Base Knowledge
Knowledge of Mathematical Analysis I (1st semester).
Teaching Methodologies
Along T classes is presented a theoretical explanation of the whole subject; in class TP and PL are solved exercises to apply the knowledge acquired. Evaluation: Students can opt for a distributed evaluation or a final exam, in wich a minimum grade of 9.5.must be obtained. In distributed evaluation a 50% atendance is mandatory. In this evaluation the PL, worth 4 points, will be distributed by 2 tests of 2 points each; a minimum of 0.75 in each one is mandatory. The remaining 16 points will be gained by T and TP evaluation, through 3 tests worth 16 points each; minimums of 6 in the first and 7 in the last 2, are required. The sum of the 2 components average need to be at least 9.5 points, otherwise the student will have to take a final exam with the whole subject. Taking this exam in the regular season, and not having the minimums required in only 1 of the 3 components, the student can take just that part on the exam.
Learning Results
Understand the basic concepts of ordinary differential equations and solve some simple first order differential equations; Knowledge of numerical and power series; Understand and apply theoretical development of multi-variable differential and integral calculus; Solve and interpret real problems. At the end of this course unit the learner is expected to be able: To explain the concepts, discuss and present each problem solution in an appropriate way; To solve practical problems with an increasing autonomy, using the subjects treated in the classroom and other related topics; To find and select relevant information from different sources such as monographs textbooks and the web.
Program
ORDINARY DIFFERENTIAL EQUATIONS: Definitions. Equation Variables separable, Homogeneous Equations. Linear equation of 1st order. Bernoulli equation.
LAPLACE TRANSFORM: Definition. Resolution of Linear Ordinary Differential Equations with Laplace Transform.
NUMERICAL SERIES: Definition, Series of nonnegative terms. Convergence criteria. Alternating series.
POWER SERIES.
DIFFERENTIAL CALCULUS IN IRn: Functions of several variables – Domains and their geometric representation. Limits iterated. Directional limits Theorems and Applications. Partial derivation. Geometric interpretation. Partial derivation rules, Schwartz Theorem. Differentials. Derivative of Composite Function. Directional derivative and partial derivatives. The gradient. Maxima, minima and saddle points. Free and constrained extrema.
DOUBLE INTEGRAL: Cartesian and polar coordinates; Change of variables in a double integral. Applications of the double integral.
TRIPLE INTEGRAL: Cylindrical coordinates. Applications.
Internship(s)
NAO