The course is the natural prosecution of the Calculus I course, and aims at giving the students a comprehensive expertise of analytical tools, to be used in the more technical courses to come. The students will learn how to handle scalar and vector-valued functions depending on several variables, compute partial derivatives, evaluate multiple integrals and integrals along lines and on surfaces. Besides the most significant theorems on the topic, stated with mathematical rigor, a large number of examples and exercises will be provided in order to teach methods and ideas.
Course Prerequisites
This course is part of the basic mathematical training of Engineering students. In order to fruitfully follow this course, the students should have followed the basic courses: Calculus I, Geometry and Linear Algebra. In particular, students should master one-variable derivatives and integrals, number series, vector and matrix calculus.
Teaching Methods
Lectures (hours/year in lecture theatre): 45 Practical class (hours/year in lecture theatre): 38 Practicals / Workshops (hours/year in lecture theatre): 0
Assessment Methods
The examination is composed by a mandatory written test and an optional oral test. In the written test the students are requested to solve exercises and to answer theory questions. The oral examination, which has to be carried out in the same session of the written one, will test theorems' statements and proofs, definitions, and fundamental examples and counterexamples.
Texts
M. Bramanti, C.D. Pagani, S. Salsa. Analisi Matematica 2. Zanichelli, Bologna, 2009. S. Salsa e A. Squellati. Esercizi di Analisi Matematica 2. Zanichelli, Bologna, 2011.
Contents
Multivariate Calculus: Basic notion of topology and metrics in n-dimensional spaces. Continuous functions. Partial and directional derivatives; differentiability. Higher order derivatives. Optimization and main results. Vector-valued functions. Curves: Definition of regular curve: main properties. Rectifiable curves and how to compute their length. Arc-length function. Arc integrals for real valued functions. Multiple integrals: Definition of a double integral in a rectangle. Extension to a Peano-Jordan measurable set. Formulas to compute a double integral. Change of variables. Geometric applications. Green and divergence theorems for two-variable functions. Triple integrals: extension of the methods considered for double integrals. Surfaces: Regular surfaces: main properties. Area of a regular surface. Surface integrals and how to compute them. Divergence and Stokes theorems for three-variable functions. Vector fields: Arc integral of a vector-valued function. Irrotational vector fields: main properties. Arc integral of an irrotational vector field: the fundamental theorem. Conditions for a vector field to be irrotational.
