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.
Written and oral examination. The written examination consists in the solution of two/three exercises to be completed in two hours. A pocket calculator, with only elementary functions, is allowed. Books, notes or tables of formulae are not. The oral examination consists in the discussion of the written part, followed by questions on the theoretical aspects of the subject. Both parts will equally contribute to assess the achievement of the didactic objectives and to determine the final grade.
The course aims to teach chemistry students the basic tenets of linear algebra and multivariable calculus. Students are expected to be able to apply the tools provided by these theories to abstract and practical problems arising in chemical practice.
Linear algebra, multivariable calculus, ordinary differential equations. Specifically:Linear systems, examples. Naive matrix representation, Gauss elimination. Linear functions, matrices. Vector spaces. Linear independence. Rank of a matrix, computation via Gauss elimination. Basis. Scalar product in R^n. Linear transformations between vector spaces and their matrix representation. Matrix-vector product. Image of basis vectors. Kernel and image of a linear transformation. Injectivity. Counting dimensions of kernel and image. Geometric interpretation of a linear system of equations. Volume transformation in R^n via linear transformations. Determinants. Examples in R^2 and R^3. Vector product. Multilinearity of determinants. Computation of determinants via Gauss elimination. Matrix singularity. Determinantal formulae: expansion via permutations, Laplace expansion. Equivalence classes of matrices. Diagonal representation. Intrinsic notion of eigenvector and eigenvalue. Explicit representation: characteristic equation, roots, algebraic multiplicity. Eigenspaces, geometric multiplicity. Diagonalization. Cayley-Hamilton theorem. Matrix calculus. Symmetric matrices and orthogonal transformations.
Multivariable calculus. Basic notions, continuity, partial derivatives. Differential as a linear transformation. Gradient, level curves. Calculus of derivatives. Implicit function theorem. Critical points, maxima and minima, saddle points. Taylor polynomial in many variables. Quadratic forms. Lagrange multipliers. Parametric curves and surfaces.
Integrals of functions of many variables. Flow, circulation, Gauss Green theorem. Basic calculus techniques. Change of variables, jacobian.
Ordinary differential equations. Classification and basic properties. Linear equations with constant and variable coefficients. Second order equations with constant coefficients. Order reduction. Variation of constants and variation of parameters. Non-linear equations of first order. Phase space. Vector fields. Separable equations, exact equations. Integrating factor.
Linear systems, examples. Naive matrix representation, Gauss elimination. Linear functions, matrices. Vector spaces. Linear independence. Rank of a matrix, computation via Gauss elimination. Basis. Scalar product in R^n. Linear transformations between vector spaces and their matrix representation. Matrix-vector product. Image of basis vectors. Kernel and image of a linear transformation. Injectivity. Counting dimensions of kernel and image. Geometric interpretation of a linear system of equations. Volume transformation in R^n via linear transformations. Determinants. Examples in R^2 and R^3. Vector product. Multilinearity of determinants. Computation of determinants via Gauss elimination. Matrix singularity. Determinantal formulae: expansion via permutations, Laplace expansion. Equivalence classes of matrices. Diagonal representation. Intrinsic notion of eigenvector and eigenvalue. Explicit representation: characteristic equation, roots, algebraic multiplicity. Eigenspaces, geometric multiplicity. Diagonalization. Cayley-Hamilton theorem. Matrix calculus. Symmetric matrices and orthogonal transformations.
Multivariable calculus. Basic notions, continuity, partial derivatives. Differential as a linear transformation. Gradient, level curves. Calculus of derivatives. Implicit function theorem. Critical points, maxima and minima, saddle points. Taylor polynomial in many variables. Quadratic forms. Lagrange multipliers. Parametric curves and surfaces.
Integrals of functions of many variables. Flow, circulation, Gauss Green theorem. Basic calculus techniques. Change of variables, jacobian.
Ordinary differential equations. Classification and basic properties. Linear equations with constant and variable coefficients. Second order equations with constant coefficoents. Order reduction. Variation of constants and variation of parameters. Non-linear equations of first order. Phase space. Vector fields. Separable equations, exact equations. Integrating factor.
Frontal lectures 24 hrs, exercise sessions 36 hrs .
The instructor can be reached for questions and appointments at alberto.setti@uninsubria.it