Subject: Calculus II
Scientific area:
Mathematics
Contact hours:
96 Hours
Number of ECTS:
7,5 ECTS
Language:
Portuguese
General Objectives:
1 - Students who successfully complete this course unit should be able to: Master the fundamental properties of real-valued functions of several variables.
2 - Recognize the concepts studied as generalizations of the corresponding notions for real-valued functions of a single variable.
3 - Understand the concepts of limit, continuity, and differentiability for scalar and vector fields, as well as their application to specific areas of engineering.
4 - Calculate multiple integrals by identifying the geometric representation of the domain and determining the appropriate coordinate system to use.
5 - Master the parameterization of curves and surfaces, and apply it to the calculation of integrals.
2 - Recognize the concepts studied as generalizations of the corresponding notions for real-valued functions of a single variable.
3 - Understand the concepts of limit, continuity, and differentiability for scalar and vector fields, as well as their application to specific areas of engineering.
4 - Calculate multiple integrals by identifying the geometric representation of the domain and determining the appropriate coordinate system to use.
5 - Master the parameterization of curves and surfaces, and apply it to the calculation of integrals.
Contents / Syllabus:
1 - Euclidean vector spaces 1.1 Vectors in a Cartesian coordinate system. 1.2 Dot product. Orthogonality. 1.3 Vector equations of lines and planes. 1.4 Distance problems. 1.5 Cross product. Areas. 1.6 Polar, cylindrical, and spherical coordinates.
2 - Vector-valued functions and parametric equations 2.1 Curves defined by parametric equations. 2.2 Calculus of vector-valued functions. 2.3 Unit vectors (tangent and normal) for parametric curves. 2.4 Arc length.
3 - Concept of real-valued functions of several variables 3.1 Definitions and graphs of functions. Domain. 3.2 Level curves and level surfaces. 3.3 Notion of limit and continuity. 3.4 Topology: open and closed sets.
4 - Partial derivatives and differentiability 4.1 Partial derivatives: geometric significance. 4.2 Higher-order partial derivatives. Schwarz's Theorem. 4.3 Sufficient condition for differentiability. 4.4 Tangent plane and normal line to graphs of functions. 4.5 Maxima and minima of functions of two variables. 4.6 Taylor's formula for functions of two variables. 4.7 Concept of differentiability: approximation by the total differential. 4.8 Concept of differentiability: approximation by the tangent plane.
5 - Gradient in vector space and its applications 5.1 Concept of differentiability: definition and calculation of the gradient vector. 5.2 Directional derivatives. 5.3 Maximum/minimum rates of change of functions. 5.4 Gradient: level curves. Concept of gradient field. 5.5 Chain rule. Parametric curves. 5.6 Implicit differentiation. 5.7 Tangent and normal lines to the level curve.
6 - Multiple integrals 6.1 Double integral. Fubini's Theorem. 6.2 Improper integrals. 6.3 Double integrals over general regions. 6.4 Double integrals in polar coordinates. 6.5 Triple integrals. 6.6 Change of variables in multiple integrals.
7 - Vector fields 7.1 Definition of a vector field. 7.2 Physical and mathematical models of vector fields. 7.3 Definition of a conservative vector field and potential function. 7.4 Relationship between vector field and gradient. 7.5 Closed and exact differential forms. 7.6 Introduction to line integrals and Green's Theorem. 7.7 Singular point: non-simply connected region.
8 - Line integrals 8.1 Line integral of a scalar function. 8.2 Line integral of a vector field. 8.3 Integration along rectangular and circular paths. 8.4 Interpretation of line integral results. 8.5 Line integrals over graphs of functions.
9 - Conservative vector fields. Path-independent integrals. 9.1 Fundamental Theorem of Line Integrals. 9.2 Determination of a potential function: function difference method. 9.3 Equivalences regarding the concept of a conservative vector field. 9.4 Main criterion for a conservative vector field.
10 - Green's Theorem 10.1 Conditions for validity. 10.2 Applications: work done by a force; plane areas. 10.3 Extension of Green's theorem. 10.4 Point singularity: specific calculation of line integrals.
11 - Surface integrals 11.1 Surface parametrizations. 11.2 Surface area. 11.3 Implicit surfaces. 11.4 Definition of the surface integral. 11.5 Methods for calculating a surface integral. 11.6 Definition of an orientable surface. Flux integral.
12 - Classical vector theorems 12.1 Three-dimensional operators: gradient, curl, and divergence. 12.2 Properties of the operators. Laplace's equation and harmonic function. 12.3 Gauss's divergence theorem in the plane. Flux integral. 12.4 Stokes' theorem in the plane. Circulation integral. 12.5 Gauss's divergence theorem in space. 12.6 Stokes's theorem in space.
2 - Vector-valued functions and parametric equations 2.1 Curves defined by parametric equations. 2.2 Calculus of vector-valued functions. 2.3 Unit vectors (tangent and normal) for parametric curves. 2.4 Arc length.
3 - Concept of real-valued functions of several variables 3.1 Definitions and graphs of functions. Domain. 3.2 Level curves and level surfaces. 3.3 Notion of limit and continuity. 3.4 Topology: open and closed sets.
4 - Partial derivatives and differentiability 4.1 Partial derivatives: geometric significance. 4.2 Higher-order partial derivatives. Schwarz's Theorem. 4.3 Sufficient condition for differentiability. 4.4 Tangent plane and normal line to graphs of functions. 4.5 Maxima and minima of functions of two variables. 4.6 Taylor's formula for functions of two variables. 4.7 Concept of differentiability: approximation by the total differential. 4.8 Concept of differentiability: approximation by the tangent plane.
5 - Gradient in vector space and its applications 5.1 Concept of differentiability: definition and calculation of the gradient vector. 5.2 Directional derivatives. 5.3 Maximum/minimum rates of change of functions. 5.4 Gradient: level curves. Concept of gradient field. 5.5 Chain rule. Parametric curves. 5.6 Implicit differentiation. 5.7 Tangent and normal lines to the level curve.
6 - Multiple integrals 6.1 Double integral. Fubini's Theorem. 6.2 Improper integrals. 6.3 Double integrals over general regions. 6.4 Double integrals in polar coordinates. 6.5 Triple integrals. 6.6 Change of variables in multiple integrals.
7 - Vector fields 7.1 Definition of a vector field. 7.2 Physical and mathematical models of vector fields. 7.3 Definition of a conservative vector field and potential function. 7.4 Relationship between vector field and gradient. 7.5 Closed and exact differential forms. 7.6 Introduction to line integrals and Green's Theorem. 7.7 Singular point: non-simply connected region.
8 - Line integrals 8.1 Line integral of a scalar function. 8.2 Line integral of a vector field. 8.3 Integration along rectangular and circular paths. 8.4 Interpretation of line integral results. 8.5 Line integrals over graphs of functions.
9 - Conservative vector fields. Path-independent integrals. 9.1 Fundamental Theorem of Line Integrals. 9.2 Determination of a potential function: function difference method. 9.3 Equivalences regarding the concept of a conservative vector field. 9.4 Main criterion for a conservative vector field.
10 - Green's Theorem 10.1 Conditions for validity. 10.2 Applications: work done by a force; plane areas. 10.3 Extension of Green's theorem. 10.4 Point singularity: specific calculation of line integrals.
11 - Surface integrals 11.1 Surface parametrizations. 11.2 Surface area. 11.3 Implicit surfaces. 11.4 Definition of the surface integral. 11.5 Methods for calculating a surface integral. 11.6 Definition of an orientable surface. Flux integral.
12 - Classical vector theorems 12.1 Three-dimensional operators: gradient, curl, and divergence. 12.2 Properties of the operators. Laplace's equation and harmonic function. 12.3 Gauss's divergence theorem in the plane. Flux integral. 12.4 Stokes' theorem in the plane. Circulation integral. 12.5 Gauss's divergence theorem in space. 12.6 Stokes's theorem in space.
Bibliography / Information Sources:
Stewart J. , 2016 , Calculus: Early Transcendentals, 8th ed , Brooks/Cole: Cengage Learning
Anton H., Bivens Irl C., Davis S., , 2012 , Calculus, 10th ed. , Wiley
Thomas G.B. Jr., Weir, M.D., Hass J., , 2014 , Thomas' calculus: Early Transcendentals, 13th ed., , Pearson
Hughes-Hallett D., Gleason A.M., McCallum W.G., , 2012 , Calculus: Single and Multivariable, 6th ed , Wiley
Larson R., Edwards B.H. , 2010 , Calculus, 9th ed , Brooks/Cole: Cengage Learning
Salas S.L, Hille E., Etgen G , 2007 , Calculus: One and Several variables, 10th ed , Hoboken, NJ, USA: John Wiley & Sons Inc
Anton H., Bivens Irl C., Davis S., , 2012 , Calculus, 10th ed. , Wiley
Thomas G.B. Jr., Weir, M.D., Hass J., , 2014 , Thomas' calculus: Early Transcendentals, 13th ed., , Pearson
Hughes-Hallett D., Gleason A.M., McCallum W.G., , 2012 , Calculus: Single and Multivariable, 6th ed , Wiley
Larson R., Edwards B.H. , 2010 , Calculus, 9th ed , Brooks/Cole: Cengage Learning
Salas S.L, Hille E., Etgen G , 2007 , Calculus: One and Several variables, 10th ed , Hoboken, NJ, USA: John Wiley & Sons Inc
Assessment Methods and Criteria:
Classification Type: Quantitativa (0-20)
Evaluation Methodology:
Classes are all the same type of theory-practical. The teaching methodology involves lectures that almost always use examples linking the tools being developed to concepts studied in specialized course units. Students are encouraged to apply and consolidate their knowledge by solving the assigned exercises. The presentation of the course content - both oral and written -aims for clarity through detailed explanations of analytical formulas derived from theorems, in-depth arguments, a logical distribution of topics, and the organization of subjects into distinct sections. Assessment is based on two midterm tests (each weighted at 50%), to be completed individually during the regular assessment period. This allows students to monitor their performance more effectively throughout the semester. However, the theoretical-practical component, assessed during the regular period via these two tests, can be retaken during the Supplementary and Special assessment periods through a Comprehensive exam covering the entire syllabus. Nevertheless, each instructor responsible for teaching and assessment may define an alternative assessment method, provided that it fully complies with the University of Madeira´s Student Learning Assessment Regulation currently in force.
Evaluation Methodology:
Classes are all the same type of theory-practical. The teaching methodology involves lectures that almost always use examples linking the tools being developed to concepts studied in specialized course units. Students are encouraged to apply and consolidate their knowledge by solving the assigned exercises. The presentation of the course content - both oral and written -aims for clarity through detailed explanations of analytical formulas derived from theorems, in-depth arguments, a logical distribution of topics, and the organization of subjects into distinct sections. Assessment is based on two midterm tests (each weighted at 50%), to be completed individually during the regular assessment period. This allows students to monitor their performance more effectively throughout the semester. However, the theoretical-practical component, assessed during the regular period via these two tests, can be retaken during the Supplementary and Special assessment periods through a Comprehensive exam covering the entire syllabus. Nevertheless, each instructor responsible for teaching and assessment may define an alternative assessment method, provided that it fully complies with the University of Madeira´s Student Learning Assessment Regulation currently in force.