Analytic and vector geometry in the plane and in space. Vector spaces. Matrices, their diagonalization and their use in solving linear systems. Functions of one or two variables. Differential and integral calculus. Applications to optimization and to the determination of the center of mass of plane figures. Outline of descriptive statistics.
Course Content - Last names E-M
Linear algebra and analytic geometry. Analysis of functions of a single variable. Basic notions of analysis of functions of two variables.
G.Anichini-G.Conti "Geometria analitica e algebra lineare" Edizione Pearson.
G.Anichini-G.Conti "Analisi Matematica I" Edizione Pearson.
G.Anichini-G.Conti "Analisi Matematica II" Edizione Pearson.
Some lecture notes and exercises are available in digital format.
G. Anichini, G. Conti e R. Paoletti "Geometria analitica e algebra lineare" Ed. Pearson
G. Anichini e G. Conti "Analisi Matematica I" Ed. Pearson
G. Anichini e G. Conti "Analisi Matematica II" Ed. Pearson
Learning Objectives - Last names A-G
The course aims to provide students with knowledge and understanding of the concepts of geometry and analysis they can encounter in their university studies.
The course also intends to strengthen the ability to apply knowledge and understanding to the performance of typical exercises and to the solution of practical problems.
Learning Objectives - Last names E-M
We provide the notions of linear algebra, geometry and mathematical analysis that are necessary for the training of future architects. We emphasize geometrical intuition and visualization of important concepts.
Prerequisites - Last names A-G
Manual skills with the concepts of algebra and geometry foreseen in the programs of all high schools. Knowledge of the definitions of trigonometric, logarithmic and exponential functions.
Prerequisites - Last names E-M
Basic mathematical concepts that are taught in secondary school. Some students must pass a preliminary evaluation test.
Teaching Methods - Last names A-G
Lectures and exercises in the classroom.
Teaching Methods - Last names E-M
Classes covering theory and exercises. We will be using the WolframAlpha computational search engine.
Further information - Last names A-G
Lecture notes, exercises for the intermediate tests or final written tests and test procedures are available on the Moodle page of the course.
Type of Assessment - Last names A-G
Written and oral exam. Intermediate tests are planned during the academic year and may substitute the final written exam.
Type of Assessment - Last names E-M
A written exam followed by an oral exam. The written exam will evaluate the ability to solve exercises. The oral exam will evaluate the comprehension of the subject matter, independence in mathematical reasoning, and the acquisition of technical language. An extra point will be awarded for passing two self-assessment tests.
Course program - Last names A-G
Vector spaces. Sum of vectors, product for a scalar, scalar product between vectors, vector product and mixed product between three-dimensional vectors; geometric interpretation of operations; angle between two vectors and orthogonal projection of one vector onto another one; linear combinations, linear dependency and independency, bases and dimension of a vector space.
Analytic geometry. Coordinate systems in the plane and in space; vector and cartesian equation of a straight line in the plane and in space; Cartesian equation of a plane in space; conditions of perpendicularity and parallelism, angles between lines and planes, distances between points, lines and planes; bundles of straight lines and bundles of planes. Notable plane figures and their symmetries: circumferences, ellipses, hyperbolas and parabolas; conics in canonical form. Change of the coordinate system, translations and rotations.
Matrices. Sum of matrices and product rows by columns, determinant of a square matrix, minor and characteristic of a matrix; transposed matrix and inverse matrix; rank of a matrix and reduction to echelon form. Use of matrices in solving linear systems. Matrix associated with a linear transformation. Eigenvalues, eigenvectors, eigenspaces and diagonalizability criterion for a square matrix.
Linear systems. Gauss method, Rouché-Capelli theorem, Cramer theorem and Kronecker theorem; homogeneous and non-homogeneous systems.
Functions of one variable. Definition and uniqueness of the limit, criteria for the non-existence of the limit, properties of limits, the theorem of sign permanence, comparison theorem; notable limits. Domain of a function; continuous functions and their properties; Weierstrass theorem, zeroes theorem and intermediate value theorem; definition of derivative and its geometric interpretation; properties of the derivative and rules of derivation; maxima and minima; Fermat, Rolle and Lagrange theorems; De l'Hôpital's theorem and Taylor's formula. Graphs of functions, symmetries, asymptotes, increasing and decreasing intervals, convexity and concavity intervals, inflection points. Definition, properties and geometric interpretation of definite and indefinite integrals; fundamental theorem of integral calculus; integration by parts and by substitution; immediate integrals and notable integrals. Application to the calculation of the areas of a flat region.
Functions of two variables. Graph of a function and its level curves; directional derivatives, partial derivatives and gradient of a function; link between the gradient and the level curves and between the gradient and the tangent plane to the graph of a function; critical points and their classification via the Hessian matrix; substitution or Lagrange multipliers method for the determination of constrained maxima and minima. Double integrals and their application to the calculus of volume, of the center of mass and of the moments of inertia of a flat sheet; change of variables in double integrals and transition to polar coordinates.
Descriptive statistics. Mean, variance and standard deviation of a probability distribution. Law of large numbers, Chebychev's inequality and Central Limit Theorem.
Course program - Last names E-M
Linear algebra: vectors, matrices, linear systems, eigenvectors and eigenvalues, diagonalization. Analytic geometry of 2-dimensional and 3-dimensional space: lines, planes, basic facts on conics and quadrics. Analysis of functions of a single variable: domain, limits, continuity, derivatives, local and global extrema, convexity, graphs, indefinite and definite integrals. Analysis of functions of two variables: partial derivatives, critical points and double integrals.