivation rules and total differential. Inverse and composite functions, parametric equations, complex forms and L’ Hospital’s rule. Analysis, monotony and extremities of functions, asymptotes. Fermat’s theorem and theorems of mean value. Sequences, number series and convergence criterions. Series of functions, uniform convergence criterions and power series. Generalized mean value theorem or Taylor’s formula and local approximation of function, binomial expansion. Taylor’s and Maclaurin’s series, binomial series and convergence. Fourier’s series and total approximation of function. Applications of derivatives with the use of method of extremities for functions of physical interest and finding the curvature of a plane curve. Indefinite integral of functions and analytic techniques of integration. Riemann’s integral and definite integral. Introduction to the numerical methods of integration and to the generalized integrals. Applications of integrals to the calculation of plane areas, curve’s length, surface areas and domain volumes by rotation. Introduction of plane vectors and the meaning of the third spatial dimension. Inner, exterior, mixed and double-exterior product, geometrical meaning. Matrix theory and square matrices, determinant and inverse matrix. Homogeneous and non homogeneous systems of linear equations, solution with Gauss’ deletion method. Spectral analysis of matrix, in spaces of finite dimensions, eigenvalues and eigenvectors or characteristic magnitudes and physical meaning, Cayley-Hamilton’s theorem. Algebraic and geometric multiplicity of eigenvalues, diagonalization of square matrix. Degenerate eigenvalues, degeneration degree and generalized eigenvectors, Jordan’s matrix. Generalization of inner product, the meaning of norm, distance and orthonormalization with Gram-Schmidt’s method.