Mathematical Analysis II

Code:

EM0010

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2007/2008 - 2S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LEM	0	Plano de estudos de transição para 2006/07	1	7	7	70	187
LGEI	0	Plano de estudos de transição para 2006/07	1	7	7	70	187
MIEIG	66	Plano de estudos de transiçao para 2006/07	1	-	7	70	187
		Syllabus since 2006/2007	1	-	7	70	187
MIEM	237	Syllabus since 2006/2007	1	-	7	70	187
		Plano de estudos de transição para 2006/07	1	-	7	70	187

Teaching language

Portuguese

Objectives

1- BACKGROUND
Almost every aspect of professional work in the world involves mathematics. A solid knowledge of mathematical analysis is required for any engineering degree namely to compute physical quantities.
2- SPECIFIC AIMS
Development of the reasoning capacity of the students and knowledge of essential mathematical concepts. Students should get solid theoretical and practical formation on the main concepts and results of differential and integral calculus of several variables, including the basic theorems of calculus. Computation of physical quantities.
3- PREVIOUS KNOWLEDGE
Functions and graphs. Knowledge of differential and integral calculus of one real variable.
4- PERCENT DISTRIBUTION
Scientific component:75%
Technological component:25%
5- LEARNING OUTCOMES
Knowledge and Understanding- Parametric curves in Rn and normal and tangent vectors; to calculate line integrals on that curves. Partial and directional derivatives for real-valued and vector-valued functions; gradient vector and jacobian matrix. The chain rule for real-valued and vector-valued functions including implicit functions. Maxima and minima of unconstrained functions of two and three variables and for constrained functions as well as using Lagrange multipliers method applied to functions with one or two constraints. Evaluation of double integrals over general regions in R2 using rectangular and polar coordinates. Establishment of the relationship between the line integral and the double integral based on the Green’s theorem. Calculation of triple integrals over regions on R3 using rectangular, cylindrical and spherical coordinates.
Engineering analysis- Differential and integral calculus of several variables.
Engineering design- Engineering problems of multiple variables.
Investigations- Practical formation on the main concepts and results of differential and integral calculus in Rn.
Engineering practice- Computation of physical quantities.
Transferable skills- Knowledge of differential and integral calculus of several real variable functions.

Program

I –Vector-valued functions depending on a real variable; parametric equations of a line in Rn. Limits, continuity, differentiation and integration of vector-valued functions and applications. Arc length evaluation.
II - Introduction to surfaces in R3: quadric, cylindrical and revolution surfaces. General notions for real-valued functions of n variables: domain and graph. Vector-valued functions of n variables; parametrical representation for curves in Rn and surfaces in R3. Introductory topological notions on Rn. Limits and continuity for scalar and vector-valued functions of n variables.
III - Differentiation: partial and directional derivatives; gradient vector; partial derivatives of higher order; total derivative or Fréchet derivative and differentiability of a scalar function of n variables. Applications of the gradient: tangent plane and maximum of a directional derivative. Differentiability of vector-valued functions of n variables–Jacobian matrix. Properties of the derivative; different cases of the chain rule. Functions defined implicitly; implicit function theorem and implicit differentiation. Taylor’s formula for scalar functions of n variables. Extrema of scalar functions of n variables; constrained extrema and Lagrange multipliers.
IV - Double integrals: over a rectangle and over more general regions in R2. Properties and geometric interpretation of double integrals. Fubini theorem – changing the order of integration. Applications of double integrals to areas and volumes, average values, center of mass and moment of inertia. Changing variables in double integrals; double integrals in polar coordinates. Triple integrals: over rectangular parallelepiped and more general regions in R3 . Properties and geometric interpretation of triple integrals. Fubini theorem – changing the order of integration for triple integrals. Applications to volumes, average values, center of mass and moment of inertia. Changing variables: triple integrals in cylindrical and spherical coordinates.
V – Line integral: definition, properties and applications. Green’s Theorem.

Mandatory literature

Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana , 2006. ISBN: 85-86804-56-8

Complementary Bibliography

Marsden, Jerrold E.; Vector Calculus, N. ISBN: 0-7167-1856-1
Apostol, Tom M.; Calculus, N. ISBN: 84-291-5001-3

Teaching methods and learning activities

In the theoretical classes the exposition of the subjects is supported by slides that are previously givem to the students. The active participation is improved with the resolution of some problems and and with the answer of questions individually posed.
In the classes for problems resolution the individual as well as the team work is improved.

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Subject Classes	Participação presencial	70,00
Examination time	Exame	8,00		2008-07-31
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Study time	Estudo autónomo	84	2008-07-31
Study time for examination	Estudo autónomo	27	2008-07-31
	Total:	111,00

Eligibility for exams

See specific school rules for frequency attainment.

Calculation formula of final grade

The final classification is obtained using a weighted mean of the distributed evaluation and final examination, where the distributed evaluation weights 25% and the final examination 75%.

Examinations or Special Assignments

none

Special assessment (TE, DA, ...)

See specific school rules of the school.

Classification improvement

See specific school rules of the school.