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. Multiple integrals. 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.
Further information - Last names N-Z
For further information, feel free to contact the lecturer.
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 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; spectral theorem for 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'Hôpital 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; 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; 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.