- Docente: Hans Joachim Rudiger Achilles
- Credits: 13
- SSD: MAT/03
- Language: Italian
- Moduli: Hans Joachim Rudiger Achilles (Modulo 1) Hans Joachim Rudiger Achilles (Modulo 2)
- Teaching Mode: In-person learning (entirely or partially) (Modulo 1); In-person learning (entirely or partially) (Modulo 2)
- Campus: Rimini
- Corso: First cycle degree programme (L) in Chemistry and Technologies for the Environment and Materials (cod. 8514)
Learning outcomes
On successful completion of the course, students will have
acquired the basic knowledge of one-variable calculus, vector
calculus and linear algebra, the first elements of multivariable
calculus, complex numbers and the most elementary methods for
solving ordinary differential equations. In particular, students
will be able to represent data or functions in graphical form, to
apply one-variable and multivariable calculus and to perform
operations on vectors and matrices.
In the computer lab, students learn the basics of computers
and to operate them. They acquire the knowledge of some basic
concepts of scientific computing, such as error analysis,
interpolation, numerical integration, nonlinear equations, systems
of linear equations and will be able to solve simple problems of
scientific computing.
Course contents
Numbers: decimal and binary representation of real numbers,
standard IEEE 754, convergent sequences, series, floating-point
numbers and arithmetic operations with them, propagation of errors,
complex numbers, complex numbers in polar form, de Moivre's
formula, Euler's identity. Vectors and matrices: scalars and
vectors, vector algebra, representation of a vector in a Cartesian
coordinate system, vector spaces, dot product, cross product in
R^3, bases and dimension, change of basis, linear maps and
matrices, determinants, systems of linear equations and
numerical methods for solving them (LU factorization, Cholesky
factorization, Jacobi method, Gauss-Seidel method), eigenvalues and
eigenvectors, numerical methods to compute them.
Functions: relations and functions, real functions and their
graphs (algebraic, exponential, logarithmic and trigonometric
functions). Limits of functions, continuous functions, bisection
method for computing the zeros of a real function, real-valued
functions of two or more real variables.
Differential calculus: derivative and differential,
numerical differentiation, geometric meaning of the
derivative
(slope of the tangent line to the graph of the function) and its
interpretation as instantaneous velocity, rules of differentiation,
Taylor's theorem, Newton's method for computing the zeros of a
differentiable function, differential calculus for functions of
several variables (partial and directional derivatives, gradient,
classification of critical points).
Integral calculus: indefinite integral, definite (Riemann)
integral, fundamental theorem of calculus, numerical integration
(midpoint formula, trapezoidal formula, Simpson formula), rules of
integration (integration by parts and by substitution),
differential equations solvable by separation of variables,
constant coefficient linear differential equations, integral
calculus for functions of several variables (multiple integrals,
line integrals, differential forms).
Approximation of functions and data: Lagrangian polynomial
interpolation, Chebyshev interpolation, trigonometric interpolation
and fast Fourier transform, approximation by spline functions, the
least-squares method.
In the computer lab, the Octave programming environment is used to
execute the introduced algorithms and to analyze their stability,
accuracy and complexity.
Readings/Bibliography
E. Steiner, The Chemistry Maths Book. Second Edition, Oxford University Press, Oxford, 2008.
A. Quarteroni, F. Saleri, Scientific Computing with MATLAB and Octave. Second Edition, Springer Berlin Heidelberg New York, 2006.
M.R. Spiegel: Theory and Problems of Advanced Calculus, Schaum's Outline Series, McGraw-Hill, 1974.
Teaching methods
Lessons accompanied by exercise classes and laboratory activities in MATLAB/Octave programming. Support by a tutor is offered for free.
Assessment methods
Written examination concerning problem solving (see http://www.dm.unibo.it/~achilles for the problems of former exams) followed by an oral examination which includes the discussion of a programming project.
Teaching tools
Blackboard (and sometimes video projector). Exercises for homework and course material are available at http://www.dm.unibo.it/~achilles/.
Links to further information
http://www.dm.unibo.it/~achilles
Office hours
See the website of Hans Joachim Rudiger Achilles