Mathematical Methods Applied to Chemistry

1] Differential calculus for functions of several real variables. Curves p: R¹→Rᵐ: orientation, arc-length , I-type line integrals and applications.
Functions f:Rⁿ→R: continuity, differenziability, second derivatives, Taylor's formula at II order.
Maps F:Rⁿ→Rᵐ: jacobian matrix, composition. Applications in R^3: polar and spherical coordinates. Surfaces. Free optimization: stationary points, hessian matrix, quadratic forms, the eigenvalues test. Constrained optimization: implicit functions, Dini's theorem. The Lagrange multipliers method.
2] Integral Calculus in 2 and 3 variables The basic properties of the Riemann integral for f:R²→R; iterated integrals for regular sets, measurable functions, sets of measure zero. Change of variables. Improper double integrals. The Riemann integral for f:Rᶟ→R: integration methods ("lines" or "layers"). I type surface integrals.
3] 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. Surface integrals of II type. The 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.
4] Differential equations: separable, linear of order 1, Bernoulli, linear of order 2. Lagrange's method. Constant coefficients linear equations. Harmonic oscillators. Few hints about PDEs.

5] Depending on the amount of time left, an extra argument might be included in the programme. The content of this extra argument may vary from year to year, and it depends on the interest of the students.