CALCULUS II

The teaching ANALISI MATEMATICA MATEMATICA 2 is one of the basic courses of the degree program in Ingegneria Fisica, and allow the student to face a mathematical problem with a consistent use of the current mathematical language.
The specific goal of this teaching is the formation of the knowledge and the skills concerning the theoretical and applicative basis of the differential and integral calculus in higher dimensions. The teaching will form the basis to deal with the mathematical models developed in the other courses of the Degree.

1. Knowledge and understanding
i) To know the basic concepts of advanced Mathematical Analysis.
ii) To know how to use differential calculus in multiple variables, to understand the concepts of limits, derivatives and integrals in multiple variables.
2. Ability to apply knowledge and understanding.
i) To know how to reason in a logical way and how to use mathematical symbolism in an appropriate way.
ii) To understand the mathematical analysis in multiple variables and to know how to set up a strategy to solve problems.
iii) To know how to recognize the role of mathematics in other sciences.
3. Ability to judge
i) Being able to evaluate the logical consistency of the results obtained, both in the theoretical field than in the case of concrete mathematical problems.
ii) Being able to recognize errors through a critical analysis of the method applied and through a control of the results obtained.
iii) To evalute the possibility of different approaches when solving mathamtical problems.
4. Communication skills
i) To know how to communicate what have been learned by using an appropriate terminology, also in written form.
ii) To know how to interact with the teacher and with the classmates in a respectful and constructive way, by asking coherent questions and by proposing other ways to solve a problem.
5. Learning skills
i) To know how to take notes in an effective way, selecting and collecting information according to their importance and priority.
ii) To know how to consult the books given by the teacher, and to know how to identify alternative references, also through the interaction with the teacher.
iii) Being able to exploit the concepts learned to correctly perform a mathematical problem.

To have achieved the goals of ANALISI MATEMATICA 1 and ALGEBRA LINEARE. In particular, it is appropriate for the students to know the concepts and the methods of the differential and integral calculus, and the basis of the linear algebra.

Differential calculus in higher dimension:
Real functions with several variables. Domains. Limits
and continuity. Derivative of a function in several variables, differentiability. Tangent plane. Total differential theorem (sufficient condition for differentiability). Schwarz's theorem.

Critical points for functions in several variables:
Quadratic forms; definite, semi-defined and indefinite matrices and their
characterization. Eigenvalue test. Free relative extremes and saddle points. Hessian matrix. Weierstrass theorem. Maxima and minima on bounded domains. Method of the Lagrange multipliers.

Double and triple integrals:
Definition of double integral and its properties. Normal domains. Reduction formulas. Change of variables. Polar and elliptic coordinates. Gauss-Green formulas. Area of a domain.
Triple integrals and properties. Integration by "fili" and by "strati". center of gravity of solids. Change of variables in triple integrals. Cylindrical and spherical coordinates. Volumes of solids.


Curves and vector fields:
Curves in the plane and in the space. Simple, regular and closed curves. Tangent vector. Curvilinear abscissa. Length of a curve. Curvilinear integrals of continuous functions. Orientation of the curve. Vector fields. Curvilinear integrals of vector fields: work of a field along a curve. Conservative vector fields and their properties: potential of a vector field and work of a conservative vector field. Connected and simply connected domains. Irrotational fields. Necessary and sufficient conditions to understand if a vector field is conservative.

Surfaces:
Cartesian and parametric surfaces and their relationships. Normal vector and tangent plane. Coordinated tangent vectors, normal vector in parametric form. Area of ​​a surface. Surface integrals. Orientated surfaces, flow of a vector field through a regular surface and its physical meaning. Flow through closed surfaces (divergence theorem). Orientation of the edge of a surface. Stokes (or rotor) theorem.

Differential Equations:
First order differential equations: separation of variables and resolution of non-homogeneous linear equations. Existence and uniqueness Cauchy theorem, maximal interval of existence. Bernoulli equations. Second order linear equations with homogeneous and non-homogeneous constant coefficients. The example of the harmonic oscillator.

M. Bramanti, C. Pagani, S. Salsa: Analisi matematica 2, Zanichelli
M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica 2Ed, McGraw-Hill

M. Strani, Esercizi svolti di Analisi Matematica 2, Esculapio
M. Bramanti, C. Pagani, S. Salsa: Esercizi di analisi matematica 2, Zanichelli
L. Moschini, R. Schianchi: Esericizi svolti di Analisi Matematica
P. Marcellini, C. Sbordone: Esercizi di matematica, Vol. 2 (Tomi 1-4), Liguori

The exam consists of two written tests, one with exercises concerning all the topics studied during the classes and another one with questions about the theory and applications. In the exam, the correctness of the exposure, the clarity and completeness of the justifications, the knowledge of the scientific language and the ability to use the tools of mathematical analysis will be evaluated.
The written test (5 exercises) will last three hours. The final grade of the written test will be defined by the sum of the grades of each exercise, which are explicitly marked next to each exercise, with a grade range for the single exercise ranging from 5 to 8. The maximum score achievable in the written test is 32.
The theoretical test, that will last 90 minutes, will consist of 3 theoretical questions (proofs, definitions, statements, applications) to be answered in a written form; the score of each question is 10, the answers will be eventually discussed with the teacher later after the correction of the test.
Those who have obtained at least 16/30 in the written test are admitted to the oral exam. The final evaluation will take into account the marks of the two tests (exercises and theory).

Lectures: theory and exercises.
Educational material will be found in the "moodle" platform.

Italian

Accommodation and support services for students with disabilities and students with specific learning impairments:
Ca’ Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding supportservices and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). In the case of disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.

written and oral