Mathematical Methods Applied to Chemistry

Goals:
We introduce concepts and computational techniques that are useful in the study of the solutions of some differential equations involved in mathematical models that describe some natural phenomena.


acquired skills:
Capability in the use of differential and integral Calculus for functions of several variables, in order to understand the qualitative behavior of certain mathematical models describing physical processes.


Course content
A) Differential calculus for functions of several real variables.
Curves p: R¹→Rᵐ: orientation, arc-length , I-type line integrals and applications.
Functions f:Rⁿ→R¹: continuità, partial derivatives, gradient, differentiability, tangent planes. Hessian matrix, Taylor's formula of order 2. Optimization: the nature of stationary points. Quadratic forms and eigenvalues.
Functions F:Rⁿ→Rᵐ: jacobian matrix, differentiability, the Chain rule. Vector fields. Surfaces in Rᶟ. Tangent plane and normal vector. Implicit functions, Dini's Theorem. Constrained optimization and the Lagrange multipliers method.
B) Integral Calculus in 2 and 3 variables.
The basic properties of double and triple integrals: regular sets and measurable functions. Iterated integration. Polar and spherical coordinates. Vector fields F:Rᶟ→Rᶟ: the differential operators grad, div, rot and their properties. Line integrals of II type: the work of a vector field along a line. Conservative and irrotational vector fields, potentials, Poincaré`s lemma, simply connected sets. I and II type surface integrals, flow of a vector field. Surfaces with boundary. Integral formulas: Gauss-Green's formula, the Divergence Theorem in dimensions 2 and 3, Stokes's theorem.
C) Differential equations
Ordinary differential equations: Cauchy's problems, exixtence/uniqueness of local/global solutions. Types of equations: separable, linear of order 1, Bernoulli, linear of order 2. Lagrange's method. Constant coefficients linear equations. Harmonic oscillators. Few hints about PDEs.

Suggested prerequisites
Istituzioni di Matematica

Reference material
- M. Bramanti, C. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli ed.;
- G. Turrell, Mathematics for Chemistry and Physics, AcademicPress, 2002.
- extra notes written by the teacher.

Assessment method
The final exam consists of two parts: a written part is a problem-solving session, containing four/five exercises; if the result is positive, this first part is then followed by a colloquium on the more abstract concepts and results.

Language of instruction: Italian

Attendance Policy: strongly recommended

Mode of teaching: frontal

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