MATHEMATICS 2
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
- Contacts/Info
It is required a good knowledge of calculus of functions of a real variable. Prerequisite: Mathematics I.
The exam is divided into three parts:
- a written test consisting in three to five exercises covering the main topics studied in the course, which will test the ability of students in applying the computational techniques learned in class;
- a second written part covering the theoretical aspects of the course, and consisting in stating and proving a few theorems seen in class; this will assess the understanding of the underlying theory and its applications to the solution of computational problems;
- an oral part, which follows immediately the second written test, consisting in the discussion of the two written tests, where the ability of expressing themselves in correct mathematical language, and to independently recognize the validity of mathematical reasoning will be assessed.
Each part will be evaluated with a grade out of thirty, and the final grade, if greater than or equal to 18, will be the arithmetic mean of the grades of the 3 parts. To be admitted to the oral exam it is necessary to have obtained a score of at least 14/30 in the first written exam.
The course aims to provide the student with fundamental methods and techniques of Linear Algebra, Differential and Integral Calculus in multiple variables and the solution techniques of the most significant ordinary differential equations. A further objective is to prepare the student for the application of these tools in problems of interest in the chemical field.
The student will have a working knowledge of the basic concepts of linear algebra, will be able to discuss continuity and differentiability of functions of multiple variables and solve simple optimization problems. They will also be able to compute integrals in R2 or R3, even using the most common variable changes. Finally, they will be able to discuss the existence and uniqueness of the Cauchy problem for the most common ordinary differential equations and to find the corresponding solutions.
Linear algebra, integral and differential calculus in several variables, ordinary differential equations.
In detail:
Linear Algebra: vector spaces, linear dependence and independence. Subspaces, bases. Rn as a vector space: Operations, dot product. Vector product in R3. Lines in Rn, and planes in R3. Dot product, dot product in R^n.
Linear transformations between vector spaces. Kernel and image of a linear map. Kernel and injectivity. Nullity and rank theorem.
Operations on matrices. Matrices and linear transformations. Determinant: Laplace formula and multilinearity properties. Invertibility and determinant. Binet's theorem. Rank of a matrix.
Linear systems and matrices. Rouchè-Capelli and Cramer theorems.
Eigenvalues and eigenvectors. Diagonalizability of a linear transformation and of the corresponding matrix. Characteristic equation of a matrix and eigenvalues. Inner product spaces. Symmetric and orthogonal matrices. Eigenvalues and eigenvectors of symmetric matrices and their diagonalizability.
Differential calculus in several variables: Continuit and directional derivatives. Differentiability and consequences. Gradient, level lines.
Differentiation theorems. Implicit function theorem. Higher order differentiability. Hessian matrix. Second order Taylor formula.
Maxima and minima. Fermat's Theorem and stationary points. Necessary and sufficient conditions for a stationary point to be extremal in terms of the sign of the eigenvalues of the Hessian. Constrained optimization. Parametric method, Lagrange multipliers. Maxima and minima on closed sets.
Multiple integrals: Definition and properties. Reduction theorem. Integrals in R2: integration by vertical and horizontal lines. Change of variables in R2: polar, elliptical, hyperbolic coordinates. Integrals in R3: integration by lines and washers. Change of variables in R3: cylindrical, polar, ellipsoidal coordinates.
Differential equations. Classification and properties. The Peano-Cauchy existence and local uniqueness theorem. separable, homogeneous, linear of the first order, Bernoulli equations. Higher order linear equations: structure of the space of solutions. Constant coefficients linear equations of order n: homogeneous and non-homogeneous equations. The similarity method, the method of variation of constants. Linear first order systems.
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Frontal lectures 24 hrs, exercise sessions 36 hrs .
The frontal lectures are devoted to the development of the theory and to the description of the computational techniques needed to solve exercises and problems which may have practical origin. The computational techniques will be strengthened and deepened in the exercise sessions where the instructor will describe the solution of further problems and exercises, some of which may be taken from problem sets assigned during the lectures or suggested by the students themselves.
The instructor can be reached for questions and appointments at alberto.setti@uninsubria.it