Real and complex numbers
Functions of one variable
Limits
Derivatives
Taylor's formula
Riemann's integral
Improper integrals
Numerical sequences and series.
Elements of point set topology
Functions of several variables
Partial and directional derivatives, continuity and differentiability, free and constrained extrema.
Double and triple integrals
Parametric curves
Line integrals
Ordinary differential equations
Method of separation of variables and linear equations
Course Content - Last names E-N
Real and complex numbers
Functions of one variable
Limits
Derivatives
Taylor's formula
Riemann's integral
Improper integrals
Numerical sequences and series.
Elements of point set topology
Functions of several variables
Partial and directional derivatives, continuity and differentiability, free and constrained extrema.
Double and triple integrals
Parametric curves
Line and surface integrals
Ordinary differential equations
Method of separation of variables and linear equations
Course Content - Last names O-Z
Real and complex numbers
Functions of one variable
Limits
Derivatives
Taylor's formula
Riemann's integral
Improper integrals
Numerical sequences and series.
Elements of point set topology
Functions of several variables
Partial and directional derivatives, continuity and differentiability, free and constrained extrema.
Double and triple integrals
Parametric curves
Line integrals
Ordinary differential equations
Method of separation of variables and linear equations
Textbooks recommended for exercises:
Benevieri P., Esercizi di Analisi Matematica, Ed. De Agostini.
Poggiolini L. e Spadini M., Esercizi e temi d'esame di Analisi Matematica II, Esculapio.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 1, Liguori
Editore.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 2, Liguori
Editore.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 1, Zanichelli.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 2, Zanichelli.
Other texts :
Bertsch M. - Dal Passo R. - Giacomelli L., Analisi Matematica, McGraw Hill, Milano 2007.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di una variabile, Pitagora Editrice, Bologna 2005.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di piu'
variabili, Pitagora Editrice, Bologna 2006.
Textbooks recommended for exercises:
Benevieri P., Esercizi di Analisi Matematica, Ed. De Agostini.
Poggiolini L. e Spadini M., Esercizi e temi d'esame di Analisi Matematica II, Esculapio.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 1, Liguori
Editore.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 2, Liguori
Editore.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 1, Zanichelli.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 2, Zanichelli.
Other texts :
Bertsch M. - Dal Passo R. - Giacomelli L., Analisi Matematica, McGraw Hill, Milano 2007.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di una variabile, Pitagora Editrice, Bologna 2005.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di piu'
variabili, Pitagora Editrice, Bologna 2006.
Textbooks recommended for exercises:
Benevieri P., Esercizi di Analisi Matematica, Ed. De Agostini.
Poggiolini L. e Spadini M., Esercizi e temi d'esame di Analisi Matematica II, Esculapio.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 1, Liguori
Editore.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 2, Liguori
Editore.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 1, Zanichelli.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 2, Zanichelli.
Other texts :
Bertsch M. - Dal Passo R. - Giacomelli L., Analisi Matematica, McGraw Hill, Milano 2007.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di una variabile, Pitagora Editrice, Bologna 2005.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di piu'
variabili, Pitagora Editrice, Bologna 2006.
Learning Objectives - Last names A-D
Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems. Knowledge of the principles of computer science and the algorithmic and numerical approach to problems.
The training goal is the acquisition of a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of some concepts of differential and integral calculus in one and several variables, improving students' skills concerning calculation techniques.
Learning Objectives - Last names E-N
Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems. Knowledge of the principles of computer science and the algorithmic and numerical approach to problems.
The training goal is the acquisition of a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of some concepts of differential and integral calculus in one and several variables, improving students' skills concerning calculation techniques.
Learning Objectives - Last names O-Z
Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems. Knowledge of the principles of computer science and the algorithmic and numerical approach to problems.
The training goal is the acquisition of a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of some concepts of differential and integral calculus in one and several variables, improving students' skills concerning calculation techniques.
Prerequisites - Last names A-D
Basic notions of high school mathematics courses. In particular: formal calculus, polynomials, algebraic equations and inequalities, elements of analytic geometry, and of trigonometry
Prerequisites - Last names E-N
Basic notions of high school mathematics courses. In particular: formal calculus, polynomials, algebraic equations and inequalities, elements of analytic geometry, and of trigonometry
Prerequisites - Last names O-Z
Basic notions of high school mathematics courses. In particular: formal calculus, polynomials, algebraic equations and inequalities, elements of analytic geometry, and of trigonometry
Teaching Methods - Last names A-D
The course has two terms consisting in theoretical lectures alternated with tutorials.
Teaching Methods - Last names E-N
The course has two terms consisting in theoretical lectures alternated with tutorials.
Teaching Methods - Last names O-Z
The course has two terms consisting in theoretical lectures alternated with tutorials.
Further information - Last names E-N
More information about the course is available at the following link http://www.dma.unifi.it/~pera
Type of Assessment - Last names A-D
The course has two parts. The exam is performed at the end of the second term. It consists in a written test and in a subsequent oral exam. At the end of the first term an intermediate written test, which is mandatory for the final exam, is performed.
The student has to develop capability in applying and understanding knowledge related to mathematical methods - with particular reference to differential and integral calculus, geometry, linear algebra, numerical calculation, linear programming, probability and statistics - to model, analyze and solve engineering problems, also with the help of IT tools.
Type of Assessment - Last names E-N
The course has two parts. The exam is performed at the end of the second term. It consists in two written tests (the first one concerning the topics developed in the first term and the other one concerning the second term) and in a subsequent oral exam.
The student has to develop capability in applying and understanding knowledge related to mathematical methods - with particular reference to differential and integral calculus, geometry, linear algebra, numerical calculation, linear programming, probability and statistics - to model, analyze and solve engineering problems, also with the help of IT tools.
Type of Assessment - Last names O-Z
The course has two parts. The exam is performed at the end of the second term. It consists in a written test and in a subsequent oral exam. At the end of the first term an intermediate written test, which is mandatory for the final exam, is performed.
The student has to develop capability in applying and understanding knowledge related to mathematical methods - with particular reference to differential and integral calculus, geometry, linear algebra, numerical calculation, linear programming, probability and statistics - to model, analyze and solve engineering problems, also with the help of IT tools.
Course program - Last names A-D
Extended program. Below there is a list of all the topics which constitute the course program.
Numbers. Sets (union, intersection, difference, empty set, complementary). Natural, integer, rational numbers. The real numbers: algebraic axioms, ordering, logical quantifiers. Inequalities. Absolute value. Powers and roots. Logarithms. Intervals. Maximum, minimum, upper and lower bounds, greatest lower bound and least upper bound. Property of completeness of the reals. Archimedean property. Density of rational numbers. Applications between sets, injective, surjective, bijective applications. Domain, codomain, image and graphic of an application.
Real functions of one variable (limits and continuity)
Real functions of real variable. Limited functions. Monotone functions. Inverse functions. Polynomials and rational functions. Main transcendent functions (exponential and logarithmic functions, trigonometric functions and their inverses, hyperbolic functions). The integer part function. Elements of topology of the real line: neighborhood of a point, accumulation points, isolated points. Absolute and relative maxima and minima.
Limits of function
Limits (finite and infinite). Uniqueness of the limit. Theorem of the permanence of the sign. Sandwich theorem. Theorem on operations for calculating limits. Indeterminate forms. Right and left limits. Limits for compositions of functions. Variable change within the limits. Theorem of existence of the limit for monotone functions. Fundamental limits and consequences. Continuity. Continuity theorems (sum, product, quotient and composition of functions). Classification of discontinuities. Theorem of intermediate values and applications. Theorem on the continuity of an inverse function. Weierstrass theorem.
Real functions of a variable: derivatives
Definition of derivative. Left and right derivative. Kinks. Geometric interpretation of the derivative. Differential. Rules of derivation (sum, product, quotient, composition and inverse function). Derivatives of the main functions. Fermat's theorem. Rolle and Lagrange theorems. Consequences of the Lagrange theorem. Hopital's theorems. Higher order derivatives. Asymptotes of a function. Convex functions in an interval. Sufficient conditions for the existence of relative maximums and minimums. Points of inflection. Studies of functions. Infinitesimals and infinities. The little-o notation. Taylor's formula with the remainder in the form of Peano. Formula of Taylor with the remainder in the form of Lagrange. MacLaurin's formula. Applications of the Taylor's formula to the calculation of limits and to some approximation problems.
Integrals in one variable
Primitive. Indefinite integrals. Integration by parts of indefinite integrals. Integration by substitution for indefinite integrals. Integration of elementary functions or deduced from elementary functions. Integration of rational functions. Some integration methods. Definition of definite integral. Negligible sets and necessary and sufficient conditions for integrability. Properties of definite integrals (linearity, monotonicity, additivity). Integration by parts for definite integrals. Integration by substitution for definite integrals. Mean theorem for the integrals. Fundamental theorem of integral calculus. Fundamental formula of integral calculus. Definition of the logarithm through the integral. Application of the definite integrals to the calculation of areas of plane figures and to the calculation of volumes of rotation solids. Improper integrals. Convergence criteria (comparison, asymptotic comparison , absolute convergence). The error function.
Real numerical sequences and series
Definition of a sequence. Limit of a sequence. Converging, diverging and indeterminate sequences. Limits theorems for sequences. Bounded sequences. Monotone sequences. Lower bound and upper bound of a sequence. Existence theorem of the limit for monotone sequences. The number e as limit of a sequence. Equivalent definitions of the convergence of a function, main theorem and some applications. N-th root, factorial and calculation of some limits connected to these notions. Numerical series. Sum of a series. Character of a series. Necessary condition for convergence. Geometric series. Harmonic series and generalized harmonic series. Series with positive terms. Comparison criterion. Asymptotic comparison criterion. Integral test and absolute convergence criterion. Series with alternate signs. Leibniz criterion. Series of functions. Simple and absolute convergence. Power series. Convergence radius and properties. Taylor series and development of some functions.
Functions of several variables
The space R ^ 2, R ^ 3, R ^ k. Norm and distance. Scalar product. Schwarz inequality. Elements of topology in R ^ k (neighborhood of a point, accumulation points, isolated points, internal points, border points, closed sets, open sets , closure and border of a set). Bounded sets. Sequences in R ^ k. Real functions of two or more real variables. Level sets. Limits (finite and infinite). Directional limits. Continuity. Continuity theorem of composite functions. Weierstrass theorem. Partial derivatives. Gradient. Directional derivatives. Differentiability and its consequences. Tangent plane. Theorem of the total differential. Partial derivatives of higher order. Schwarz's theorem. Relative maxima and minima. Necessary condition for extreme points (Fermat's theorem). Search for absolute maxima and minima in closed and bounded sets. Hessian matrix (in two variables). Sufficient conditions for extreme points. Vector functions of several variables. Limits and continuity. Jacobian matrix. Polar, spherical, cylindrical coordinates and their Jacobian.
Multiple integrals
Double integrals. Negligible sets in R ^ 2 and necessary and sufficient condition for integrability in a rectangle. Reduction theorem for double integrals (by Fubini). Integral in a bounded set of R ^ 2. Property of the double integrals (linearity, monotonicity, additivity). Reduction formulas deducible from Fubini's Theorem. Change of variables formula for double integrals. Applications of double integrals to the calculation of masses, centers of mass and moments of inertia. Calculation of the integral of the Gauss function using double integrals. Triple integrals. Negligible sets in R ^ 3 and necessary and sufficient condition for integrability. Reduction theorem for triple integrals (by Fubini). Integral in a bounded set of R ^ 3. Property of the triple integrals (linearity, monotonicity, additivity). Reduction formulas for triple integrals (formula of "spaghetti" and "slices"). Change of variable formulas for triple integrals. Applications of triple integrals to the calculation of masses and centers of mass.
Complex numbers
The complex numbers. Real part and imaginary part. Sum, product, quotient of complex numbers. Conjugate of a complex number. Module. Complex plane. Trigonometric form of a complex number. Exponential form of a complex number. Euler formulas. N-th power of a complex number. N-th roots of a complex number and their calculation. Polynomials in the complex field. Fundamental theorem of algebra.
Ordinary differential equations
First order equations in normal form. Definition of a solution. Maximal solutions. Cauchy problem. Peano's existence theorem. Existence and uniqueness theorem (of the maximal solution to the problem of Cauchy). Equations with separable variables. Integration of linear, first order equations with continuous coefficients. Equations of n-th order in normal form. Linearly independent functions. Integration of linear differential equations of second order with constant coefficients. Methods of variation of constants and other practical methods for determining the particular solution of non-homogeneous linear equations. Example of the harmonic oscillator. Curvilinear curves and integrals Parametric curves in R ^ k. Closed and simple curves, curve support. Regular and piecewise regular curves. Definition of non-oriented curvilinear integral and its linearity, monotonicity, additive properties. Rectifiable curves. Theorem of the reduction of a not oriented curvilinear integral to a simple integral. Change of parameters and equivalent curves. Independency of a not oriented integral from the parameterization. Applications to the calculation of masses, centers of mass and moments of inertia of a curve. Work of a vector field. Gauss-Green formulas.
Course program - Last names E-N
Extended program. Below there is a list of all the topics which constitute the course program.
Numbers. Sets (union, intersection, difference, empty set, complementary). Natural, integer, rational numbers. The real numbers: algebraic axioms, ordering, logical quantifiers. Inequalities. Absolute value. Powers and roots. Logarithms. Intervals. Maximum, minimum, upper and lower bounds, greatest lower bound and least upper bound. Property of completeness of the reals. Archimedean property. Density of rational numbers. Applications between sets, injective, surjective, bijective applications. Domain, codomain, image and graphic of an application.
Real functions of one variable (limits and continuity)
Real functions of real variable. Limited functions. Monotone functions. Inverse functions. Polynomials and rational functions. Main transcendent functions (exponential and logarithmic functions, trigonometric functions and their inverses, hyperbolic functions). The integer part function. Elements of topology of the real line: neighborhood of a point, accumulation points, isolated points. Absolute and relative maxima and minima.
Limits of function
Limits (finite and infinite). Uniqueness of the limit. Theorem of the permanence of the sign. Sandwich theorem. Theorem on operations for calculating limits. Indeterminate forms. Right and left limits. Limits for compositions of functions. Variable change within the limits. Theorem of existence of the limit for monotone functions. Fundamental limits and consequences. Continuity. Continuity theorems (sum, product, quotient and composition of functions). Classification of discontinuities. Theorem of intermediate values and applications. Theorem on the continuity of an inverse function. Weierstrass theorem.
Real functions of a variable: derivatives
Definition of derivative. Left and right derivative. Kinks. Geometric interpretation of the derivative. Differential. Rules of derivation (sum, product, quotient, composition and inverse function). Derivatives of the main functions. Fermat's theorem. Rolle and Lagrange theorems. Consequences of the Lagrange theorem. Hopital’s theorems. Higher order derivatives. Asymptotes of a function. Convex functions in an interval. Sufficient conditions for the existence of relative maximums and minimums. Points of inflection. Studies of functions. Infinitesimals and infinities. The little-o notation. Taylor's formula with the remainder in the form of Peano. Formula of Taylor with the remainder in the form of Lagrange. MacLaurin's formula. Applications of the Taylor’s formula to the calculation of limits and to some approximation problems.
Integrals in one variable
Primitive. Indefinite integrals. Integration by parts of indefinite integrals. Integration by substitution for indefinite integrals. Integration of elementary functions or deduced from elementary functions. Integration of rational functions. Some integration methods. Definition of definite integral. Negligible sets and necessary and sufficient conditions for integrability. Properties of definite integrals (linearity, monotonicity, additivity). Integration by parts for definite integrals. Integration by substitution for definite integrals. Mean theorem for the integrals. Fundamental theorem of integral calculus. Fundamental formula of integral calculus. Definition of the logarithm through the integral. Application of the definite integrals to the calculation of areas of plane figures and to the calculation of volumes of rotation solids. Improper integrals. Convergence criteria (comparison, asymptotic comparison , absolute convergence). The error function.
Real numerical sequences and series
Definition of a sequence. Limit of a sequence. Converging, diverging and indeterminate sequences. Limits theorems for sequences. Bounded sequences. Monotone sequences. Lower bound and upper bound of a sequence. Existence theorem of the limit for monotone sequences. The number e as limit of a sequence. Equivalent definitions of the convergence of a function, main theorem and some applications. N-th root, factorial and calculation of some limits connected to these notions. Numerical series. Sum of a series. Character of a series. Necessary condition for convergence. Geometric series. Harmonic series and generalized harmonic series. Series with positive terms. Comparison criterion. Asymptotic comparison criterion. Integral and absolute convergence criterion. Series with alternate signs. Leibniz criterion. Series of functions. Simple and absolute convergence. Power series. Convergence radius and properties. Taylor series and development of some functions.
Functions of several variables
The space R ^ 2, R ^ 3, R ^ k. Norm and distance. Scalar product. Schwarz inequality. Elements of topology in R ^ k (neighborhood of a point, accumulation points, isolated points, internal points, border points, closed sets, open sets , closure and border of a set). Bounded sets. Sequences in R ^ k. Real functions of two or more real variables. Level sets. Limits (finite and infinite). Directional limits. Continuity. Continuity theorem of composite functions. Weierstrass theorem. Partial derivatives. Gradient. Directional derivatives. Differentiability and its consequences. Tangent plane. Theorem of the total differential. Partial derivatives of higher order. Schwarz’s theorem. Relative maxima and minima. Necessary condition for extreme points (Fermat's theorem). Search for absolute maxima and minima in closed and bounded sets. Hessian matrix (in two variables). Sufficient conditions for extreme points. Vector functions of several variables. Limits and continuity. Jacobian matrix. Polar, spherical, cylindrical coordinates and their Jacobian.
Multiple integrals
Double integrals. Negligible sets in R ^ 2 and necessary and sufficient condition for integrability in a rectangle. Reduction theorem for double integrals (by Fubini). Integral in a bounded set of R ^ 2. Property of the double integrals (linearity, monotonicity, additivity). Reduction formulas deducible from Fubini's Theorem. Change of variables formula for double integrals. Applications of double integrals to the calculation of masses, centers of mass and moments of inertia. Calculation of the integral of the Gauss function using double integrals. Triple integrals. Negligible sets in R ^ 3 and necessary and sufficient condition for integrability. Reduction theorem for triple integrals (by Fubini). Integral in a bounded set of R ^ 3. Property of the triple integrals (linearity, monotonicity, additivity). Reduction formulas for triple integrals (formula of "spaghetti" and "slices"). Change of variable formulas for triple integrals. Applications of triple integrals to the calculation of masses and centers of mass.
Complex numbers
The complex numbers. Real part and imaginary part. Sum, product, quotient of complex numbers. Conjugate of a complex number. Module. Complex plane. Trigonometric form of a complex number. Exponential form of a complex number. Euler formulas. N-th power of a complex number. N-th roots of a complex number and their calculation. Polynomials in the complex field. Fundamental theorem of algebra.
Ordinary differential equations
First order equations in normal form. Definition of a solution. Maximal solutions. Cauchy problem. Peano's existence theorem. Existence and uniqueness theorem (of the maximal solution to the problem of Cauchy). Equations with separable variables. Integration of linear, first order equations with continuous coefficients. Equations of n-th order in normal form. Linearly independent functions. Integration of linear differential equations of second order with constant coefficients. Methods of variation of constants and other practical methods for determining the particular solution of non-homogeneous linear equations. Example of the harmonic oscillator. Curvilinear curves and integrals Parametric curves in R ^ k. Closed and simple curves, curve support. Regular and piecewise regular curves. Definition of non-oriented curvilinear integral and its linearity, monotonicity, additive properties. Rectifiable curves. Theorem of the reduction of a not oriented curvilinear integral to a simple integral. Change of parameters and equivalent curves. Independency of a not oriented integral from the parameterization. Applications to the calculation of masses, centers of mass and moments of inertia of a curve. Work of a vector field. Gauss-Green formulas.
Course program - Last names O-Z
Extended program. Below there is a list of all the topics which constitute the course program.
Numbers. Sets (union, intersection, difference, empty set, complementary). Natural, integer, rational numbers. The real numbers: algebraic axioms, ordering, logical quantifiers. Inequalities. Absolute value. Powers and roots. Logarithms. Intervals. Maximum, minimum, upper and lower bounds, greatest lower bound and least upper bound. Property of completeness of the reals. Archimedean property. Density of rational numbers. Applications between sets, injective, surjective, bijective applications. Domain, codomain, image and graphic of an application.
Real functions of one variable (limits and continuity)
Real functions of real variable. Limited functions. Monotone functions. Inverse functions. Polynomials and rational functions. Main transcendent functions (exponential and logarithmic functions, trigonometric functions and their inverses, hyperbolic functions). The integer part function. Elements of topology of the real line: neighborhood of a point, accumulation points, isolated points. Absolute and relative maxima and minima.
Limits of function
Limits (finite and infinite). Uniqueness of the limit. Theorem of the permanence of the sign. Sandwich theorem. Theorem on operations for calculating limits. Indeterminate forms. Right and left limits. Limits for compositions of functions. Variable change within the limits. Theorem of existence of the limit for monotone functions. Fundamental limits and consequences. Continuity. Continuity theorems (sum, product, quotient and composition of functions). Classification of discontinuities. Theorem of intermediate values and applications. Theorem on the continuity of an inverse function. Weierstrass theorem.
Real functions of a variable: derivatives
Definition of derivative. Left and right derivative. Kinks. Geometric interpretation of the derivative. Differential. Rules of derivation (sum, product, quotient, composition and inverse function). Derivatives of the main functions. Fermat's theorem. Rolle and Lagrange theorems. Consequences of the Lagrange theorem. Hopital's theorems. Higher order derivatives. Asymptotes of a function. Convex functions in an interval. Sufficient conditions for the existence of relative maximums and minimums. Points of inflection. Studies of functions. Infinitesimals and infinities. The little-o notation. Taylor's formula with the remainder in the form of Peano. Formula of Taylor with the remainder in the form of Lagrange. MacLaurin's formula. Applications of the Taylor's formula to the calculation of limits and to some approximation problems.
Integrals in one variable
Primitive. Indefinite integrals. Integration by parts of indefinite integrals. Integration by substitution for indefinite integrals. Integration of elementary functions or deduced from elementary functions. Integration of rational functions. Some integration methods. Definition of definite integral. Negligible sets and necessary and sufficient conditions for integrability. Properties of definite integrals (linearity, monotonicity, additivity). Integration by parts for definite integrals. Integration by substitution for definite integrals. Mean theorem for the integrals. Fundamental theorem of integral calculus. Fundamental formula of integral calculus. Definition of the logarithm through the integral. Application of the definite integrals to the calculation of areas of plane figures and to the calculation of volumes of rotation solids. Improper integrals. Convergence criteria (comparison, asymptotic comparison , absolute convergence). The error function.
Real numerical sequences and series
Definition of a sequence. Limit of a sequence. Converging, diverging and indeterminate sequences. Limits theorems for sequences. Bounded sequences. Monotone sequences. Lower bound and upper bound of a sequence. Existence theorem of the limit for monotone sequences. The number e as limit of a sequence. Equivalent definitions of the convergence of a function, main theorem and some applications. N-th root, factorial and calculation of some limits connected to these notions. Numerical series. Sum of a series. Character of a series. Necessary condition for convergence. Geometric series. Harmonic series and generalized harmonic series. Series with positive terms. Comparison criterion. Asymptotic comparison criterion. Integral test and absolute convergence criterion. Series with alternate signs. Leibniz criterion. Series of functions. Simple and absolute convergence. Power series. Convergence radius and properties. Taylor series and development of some functions.
Functions of several variables
The space R ^ 2, R ^ 3, R ^ k. Norm and distance. Scalar product. Schwarz inequality. Elements of topology in R ^ k (neighborhood of a point, accumulation points, isolated points, internal points, border points, closed sets, open sets , closure and border of a set). Bounded sets. Sequences in R ^ k. Real functions of two or more real variables. Level sets. Limits (finite and infinite). Directional limits. Continuity. Continuity theorem of composite functions. Weierstrass theorem. Partial derivatives. Gradient. Directional derivatives. Differentiability and its consequences. Tangent plane. Theorem of the total differential. Partial derivatives of higher order. Schwarz's theorem. Relative maxima and minima. Necessary condition for extreme points (Fermat's theorem). Search for absolute maxima and minima in closed and bounded sets. Hessian matrix (in two variables). Sufficient conditions for extreme points. Vector functions of several variables. Limits and continuity. Jacobian matrix. Polar, spherical, cylindrical coordinates and their Jacobian.
Multiple integrals
Double integrals. Negligible sets in R ^ 2 and necessary and sufficient condition for integrability in a rectangle. Reduction theorem for double integrals (by Fubini). Integral in a bounded set of R ^ 2. Property of the double integrals (linearity, monotonicity, additivity). Reduction formulas deducible from Fubini's Theorem. Change of variables formula for double integrals. Applications of double integrals to the calculation of masses, centers of mass and moments of inertia. Calculation of the integral of the Gauss function using double integrals. Triple integrals. Negligible sets in R ^ 3 and necessary and sufficient condition for integrability. Reduction theorem for triple integrals (by Fubini). Integral in a bounded set of R ^ 3. Property of the triple integrals (linearity, monotonicity, additivity). Reduction formulas for triple integrals (formula of "spaghetti" and "slices"). Change of variable formulas for triple integrals. Applications of triple integrals to the calculation of masses and centers of mass.
Complex numbers
The complex numbers. Real part and imaginary part. Sum, product, quotient of complex numbers. Conjugate of a complex number. Module. Complex plane. Trigonometric form of a complex number. Exponential form of a complex number. Euler formulas. N-th power of a complex number. N-th roots of a complex number and their calculation. Polynomials in the complex field. Fundamental theorem of algebra.
Ordinary differential equations
First order equations in normal form. Definition of a solution. Maximal solutions. Cauchy problem. Peano's existence theorem. Existence and uniqueness theorem (of the maximal solution to the problem of Cauchy). Equations with separable variables. Integration of linear, first order equations with continuous coefficients. Equations of n-th order in normal form. Linearly independent functions. Integration of linear differential equations of second order with constant coefficients. Methods of variation of constants and other practical methods for determining the particular solution of non-homogeneous linear equations. Example of the harmonic oscillator. Curvilinear curves and integrals Parametric curves in R ^ k. Closed and simple curves, curve support. Regular and piecewise regular curves. Definition of non-oriented curvilinear integral and its linearity, monotonicity, additive properties. Rectifiable curves. Theorem of the reduction of a not oriented curvilinear integral to a simple integral. Change of parameters and equivalent curves. Independency of a not oriented integral from the parameterization. Applications to the calculation of masses, centers of mass and moments of inertia of a curve. Work of a vector field. Gauss-Green formulas.