The Calculus II is part of the subjects that make up the Mathematics module for the degree of Engineering. These subjects are basic for the scientific and technical education of the student when promoting the development of their capacities of abstraction and scientific rigor, as well as those of analysis and synthesis.

The differential calculation of several variables allows the analysis of the optimization of functions and acquire quantitative techniques essential for the allocation of resources, decision-making and management in various problems that the future engineer may pose throughout his professional life. With the contribution of the integral calculation, it will help not only the resolution of multiple problems in the world of science and engineering, but also a better understanding of the knowledge and instrumental and analytical techniques that can be used in them.

The subject, as a whole, will allow to understand more deeply other subjects studied previously (Calculus I, Algebra, Physics, ...) and will facilitate the study of new ones, both basic and specific.

Course competences
Code	Description
A01	To understand and have knowledge in an area of study that moves on from the general education attained at secondary level and usually found at a level that, while supported in advanced text books, also includes some aspects that include knowledge found at the cutting edge of the field of study.
A02	To know how to apply knowledge to work or vocation in a professional manner and possess the competences that are usually demonstrated by the formulation and defence of arguments and the resolution of problems in the field of study.
A03	To have the capability to gather and interpret relevant data (normally within the area of study) to make judgements that include a reflection on themes of a social, scientific or ethical nature.
A07	Knowledge of Information Technology and Communication (ITC).
A08	Appropriate level of oral and written communication.
A12	Knowledge of basic materials and technologies that assist the learning of new methods and theories and enable versatility to adapt to new situations.
A13	Ability to take the initiative to solve problems, take decisions, creativity, critical reasoning and ability to communicate and transmit knowledge, skills and abilities in Mechanical Engineering.
A17	Ability to apply principles and methods of quality control.
B01	Ability to solve mathematical problems that occur in engineering. Aptitude to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential and partial differential equations; numerical methods; numerical algorithms; statistics and optimization.
CB01	Prove that they have acquired and understood knowledge in a subject area that derives from general secondary education and is appropriate to a level based on advanced course books, and includes updated and cutting-edge aspects of their field of knowledge.
CB02	Apply their knowledge to their job or vocation in a professional manner and show that they have the competences to construct and justify arguments and solve problems within their subject area.
CB03	Be able to gather and process relevant information (usually within their subject area) to give opinions, including reflections on relevant social, scientific or ethical issues.
CB04	Transmit information, ideas, problems and solutions for both specialist and non-specialist audiences.
CB05	Have developed the necessary learning abilities to carry on studying autonomously

Course learning outcomes
Description
Know the use of the functions of one and various variables including its derivation, integration and graphic representation
Know the main approaches for resolution through using numerical methods, to use some statistical software packages at user level, data processing, mathematical calculus and vizualization, set out algorithms and program through programming language of a high level, vizualize functions, geometric figures and data, design experiments, analyze data and interpret results
Be familiar with the concepts of differential geometry and use them appropriately.
To know the tundamentals and applications of Optimization
Additional outcomes
Description

• Unit 1: Differential Geometry
• Unit 2: Multivariate functions: Limits and Continuity
• Unit 3: Multivariate functions: Differential Calculus
• Unit 4: Optimization of scalar functions
• Unit 5: Multivariate functions: Multiple Integrals
• Unit 6: Vector Analysis
• Unit 7: Introduction to partial differential equations

Training Activity	Methodology	Related Competences (only degrees before RD 822/2021)	ECTS	Hours	As	Com	R	Description *
Class Attendance (theory) [ON-SITE]	Combination of methods	A01 A02 A03 A07 A12 B01 CB01 CB02 CB03 CB04 CB05	1	25	N	N	N
Individual tutoring sessions [ON-SITE]	Problem solving and exercises	A01 A02 A03 A08 A13 A17 B01 CB01 CB02 CB03 CB04 CB05	0.2	5	N	N	N
Problem solving and/or case studies [ON-SITE]	Problem solving and exercises	A02 A07 A13 B01 CB01 CB02 CB03 CB04 CB05	0.6	15	Y	N	N
Workshops or seminars [ON-SITE]	Workshops and Seminars	A02 A08 A12 A13 A17 B01 CB01 CB02 CB03 CB04 CB05	0.1	2.5	N	N	N
Computer room practice [ON-SITE]	Practical or hands-on activities	A02 A07 B01 CB01 CB02 CB03 CB04 CB05	0.3	7.5	Y	N	N
Final test [ON-SITE]	Assessment tests	A01 A02 A03 A07 A08 A12 A13 A17 B01 CB01 CB02 CB03 CB04 CB05	0.2	5	Y	Y	Y
Other off-site activity [OFF-SITE]	Self-study	A02 A03 A08 B01 CB01 CB02 CB03 CB04 CB05	3.6	90	N	N	N
Total:			6	150
Total credits of in-class work: 2.4			Total class time hours: 60
Total credits of out of class work: 3.6			Total hours of out of class work: 90

As: Assessable training activity
Com: Training activity of compulsory overcoming
R: Rescheduling training activity

	Grading System
Evaluation System	Face-to-Face	Self-Study Student	Description
Assessment of activities done in the computer labs	5.00%	5.00%	For the evaluation of the practices in the computer room, with application of specific software, the delivery of the work carried out in the same ones and a documentation with the resolution of the same will be valued.
Final test	70.00%	70.00%	Finally, there will be a written test that will consist of questions, theoretical questions and problems whose evaluation criteria will be similar to those of the academic works described above.
Assessment of problem solving and/or case studies	5.00%	5.00%	For the evaluation of the academic works carried out by the students in class, a memory should be given where the approach of the problem will be assessed, the use of appropriate terminology and notation to express the mathematical ideas and relationships used, the choice of the most appropriate procedure for each situation, the justification of the different steps of the procedure used, the results obtained and the cleaning and presentation of the document.
Progress Tests	20.00%	20.00%	Partial exams to prove the way the students are learning the subject.
Total:	100.00%	100.00%

Not related to the syllabus/contents
Hours	hours
Individual tutoring sessions [PRESENCIAL][Problem solving and exercises]	5
Workshops or seminars [PRESENCIAL][Workshops and Seminars]	2.5
Computer room practice [PRESENCIAL][Practical or hands-on activities]	7.5
Final test [PRESENCIAL][Assessment tests]	5

Unit 1 (de 7): Differential Geometry
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	1.5
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	1
Other off-site activity [AUTÓNOMA][Self-study]	4

Unit 2 (de 7): Multivariate functions: Limits and Continuity
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	3.5
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	2
Other off-site activity [AUTÓNOMA][Self-study]	12

Unit 3 (de 7): Multivariate functions: Differential Calculus
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	6.5
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	4
Other off-site activity [AUTÓNOMA][Self-study]	24

Unit 4 (de 7): Optimization of scalar functions
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	3
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	1.5
Other off-site activity [AUTÓNOMA][Self-study]	12

Unit 5 (de 7): Multivariate functions: Multiple Integrals
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	6
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	3.5
Other off-site activity [AUTÓNOMA][Self-study]	24

Unit 6 (de 7): Vector Analysis
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	3
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	2
Other off-site activity [AUTÓNOMA][Self-study]	10

Unit 7 (de 7): Introduction to partial differential equations
Activities	Hours
Class Attendance (theory) [PRESENCIAL][Combination of methods]	1.5
Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises]	1
Other off-site activity [AUTÓNOMA][Self-study]	4

Global activity
Activities	hours

Author(s)	Title	Book/Journal	Citv	Publishing house	ISBN	Year	Description	Link	Catálogo biblioteca
A. Garcia, A. López, G. Rodríguez, S. Romero, A. de la Villa	Calculo II. Teoría y problemas de funciones de varias variables		Madrid	CLAGSA	84-921847-0-1	1996
APOSTOL, T.	Calculus			Ed. Reverté		1995
ARANDA, E; PEDREGAL, P.	Problemas de cálculo vectorial			Lulu.com		2004
BURGOS, J.	Cálculo infinitesimal de varias variables.			McGraw-Hill
DEMIDOVICH, B.	5000 problemas de análisis matemático.			Ed. Paraninfo.
GARCIA, A.; LOPEZ, A.; RODRIGUEZ, G; ROMERO, S; DE LA VILLA, A.	Cálculo II.			Ed. Clagsa		2002
GRANERO	Cálculo infinitesimal			McGraw-Hill.
LARSON , R; HOSTETLER, R; EDWARDS, B;	Cálculo y geometría analítica			Ed. McGraw Hill
LOPEZ DE LA RICA, A ; DE LA VILLA, A.	Geometría diferencial.			CLAGSA.
PERAL ALONSO, I.	Primer curso de ecuaciones en derivadas parciales			Ed. Addison-Wesley/Universidad autónoma de Madrid
SALAS, S; HILLE, E.	Calculus			Ed. Reverté.
STEWART, J.	Cálculo multivariable			THOMSON
ZILL, D.	Ecuaciones diferenciales.			THOMSON