Calculus II

Base Knowledge

Knowledge of Mathematics I of the Bachelor’s degree in Civil Engineering

Teaching Methodologies

This curricular unit is essentially formative and attempts to coordinate the fundamental theoretical knowledge with the necessary developments in the following curricular units of the study plan. At this level the intuitive understanding of concepts and the ability to calculate is encouraged.

In classes, a brief explanation of the subject is given followed by the resolution of application exercises.

Students are also invited to participate in a discussion forum created on Moodle where they must post homework and where they can ask questions and comment on the exercises solved by their classmates.


Learning Results

Objectives: Perform basic matrix operations; Calculate determinants, vectors and eigenvalues; Solve linear differential equations of order n; Understand and apply concepts related to vector spaces; Solve linear systems using matrix theory; Solve real problems whose model is given by matrices and systems.

Generic Skills Compare critically the results obtained analytically with those obtained computationally; Expose the solution of problems in a clear and simple way; Explain the concepts and the solution of problems adequately; Solve, in an autonomous way, practical problems using, the subjects treated in classes.

Specific Skills: Development of critical spirit, coordination and exposition skills, reflection and research attitudes, aiming at the acquisition of basic knowledge indispensable for the set of disciplines of the Civil Engineering course


CHAPTER I – Introduction to the study of ordinary differential equations

           1.1 Definition

           1.2 First Order Differential Equations

                 1.2.1 Separable variable equation

                 1.2.2 First order linear equation

          1.3 Linear differential equations of order n

                1.3.1 Homogeneous linear equation with constant coefficients

                1.3.2 Complete linear equation with constant coefficients

CHAPTER II – Matrices

         2.1 Definitions

         2.2 Operations with matrices and properties

CHAPTER III – Systems of Linear Equations

         3.1 Condensation of matrices and characteristic

         3.2 Systems of linear equations

         3.3 Classification and solution of systems of linear equations by condensation

         3.4 Inverse matrix

CHAPTER IV – Determinants

         4.1 Definitions and Properties

         4.2 Cramer’s Rule

CHAPTER V – Vector Spaces

         5.1 Definitions and examples

         5.2 Vector subspaces

         5.3 Vector subspace generated by a set of vectors

         5.4 Linear dependence and independence

         5.5 Basis and dimension

CHAPTER VI – Linear transformations

        6.1 Definition and examples

        6.2 Kernel and Image of a Linear Transformation

        6.3 Matrix of a Linear Transformation

        6.4 Inverse of a linear transformation

CHAPTER VI – Eigenvalues and Eigenvectors

        7.1 Definitions, computation and properties

        7.2 Diagonalization

        7.3 Cayley-Hamilton Theorem

Curricular Unit Teachers




Main Bibliography

  • Anton, H. & Rorres, C. (2005). Elementary Linear Algebra with applications. (9ª ed.). John Wiley & Sons. Cota Biblioteca: 3-1-99 (ISEC) – 11300;
  • Cabral, I., Perdigão, C. & Santiago, C. (2018). Álgebra Linear – Teoria, Exercícios resolvidos e Exercícios propostos com soluções. (5ª ed.). Escolar Editora. Cota Biblioteca:  3-1-27 (ISEC) – 15011;
  • Fidalgo, C. (2016). Álgebra Linear, DFM, Instituto Superior de Engenharia de Coimbra. Cota Biblioteca: 3-1-116 (ISEC) – 13179;
  • Graham, A. (2018). Matrix Theory and Applications for Scientists and Engineers. Dover Books on Mathematics. Cota Biblioteca: 3-1-58 (ISEC) – 03750;
  • James, G. & Dyke, P. (2020). Modern Engineering Mathematics. (6ª ed.). Pearson. Cota Biblioteca: 3-2-193 (ISEC) – 08334, 3-2-220 (ISEC) – 08838, 3-2-221 (ISEC) – 08839;
  • Kreyszig, E. (2011). Advanced Engineering Mathematics (10ª ed.). John Wiley & Sons. Cota Biblioteca: 3-7-95 (ISEC) – 17207, 3-7-51 (ISEC) – 11749, 3-7-52 (ISEC) – 11750, 3-7-53 (ISEC) – 11751;
  • Monteiro, A., Marques, C. & Pinto, G. (2000). Álgebra Linear e Geometria Analítica. Problemas e Exercícios. McGraw-Hill. Cota Biblioteca: 3-1-75 (ISEC) – 08697, 3-1-76 (ISEC) – 08698.

Complementary Bibliography

  • Nicholson, W. (1993). Elementary Linear Algebra with Applications. (2ª ed.). PWS Publishing Company;
  • Santana, A. & Queiró, J. (2018). Introdução à Álgebra Linear. Gradiva.