Course: 2024/2025

Calculus II

(13406)

Requirements (Subjects that are assumed to be known)

Calculus I
Linear Algebra

The student will be able to formulate, solve and understand mathematically the 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 vectorial) several variables functions and its continuity, differentiability and integrability properties. The student must solve optimization problems with and without restrictions and will apply the main integration theorems to compute areas and volumes, inertial moments and heat flow.

Skills and learning outcomes

Description of contents: programme

1. Differential calculus on several variables:
1.1 Functions of several variables. Limits and continuity.
1.2 Derivatives. Differenciability.
1.3 Vectorial functions and differential operators.
1.4 Chain rule and directional derivatives.
2. Local study of functions of several variables.
2.1 Derivatives of higher order.
2.2 Extrems of functions of several variables.
2.3 Conditioned extrems.
3. Integration on Rn:
3.1 Multiple integral.
3.2 Changes of variable on multiple integrals.
3.3 Applications.
4. Line and surface integrals:
4.1 Line integrals and conservative fields.
4.2 Surface integrals.
4.3 Green, Stokes and Gauss theorems.

Learning activities and methodology

The docent methodology will include:
- 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.
- The use of Artificial Intelligence tools is not allowed in this course.

Assessment System

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

Calendar of Continuous assessment

Extraordinary call: regulations

Basic Bibliography

- MARSDEN, TROMBA. CALCULO VECTORIAL. ADDISON WESLEY.
- SALAS, HILLE, ETGEN. CALCULUS, VOLUMEN II. REVERTE.
- SPIEGEL. MATEMATICAS AVANZADAS PARA INGENIERIA Y CIENCIAS. MC GRAW HILL (SERIE SCHAUM).
- UÑA, SAN MARTIN, TOMEO. PROBLEMAS RESUELTOS DE CALCULO EN VARIAS VARIABLES. THOMSON.

Additional Bibliography

- APOSTOL. CALCULUS. REVERTE.
- BRADLEY, SMITH. CALCULO DE VARIAS VARIABLES (VOLUMEN 2). PRENTICE HALL.
- BURGOS. CALCULO INFINITESIMAL DE VARIAS VARIABLES. MC GRAW HILL.
- LARSON, HOSTETLER, HEYD. CALCULO II. PIRAMIDE.
- LIASHKO, BOIARCHUK, GAI, GOLOVACH. ANTI-DEMIDOVICH (VOLUMENES 3 Y 4). URSS.
- STEWART,. CALCULO: CONCEPTOS Y CONTEXTOS. THOMSON.
- WREDE, SPIEGEL. CALCULO AVANZADO. MC GRAW HILL (SEIRE SCHAUM).

The course syllabus may change due academic events or other reasons.