Calculus II

This course is essentially formative and tries to coordinate with the fundamental theoretical developments needed in the following courses in the curriculum. At this level is promoted intuitive understanding of the concepts and calculation capacity.In Theoretical-Practical lessons the expository and questioning method is used during the explanation of the theoretical subjects and exercises are solved in groups and individually.
Assessment: a Final Exam, rated for 20 points, is performed ; The approval requires that the mark is greater than or equal to 9,5 points.

Objectives: Perform basic matrix operations. Compute matrix determinants, eigenvalues and eigenvectors. Solve linear differential equations of order n. Solve 1st order linear differential systems of differential equations. Understand and apply concepts related to vector spaces. Solve and interpret linear systems using matrix theory.
Generic skills: Develop algorithms using a logical and structured reasoning. Base problem solving on mathematics. Compare, with criticism, the results obtained by analytical means with the ones obtained by computational means. Expose, using documents, the problems’ solution in a clear and simple way. Explain the concepts and problems’ solution in an appropriated way.
Specific skills: Development of critical thinking, ability to coordinate and exposure, attitudes of reflection and research, for the acquisition of basic knowledge necessary for all subjects of the civil engineering course, particularly in the context of Linear Algebra

1. Matrices and Linear Systems: Introduction; Matrix operations and their properties; Row echelon form and rank; Classification and geometry of linear systems; Gaussian elimination; Homogeneous systems; Matrix inversion: Gauss-Jordan method;
2. Determinants: Definition and properties; Cramer’s rule.
3. Linear Spaces: Definition, Examples and Properties; Subspaces; Linear combinations; Linear expansion; Linear independence; Basis and dimension.
4. Eigenvalues: Eigenvalues, eigenvectors and their properties; Diagonalization; Cayley-Hamilton Theorem.
5. Linear Differential Equations of order n: Basic definitions. Linear homogeneous equations with constant coefficients; Nonhomogeneous equations.
6. First-Order Linear Systems of Differential Equations: Basic definitions. Initial condition problems. Superposition principle. Solution of a 1st order linear system using eigenvalues and eigenvectors. Matrix exponential. Nonhomogeneous systems.

NAO