Calculus II

Code:

EQ0063

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Physical Sciences (Mathematics)

Instance: 2011/2012 - 2S

Active?	Yes
Responsible unit:	Department of Chemical Engineering
Course/CS Responsible:	Master in Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEQ	99	Syllabus	1	-	6	56	162

Teaching language

Portuguese

Objectives

1. BACKGROUND
Analytic Geometry and the extension of basic concepts of Calculus to functions of several variables (partial differentiation and multiple integrals) are everyday tools in engineering calculations. Topics in Vector Analysis allows the student to understand essential theorems (Green, Divergence and Stokes) which provide the basis for understanding important principles in engineering.
2. SPECIFIC AIMS
- Get acquainted to different coordinate systems;
- Learn about the representation of curves and surfaces in the 3D space;
- Know how to calculate partial derivatives for functions of several variables (either defined by composition or implicit);
- To locate extreme values of functions (unconstrained and constrained);
- Perform double and triple integrals evaluation (in rectangular, cylindrical and spherical coordinates systems) and understand its pratical applications;
- Learn avout Vector Analysis, how to calculate Line and Surface Integrals and understand its pratical applications.
3. PREVIOUS KNOWLEDGE
EQ0059: Differential and Integral Calculus; Definite Integrals;
EQ0058: Vector Algebra.
4. PERCENT DISTRIBUTION
Scientific Component (stablishes and develops scientific basis): 80%
Technological Component (apply to design and process operation): 20%
5. LEARNING OUTCOMES
After this couse, the student is able to:
- Use polar, cylindrical and sperical coordinate systems;
- Represent curves (as well as their projection over coordinate plans) and surfaces in the 3D space;
- Calculate partial derivatives for functions of several variables (either defined by composition or implicit);
- Locate extreme values of functions (unconstrained and constrained);
- Calculate double and triple integrals (in rectangular, cylindrical and spherical coordinates) and understand its pratical applications (calculation of area, volume, centroid);
- Understand Vector Analysis, calculate line and surface integrals and understand its pratical applications (calculation of work done by a force and the flux of a vector field).

Program

1. POLAR COODINATES. PARAMETRIC CURVES ON THE PLANE
Polar coordinates on the plane; area in polar coordinate; parametric curves on the plane; functions defined by parametric equations: its derivation and integration

2. CURVES AND SURFACES IN SPACE. VECTOR FUNCTIONS
Curves in three-dimensional surfaces; Cylindrical and spherical coordinates in three-dimensional space; Analytical representation of curves in three-dimensional space; Vector functions: derivation and integration

3. INTRODUCTION TO MULTIVARIABLE FUNCTIONS
Function of two variables; Function of three variables

4. MULTIVARIABLE FUNCTIONS: PARTIAL DERIVATIVES
Partial derivatives; Differentiation of functions of two or three variables; Implicit functions and their derivative; Functions of two or more variables: minimum and maximum; Directional derivative and gradient; Maximum and/or minimum with restrictions: Lagrange multiplier method

5. MULTIVARIABLE FUNCTIONS: MULTIPLE INTEGRALS
Double integrals in rectangular domains (RxR); Double integrals in limited arbitrary domains (RxR); Application of double integrals; Double integrals in polar coordinates; Parametric surfaces and superficial area; Triple integrals in limited arbitrary domains of three-dimensional space; Application of triple integrals; Triple integrals in cylindrical and spherical coordinates; Improper multiple integrals

6. TOPICS IN VECTOR ANALYSIS
Line integrals: definition and evaluation; Line integrals in vector fields; Conservative fields; Green’s theorem; Surface integrals: definition and evaluation; Surface integrals and vector fields; Vector field flow; Gauss’ and Stokes’ theorems

Mandatory literature

J. M. Mendonça; Matemática II, 2008

Complementary Bibliography

Anton, Howard; Calculus. ISBN: 0-471-48237-4
Ron Larson, Robert P. Hostetler, Bruce H. Edwards; Cálculo. ISBN: 85-86804-56-8 (vol. 1)

Teaching methods and learning activities

General theoretical-practical classes will be based on the presentation of theoretical themes, along with examples resolution.
Theoretical-practical classes will be based on problem solving and application of the themes taught in general theoretical-practical classes.

keywords

Physical sciences > Mathematics > Mathematical analysis
Physical sciences > Mathematics > Mathematical analysis > Functions

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	56,00
Study time for examinations	Exame	42,00
Minitest 1	Exame	1,50
Minitest 2	Exame	1,50
First Examination	Exame	2,50
Second Examination	Exame	2,50
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Regular Study	Estudo autónomo	56
	Total:	56,00

Eligibility for exams

A positive term grade is attained if (which means admittance to examination):

i) The student misses no more than 25% of total classes without an acceptable justification;
ii) The student attained a positive term grade in the imediate previous school year (2010/11); in this case, students are excused to attend classes and mini-tests;
iii) The student have Term Grade, TG, equal or greater than 5 (on a 0 to 20 scale);
iv) Special cases of working students, military, etc;

The Term Grade , TG, is the arithmetic mean of the two mini-tests grades , T1 and T2: TG = 0.5*(T1+T2).

A positive term grade is mandatory to access final examinations.

Calculation formula of final grade

Final grade (FG) is calculated according to:

if TG > EG : FG= 0.35 TG + 0.65 EG;
if TG <= EG: FG= EG

where EG stands for final examination grade.

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to the rules defined in FEUP.

Classification improvement

According to the rules defined in FEUP.

Observations

The use of a graphing calculator is recommended in the Calculus module.