MATHEMATICS 2

The objectives of the course are the following:

Knowledge and understanding: the student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures among which integral calculus for real functions of one or more variables, differential equations, multivariable calculus.

Applying knowledge and understanding: by means of examples related to applied sciences, the student will learn the central role of Mathematical Analysis within science and not only as an abstract topic. This will expand his cultural horizon.

Making judgements: the student will tackle with rigour some simple meaningful methods of Mathematical Analysis. This will sharpen his logical ability. Many proofs will be exposed in an intuitive and schematic way, to make them more usable also to students that are not committed to Mathematics.

Communication skills: By studying Mathematics and doing guided exercitations , the student will learn to communicate with clarity and rigour both, verbally and in writing. The student will learn that the use of a properly structured language is the key point to clear and effective scientific, and non-scientific, communication.

Learning skills: the students, in particular the more willing, will be stimulated to examine in depth some topics, alone or working in team.

Traditional (teacher up front) lessons.

1. Integrals. Areas and distances. The definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts. Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial fractions. Strategy for integration. Impropers integrals. Applications of integration.
2. Differential equations. Definitions and terminology. Initial-Value problems. Separable equations. Linear equations. Elements of theory of linear equations: initial-value and boundary-value problems, homogeneous equations, nonhomogeneous equations. Homogeneous linear equations with constant coefficients. Undetermined coefficients. Variation of parameters.
3. Partial derivatives. Functions of several variables. Limits and continuity. Partial derivatives. Tangent planes and linear approximations. The Chain Rule. Directional derivatives and the gradient vector. Maximum and minimum values. Lagrange multipliers.
4. Multiple integrals. Double integrals over rectangles. Double integrals over general regions. Double integrals in polar coordinates. Applications of double integrals. Surface area. Triple integrals. Triple integrals in cylindrical coordinates. Triple integrals in spherical coordinates. Change of variables in multiple integrals.

1. C. Canuto, A. Tabacco – Mathematical Analysis I – Springer (2015).
2. C. Canuto, A. Tabacco – Mathematical Analysis II – Springer (2015).
3. J. Stewart – Calculus. Early Transcendentals – Eight Edition, Cengage Learning (2016).