Vectorial calculus. Vector spaces. Analytic geometry. Matrices. Linear system. Limits of functions. Real functions of a real variable. Conics and quadrics. Matrices and linear transformations. Functions of two variables. Complex numbers. Integrals and double integrals. Linear differential equations with constant coefficients.
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.
Complex numbers.
Functions of two or more variables.
Calculus in several variables.
Multiple integrals.
Differential equations.
Course Content - Last names N-Z
Vector spaces. Plane and space analytic geometry.
Matrices, determinant, rank, and characteristic. Linear systems. Properties of continuous functions, of derivatives.
Definite and non definite integrals. Eigenvalues, eigenvectors, diagonalizable matrices. Two variables functions, gradient, and Hessian matrix. Constrained maximum and minimum points. Complex numbers. Linear differential equations with constant coefficients.
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 E-M
The course aims at providing basic tools in geometry and analysis. These tools contribute to an architect's general culture and they 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 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 E-M
Lectures and problem sessions in class.
Teaching Methods - Last names N-Z
Lessons and exercise sessions delivered in classroom.
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
Vectorial calculus: Operations among vectors: addition, product by a real number, scalar product, vectorial product and mixed product; geometric interpretation of the operations; angle between two vectors and orthogonal projection of a vector onto another one.
Vector spaces: linear combinations, generators, linear dependence and independence, basis and dimension of vector subspaces.
Analytic geometry: Reference frames in the plane and in the 3d-space; vectorial and Cartesian equation of a line in the plane and in the 3d-space; Cartesian equation of a plane in 3d-space; parallelism and orthogonality, angles between lines and planes, distances among points, lines and planes; families of lines and families of planes. Basic planar sets and their symmetries: circle, ellipse, hyperbola and parabola; conics in canonical form.
Matrices: Operations among matrices, determinant of a square matrix, submatrices and characteristic of a matrix; transpose and inverse matrix; rank of a matrix and row echelon form.
Linear systems: Gauss method, Rouché-Capelli Theorem, Cramer Theorem; homogeneous and nonhomogeneous systems
Limits of functions: Definition of a limit at a point or at infinity; right and left limit; uniqueness of the limit; criterions for the nonexistence of a limit, properties of limits; sign permanence Theorem, comparison Theorem; fundamental limits.
Real functions of a real variable: Domain and graphic of a function; continuous functions and their properties; Weierstrass Theorem, zeros Theorem and intermediate value Theorem; definition of the derivative and its geometrical interpretation; properties of derivative and differentiation rules; relative maxima and minima; Feramt Theorem, Rolle Theorem and Lagrange Theorem; de l’Hospital Theorem and Taylor formula.
Graphics of functions: Symmetries, asymptotes, increasing and decreasing intervals, convexity and concavity intervals, inflection points.
Conics and quadrics: Quadrics in canonical form; ruled quadrics. Change of reference orthogonal frame, translations and rotations. Schemes for the affine classification of conics and quadrics; centers and planes of symmetries.
Matrices and linear transformations: Matrix associated to a linear transformation; eigenvalues, eigenvectors, eigenspaces and a criterion for a square matrix diagonalization; eigenvalues as maxima or minima of a quadratic form, spectral Theorem for real symmetric matrices.
Functions of two variables: Graph of a function and its level curves; directional derivative, partial derivative and gradient of a function; relation between the gradient and the level curves of a function an between the gradient and the tangent plane to the graph of a function; stationary points and their classification via the Hessian matrix; second directional derivative and their representation as the quadratic form associated to the Hessian matrix; substitution or Lagrange multipliers methods for finding constrained maxima and minima.
Complex numbers: Operations among complex numbers, the complex exponential function, geometric interpretation of sums and products of complex numbers; complex conjugate roots of a real polynomial.
Integrals: Definition and properties of indefinite integrals; fundamental Theorem of calculus; integration by parts and by substitution; elementary and basic integrals. Definite integrals and their application to calculate the area of a planar region; double integrals and their application to calculate volume, center of mass and inertia moments of a plate; change of variables in double integrals and polar coordinates.
Linear differential equations with constant coefficients: The vector space of solutions of a homogeneous equation and its dimension; characteristic polynomial and generators of the solution vector space; particular solution for equations with known term of a convenient form; motion equation in free fall state with or without air resistance; motion equation of a simple pendulum, undamped or damped forced motion; resonance.
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: 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.
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.
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; Rolle’s, Lagrange’s, Cauchy’s and de l’Hôpital’s theorems; differentiability and monotonicity; Taylor’s formula; convexity and concavity; applications of derivatives.
Integration: definition of integral; main properties of the integral; fundamental theorem of calculus; direct integration; integration by parts; integration by substitution; integration for some classes of functions; areas, volumes, center of mass.
Plane analytic geometry: polar coordinates; change of frame; circumference; ellipse; hyperbola; parabola; conic sections in general.
Analytic geometry in space: curves and surfaces in space, mentioned; quadrics.
Vector spaces and eigenvalues: vector spaces; vector subspaces; base and dimension; linear maps; linear maps and matrices; change of base; eigenvectors and eigenvalues; matrix diagonalization; Hermitian and symmetric matrices.
Complex numbers: algebraic, geometric and trigonometric representation; de Moivre’s formula; n-th roots of a complex number.
Functions of two or more variables: topology of the plane, mentioned; functions of two variables; graph; contour lines; limits and continuity.
Calculus in several variables: partial derivatives; differential; equation of the tangent plane; derivative of a composition of functions; directional derivatives; gradient; higher order partial derivatives; local maxima and minima; constrained maxima and minima; global maxima and minima.
Multiple integrals: definition of double integral; double integral calculus; change of variables; applications; centroids.
Differential equations: general theory; solution of an ordinary differential equation; equations in normal form; Cauchy problems; first-order linear differential equations; n-th order linear differential equations with constant coefficients, general solution; characteristic equation; solution in special cases with convenient known terms.
Course program - Last names N-Z
Matrices: operations, determinant of 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: operations among vectors: 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; orthogonality and parallelism conditions, angles between lines and planes, distances between points, lines, and planes; families of lines and planes. Special plane curves and their symmetries: circles, ellipses, hyperbola, parabola; canonical form of conics. Vector spaces: linear combination and generators, linear dependence and independence, basis and dimension of vector subspaces.
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.
One variable real functions: domain and graph of a function; continuous functions and their properties; Weierstrass theorem, zeroes theorem, and intermediate values theorem; definition of derivative and its geometric interpretation; properties of derivative and derivation rules; local maximum and minimum points; Fermat, Rolle, Lagrange theorems; de l'Hospital theorem and Taylor formula.
Graphs of functions.
Symmetries, asymptotes, monotonicity and convexity of functions, inflection points. Non definite integrals: definition and properties; fundamental theorem of calculus; integration by parts and by substitution; immediate and special integrals.
Conics and quadrics: canonical forms; ruled quadrics. Change of orthogonal Cartesian reference system, translations and rotations. Tables of affine classification of conics and quadrics; centre and planes of symmetry.
Matrices and linear transformation: correspondence; eigenvalues, eigenvectors, criterion to diagonalize a square matrix; eigenvalues as extremal values of a quadratic form; spectral theorem for real symmetric matrices. Two variables functions: graph and level curves; directional derivatives, partial derivatives and gradient; correspondence between gradient, level curves, and tangent plane to a graph; critical points and their classification via the Hessian matrix; second directional derivatives and their representation as quadratic form corresponding to the Hessian matrix; constrained maximum and minimum points: substitution and Lagrange methods.
Complex numbers: operations, polar representation, and their geometric interpretations; conjugate roots of a polynomial with real coefficients; complex exponential function.
Definite integrals: application to measure of the area and volume, center of mass, and moment of inertia; change of variable in double integrals and polar co ordinates.
Linear differential equations with constant coefficients: the vector space of solutions of a homogeneous equation and its dimension; characteristic polynomial and generators of the space of solutions of the homogeneous equation; a solution for the non homogeneous case given a convenient non homogeneous term.