ITA | ENG
B015496 - MATHEMATICS I
Main information
Teaching Language
Course Content
Learning Objectives
Prerequisites
Type of Assessment
Course program
Academic Year 2013-14
Course year
First year - First Semester
Belonging Department
Physics and Astronomy
Course Type
Single education field course
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
9
Teaching Hours
72
Teaching Term
23/09/2013 ⇒ 20/12/2013
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
italian
Course Content
- 6 CFU (teacher: Emanuele Paolini): Real numbers. Function of one variable. Limits. Derivatives. Taylor polinomial. Functions of several variables. Partial derivatives.
- 3 CFU (teacher Enrico Sbarra): Gaussian Elimination. Vector spaces. Matrices and determinants. Linear maps. Analytic Geometry of Space. Diagonalization of a matrix
- 3 CFU (teacher Enrico Sbarra): Gaussian Elimination. Vector spaces. Matrices and determinants. Linear maps. Analytic Geometry of Space. Diagonalization of a matrix
Learning Objectives
Knowledge acquired:
The course will offer knowledge on the concepts and theorems regarding real numbers, functions of one variable and functions of several variables. The course will also offer knowledge on Linear Algebra and Analytic Geometry of Space.
Competence acquired
The student will be able to compute limits and derivatives. By means of these tools he will be able to study the trend of functions in one or several variables. The student will also be able to study and solve linear systems; to identify vector spaces or subspaces and linear maps; to find eigenvalues and eigenvectors of a matrix; to write equations of lines and planes; to use the concept of distance for analytic representation of geometric entities.
Skills acquired (at the end of the course):
The student will acquire the ability to study and solve nonlinear equations and inequalities and to solve optimization problems with one or several variables. The student will also acquire the ability to use linear system theory and Gauss algorithm in several applications; he will be able to use geometric vectors to solve problems in analytic geometry of space; will be able to determine if a given matrix is diagonalizable.
The course will offer knowledge on the concepts and theorems regarding real numbers, functions of one variable and functions of several variables. The course will also offer knowledge on Linear Algebra and Analytic Geometry of Space.
Competence acquired
The student will be able to compute limits and derivatives. By means of these tools he will be able to study the trend of functions in one or several variables. The student will also be able to study and solve linear systems; to identify vector spaces or subspaces and linear maps; to find eigenvalues and eigenvectors of a matrix; to write equations of lines and planes; to use the concept of distance for analytic representation of geometric entities.
Skills acquired (at the end of the course):
The student will acquire the ability to study and solve nonlinear equations and inequalities and to solve optimization problems with one or several variables. The student will also acquire the ability to use linear system theory and Gauss algorithm in several applications; he will be able to use geometric vectors to solve problems in analytic geometry of space; will be able to determine if a given matrix is diagonalizable.
Prerequisites
Courses required: None.
Courses recommmended: None
Courses recommmended: None
Type of Assessment
Exam modality: written and oral examination
Course program
Recapitulation on equations and inequalities. Numbers: natural, integers, rational. Real numbers, properties. Axiom of completion. Elementary functions: linear functions, absolute value, power, exponential, logarithm, trigonometric functions and their inverse. Properties and computation of limits. Comparison theorem, sign persistence, product of limited by infinitesimal function. Continuous functions. Theorem of zeroes, bisection method. Weierstrass Theorem. Derivatives, computation of derivatives. Increasing and decreasing functions. Extremal points. Convexity and concavity. Graph of a function. Taylor polinomial. Taylor formula in the computation of limits. Functions of several variables: limits, continuity. Partial derivatives, gradient, Schwarz Theorem. Local maxima and minima, Hessian. Differentiability.
Geometric vectors. Vector addition and scalar multiplication and characteristic properties. Linear combinations, linear dependence and independence. Parallel and coplanar vectors and equivalent conditions. Theorem “Given tree linearly independent geometric vectors, any given vector is uniquely expressed as a linear combination of those”. Dot, cross and scalar triple product and characteristic properties. Product computation with respect to orthonormal basis.
Vector spaces: definitions, properties, examples.
Linear systems. Gaussian Elimination. Consistent systems and characteristic properties. Rouché-Capelli Theorem.
Matrix multiplication and characteristic properties.
Invertible matrices and equivalent properties; a method of matrix inversion.
How to select a basis from a set of generator with Gaussian Elimination.
Linear and affine subspaces.
Determinant of a square matrix and characteristic properties. The determinant of a non-singular matrix. Cramer Theorem.
Linear maps. Injective, surjective, bijective linear maps. The linear map defined by a matrix. Kernel, image and the rank-nullity theorem.
Affine Geometry: Cartesian coordinate systems for a line, a plane and a three-dimensional space. Parametric and Cartesian equations of the line and planes; conditions for line-line and line-plane intersection or parallelism.
Elements of Euclidean Geometry: distance formulas and condition of orthogonality.
Eigenvalues and eigenvectors of a square matrix. Analitic determination of eigenvalues. The real and the complex case. Diagonalization of a matrix; the case of matrices with distinct eigenvalues. The general case: algebraic and geometric multiplicities.
Geometric vectors. Vector addition and scalar multiplication and characteristic properties. Linear combinations, linear dependence and independence. Parallel and coplanar vectors and equivalent conditions. Theorem “Given tree linearly independent geometric vectors, any given vector is uniquely expressed as a linear combination of those”. Dot, cross and scalar triple product and characteristic properties. Product computation with respect to orthonormal basis.
Vector spaces: definitions, properties, examples.
Linear systems. Gaussian Elimination. Consistent systems and characteristic properties. Rouché-Capelli Theorem.
Matrix multiplication and characteristic properties.
Invertible matrices and equivalent properties; a method of matrix inversion.
How to select a basis from a set of generator with Gaussian Elimination.
Linear and affine subspaces.
Determinant of a square matrix and characteristic properties. The determinant of a non-singular matrix. Cramer Theorem.
Linear maps. Injective, surjective, bijective linear maps. The linear map defined by a matrix. Kernel, image and the rank-nullity theorem.
Affine Geometry: Cartesian coordinate systems for a line, a plane and a three-dimensional space. Parametric and Cartesian equations of the line and planes; conditions for line-line and line-plane intersection or parallelism.
Elements of Euclidean Geometry: distance formulas and condition of orthogonality.
Eigenvalues and eigenvectors of a square matrix. Analitic determination of eigenvalues. The real and the complex case. Diagonalization of a matrix; the case of matrices with distinct eigenvalues. The general case: algebraic and geometric multiplicities.