- Docente: Enrico Obrecht
- Credits: 6
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Engineering Management (cod. 0925)
Course contents
LIMITS AND CONTINUITY FOR REAL AND VECTOR FUNCTIONS OF SEVERAL
REAL VARIABLES. Definition of a neighbourhood of a point in
R^n, of bounded, open and closed sets. Definition of limit
and continuity in a point for real and vector functions of several
real variables. The Weierstrass theorem and the Bolzano
theorem.
DIFFERENTIAL CALCULUS FOR REAL AND VECTOR FUNCTIONS OF SEVERAL
REAL VARIABLES. Definition of partial and directional derivatives,
jacobian matrix, gradient, differential. The equation of the
tangent plane of a graph. The theorem on the differentiability of
functions of class C^1. The chain rule. Partial derivatives of
higher order. The Schwartz theorem on mixed derivatives. The
Hessian matrix. The Taylor formula of the second order. Local
extrema for real functions of several variables: definitions,
Fermat theorem, sufficient conditions depending on the Hessian
matrix. Constrained extrema: the Lagrange multiplier theorem.
Convex functions in R^n: definitions, characterization of
convexity for C^2 functions.
COMPLEX NUMBERS. Definition of complex numbers and of
operations on them. Algebraic, trigonometric and exponential form
of a complex number.
De Moivre formula for n-th roots of a complex number.
DIFFERENTIAL EQUATIONS. The Cauchy problem for ordinary
differential equations. Theorems on existence and uniqueness of
solutions. Differential equations with separable variables. Linear
first order differential equations. Linear second order
differential equations: general integral for homogeneous and non
homogeneous equations; explicit solutions for the constant
coefficient case.
DOUBLE INTEGRALS. Definition of a double integral. The
reduction theorem. The change of variables theorem.
IMPROPER INTEGRALS IN R AND R^2. Definition of
an improper integral. Absolute integrability. The comparison
criterion.
NUMERICAL SERIES. Definition of a series, convergence and
absolute convergence. Convergence criteria for series with
nonnegative terms. Leibniz criterion for alternating series.
Integral criterion.
Teaching methods
Lessons and exercises in classroom.
Assessment methods
Written and oral examination. The written examination may be replaced by two partial written examinations.
Links to further information
http://www.dm.unibo.it/~obrecht/
Office hours
See the website of Enrico Obrecht