


Politecnico di Torino  
Academic Year 2007/08  
01AJJCG, 01AJJCH, 01AJJCW Advanced calculus I 

1st degree and Bachelorlevel of the Bologna process in Electronic Engineering  Vercelli Master of sciencelevel of the Bologna process in Electronic Engineering  Vercelli 1st degree and Bachelorlevel of the Bologna process in Computer Engineering  Vercelli 





Objectives of the course
The goal of this course is to learn multivariate calculus which is fundamental for engineering applications.

Expected skills
The student will be able to study multivariate functions and to compute integrals both on curves or surfaces and on multivariate domains.

Prerequisites
The students are expected to know results and techniques from calculus in one variable, linear algebra (matrix, vector spaces, eigenvalues and quadratic form), trigonometry and plane geometry.

Syllabus
 Space curves, tangent line and length of a curve. Integration along a curve.
 Topology of real ndimensional spaces; continuity of multivariate functions, gradients and directional derivatives. Maxima and Minima, constrained Extrema and Lagrange multipliers.  Parametric surfaces, rotational surfaces, graphs of two variable functions, plane and spatial regular transformations.  Vector fields, rotational fields. Integrals on curves, exact differentials.  Multiple integration: double and triple integrals. Integrals in polar, cylindrical and spherical coordinates. GaussGreen formula and Stoke's Theorem. 
Laboratories and/or exercises
Exercise sections are done during the course. Exercises with complete solutions can be find in my personal web page.

Bibliography
R.A. Adams, Calculus: a Complete Corse, Pearson Educational Canada Inc.

Revisions / Exam
The examination consists in a written proof based on resolution of exercises and in an oral discussion about the main theorems proved during the course.
No calculator, no texts or notes can be used during the examination. 
