Course: 2019/2020

Calculus II

(13970)

Students are expected to have completed

Calculus I

Competences and skills that will be acquired and learning results. Further information on this link

The student will be able to formulate, solve and understand mathematical problems arising in engineering. To do so it is necessary, in this second course of Calculus, to be familiar with the n-dimensional euclidean space, in particular in dimension 3, and with its more usual subsets. He/she must be able to manage (scalar and vector) functions of several variables, their continuity, differentiability and integrability properties. The student must be able to solve optimization problems with and without restrictions and will apply the main integration theorems to compute areas and volumes, inertia moments and heat flow. He/she must know the concepts of ordinary differential equation and differential equations problem. The student will be able to solve the main first and second order differential equations.

Description of contents: programme

The Euclidean space. Functions of several variables. Continuity and differentiability. Polar, spherical and cylindrical coordinates. The chain rule. Directional derivatives. Gradient, divergence and curl. Free and conditional optimization. Multidimensional Iterated integration. Changes of variables. Integration along trajectories. Integration on surfaces. Computation of areas, volumes, centers of mass , moments of inertia and. other applications of the integral. Theorems of Green, Stokes and Gauss. Introduction to differential equations. Laplace transform.

Learning activities and methodology

The academic methodology includes:
- Master classes, where the knowledge that the students must acquire will be presented. To make easier the development of the class, the students will have written notes and also will have the basic texts of reference that will facilitate their subsequent work.
- Resolution of exercises by the student that will serve as self-evaluation and to acquire the necessary skills.
- Problem classes, in which problems proposed to the students are discussed and developed.
- Partial controls.
- Final control.
- Tutorials.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- MARSDEN, TROMBA. VECTOR CALCULUS. W. H. FREEMAN. 2003
- NAGLE. FUNDAMENTALS OF DIFERENTIAL EQUATIONS. PEARSON-ADDISON WESLEY. 2008
- SALAS, S.. CALCULUS: ONE AND SEVERAL VARIABLES. WILEY. 2007
- UÑA, SAN MARTIN, TOMEO. PROBLEMAS RESUELTOS DE CALCULO EN VARIAS VARIABLES. THOMSON.
- ZILL D.. A FIRST COURSE IN DiFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS. BROOKS/COLE. 2013

Additional Bibliography

- APOSTOL. . CALCULUS. John Wiley & Sons. .