Analytic and vector geometry in the plane and in the space. Vector spaces.
Matrices, their diagonalization and their use in the resolution of linear systems.
Functions of one or two variables. Differential and integral calculus.
Applications to the optimization and determination of the centroid of planar figures.
Course Content - Last names E-M
Matrices and determinants. Linear systems. Coordinates and vectors. Plane analytic geometry. Analytic geometry in space. Vector spaces and eigenvalues. Functions of one variable. Limits and continuity. Differentiation. Integration. Functions of two or more variables, mentioned. Calculus in several variables, mentioned. Multiple integrals, mentioned.
Course Content - Last names N-Z
Matrices, determinant and characteristic. Linear systems. Analytic geometry in the plane and in the space. Functions in one variable. Limits and continuity. Differential calculus. Integration.
Vector spaces. Eigenvalues, eigenvectors, diagonalizable matrices. Functions of two variables: elements of differential calculus and integration.
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 notes and series of exercises are available in digital format.
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.
Learning Objectives - Last names A-D
The course aims to provide students with knowledge and ability to understand the concepts of geometry and analysis that they can find during their university career.
The course also aims 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
The course aims at providing basic tools in geometry and analysis. These tools are prerequisites for subsequent courses in science and technology.
Learning Objectives - Last names N-Z
The course aims to provide basic tools in geometry and calculus which are both particularly useful for architects cultural knowledge and necessary background for the comprehension of courses of the scientific and technical area.
Prerequisites - Last names A-D
Good command of the concepts of algebra and geometry foreseen in the programs of all upper secondary schools. Knowledge of the definitions of trigonometric, logarithmic and exponential functions.
Prerequisites - Last names E-M
Basic notions of algebra and geometry taught in high school. Trigonometry. Logarithms and exponentials. Solution of algebraic, trigonometric, logarithmic and exponential equations.
Prerequisites - Last names N-Z
Basic notions of algebra and geometry usually taught in high schools.
Trigonometry. Logarithms and exponential functions.
Techniques for solving algebraic, trigonometric, logarithmic, and exponential equations.
Teaching Methods - Last names A-D
Lectures and exercises in the classroom.
Teaching Methods - Last names E-M
Lectures and problem sessions in class.
Teaching Methods - Last names N-Z
Lessons and exercise sessions delivered in classroom.
Further information - Last names A-D
Handouts, training exercises for the intermediate tests or the final written exam and test procedures are available on the Moodle page of the course.
Further information - Last names N-Z
For further information, feel free to contact the lecturer.
Type of Assessment - Last names A-D
Written test and oral exam. Intermediate tests are scheduled during the academic year, the exceeding of which exonerates from the final written.
Type of Assessment - Last names E-M
Written and oral examination. Tests passed throughout the academic year can substitute for the written exam.
Type of Assessment - Last names N-Z
Written and oral final exams, plus intermediate tests during the semesters.
Course program - Last names A-D
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 a vector on another vector; linear combinations, linear dependence and independence, bases and dimension of a vector subspace.
Analytic geometry. Reference systems in the plane and in the space; vector and Cartesian equation of a straight line in the plane and in the space; Cartesian equation of a plane; conditions for perpendicularity and parallelism, angles between lines and planes, distances between points, lines and planes; bundles of lines and bundles of planes. Notable plane figures and their symmetries: circumferences, ellipses, hyperboles and parabolas; conics in canonical form. The quadrics in canonical form; the ruled quadrics. Change of an orthogonal Cartesian reference system, translations and rotations.
Matrices. Sum of matrices and product rows per column, determinant of a square matrix, minor and characteristic of a matrix; transposed matrix and inverse matrix; rank of a matrix and row echelon form. Matrix associated to a linear transformation. Eigenvalues, eigenvectors, autospaces and diagonalization criterion of a square matrix.
Linear systems. Gauss Method, Rouché-Capelli Theorem, Cramer's Theorem and Kronecker's Theorem; homogeneous and non-homogeneous systems.
Functions of one variable. Definition and uniqueness of the limit, criteria for non-existence of the limit, properties of limits, theorem of sign permanence, theorem of comparison; notable limits. Domain of a function; continuous functions and their properties; Weierstrass theorem, zero theorem and intermediate value theorem; definition of derivative and its geometric interpretation; derivative properties and derivation rules; relative maximum and minimum; Fermat, Rolle and Lagrange theorems; de l'Hôpital theorem. Graphs of functions, symmetries, asymptotes, monotonicity intervals, intervals of convexity and concavity, points of inflection. Definition, properties and geometric interpretation of definite and indefinite integrals; fundamental theorem of integral calculus; integration by parts and by substitution; immediate integrals and notabel integrals. Application to the determination of areas of planar regions.
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 using the Hessian matrix; substitution or Lagrange multipliers method for determining the constrained maxima and minima. Double integrals and their application to the determination of the volume of a solid, of the centroid and moments of inertia of a flat plate; change of variables in double integrals and transition to polar coordinates.
Course program - Last names E-M
Matrices and determinants: matrices; operations on matrices; matrix product; determinants; linear combinations; characteristic and rank of a matrix.
Linear systems: linear equations; linear systems; Rouché-Capelli’s theorem; Cramer’s rule; Gauss’ method; homogenous linear systems; inverse of a matrix.
Coordinates and vectors: Cartesian coordinates in plane and in space; vectors; scalar product; vector product; triple product; linear combinations.
Plane analytic geometry: polar coordinates; change of frame; equation of a straight line; parallel or perpendicular lines; angle between two lines; equation of a straight line in explicit form; distance of a point from a straight line; circumference; ellipse; hyperbola; parabola; conic sections in general.
Analytic geometry in space: parametric equation of a straight line; equation of a plane; parallel or perpendicular planes; equations of a straight line, sheaf of planes; parallel or perpendicular lines; lines parallel or perpendicular to a plane; angles in space; distance of a point from a straight line or from a plane; skew lines; curves and surfaces in space, mentioned; quadrics.
Vector spaces and eigenvalues: vector spaces; vector subspaces; base and dimension; linear maps; eigenvectors and eigenvalues; matrix diagonalization; spectral theorem for symmetric matrices.
Functions of one variable: review of real numbers; review of functions; topology of the real line; examples of real functions of one real variable and their graphs.
Limits and continuity: limit of a function; continuous functions; properties of continuous functions; asymptotes.
Differentiation: derivative of a function; higher derivatives; differentiation rules; local maxima or minimum points; some properties; differentiability and monotonicity; convexity and concavity; applications of derivatives.
Integration: definition of integral; main properties of the integral; fundamental theorem of calculus; some integration techniques; areas, volumes, center of mass.
Functions of two or more variables, mentioned: domain; graph; contour lines.
Calculus in several variables, mentioned: partial derivatives; gradient; critical points.
Multiple integrals: definition of double integral; double integral calculus; change of variables; applications; centroids.
Course program - Last names N-Z
Matrices: operations, determinant of a square matrix, minors and characteristic of a matrix;
transpose matrix and inverse matrix; rank of a matrix and row reduction.
Linear systems: Gaussian method, Rouché-Capelli theorem, Cramer theorem; homogeneous and non-homogeneous systems.
Vector algebra: vector operations: sum, product by a scalar, scalar product, vector product,
and mixed product; geometric interpretation of the operations; angle between two vectors and
orthogonal projection of a vector onto another vector.
Analytic geometry: plane and space reference systems;
vectorial and Cartesian equation of a line in the plane; vectorial and Cartesian equation of a line in the space;
vectorial and Cartesian equation of a plane in the space;
orthogonality and parallelism conditions, angles between lines and planes, distances between points,
lines, and planes; families of lines and planes.
Vector spaces: linear combination and generators, linear dependence and independence,
basis and dimension of vector subspaces.
Matrices and linear transformations: correspondence; eigenvalues, eigenvectors, criterion to diagonalize a
square matrix; spectral theorem for real symmetric matrices.
Conics and quadrics: circles, ellipses, hyperbola, parabola, conics in general; quadrics in canonical forms.
Change of orthogonal Cartesian reference system, translations and rotations.
Affine classification of conics and quadrics; centre and planes of symmetry.
One real variable real-valued functions: domain and graph of a function; continuous functions and their properties;
Weierstrass theorem, zeroes theorem, and intermediate values theorem.
Limits of functions: definition of limit at finite points and at infinite;
uniqueness of limit; right and left limits; criteria for the non existence of limits;
properties of limits, permanence of sign in the limit, limit of bounded functions; special limits.
Definition of derivative and its geometric interpretation; properties of derivative and derivation rules;
local maximum and minimum points; Fermat, Rolle, Lagrange theorems; de l'Hôpital theorem and Taylor formula.
Graph of a function: monotonicity, convexity, infection points, asymptotes.
Integration in one variable: definition and properties; fundamental theorem of calculus;
some techniques of integration.
Two real variables real-valued functions: graph and level curves;
directional derivatives, partial derivatives and gradient; critical points and their
classification via the Hessian matrix. Elements of integration in two variables.