Calculus II

Code	12809
Year	1
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Not applicable, but it is advisable to have prior knowledge of Calculus I (limits/continuity on R, differentiation and integration in 1D), basic notions of linear algebra/vectors, and fluent algebraic manipulation.
Mode of delivery	Face to Face with recourse to e-learning.
Work placements	N/A
Learning outcomes	The aim is for students to understand and apply differential and integral calculus in higher dimensions, as well as basic ODEs. In particular: (i) analyse functions Rn?R? with respect to limits and continuity; (ii) compute partial/directional derivatives, the gradient and tangent planes, verifying differentiability; (iii) apply the chain rule, higher-order derivatives, Schwarz’s theorem and the implicit function theorem; (iv) identify extrema (free and constrained, via Lagrange multipliers); (v) set up and solve double/triple integrals, perform changes of coordinates, and apply them to areas/volumes; (vi) model and solve separable/linear ODEs and second-order linear ODEs with constant coefficients. These objectives are articulated with TP classes: some focused on foundations and examples; others on guided problem-solving, Moodle tasks, assignments, and in-class presentations.
Syllabus	1. Ordinary Differential Equations 1.1. Definition, examples and applications. Separation of variables. 1.2. Linear equations, integrating factor method, and Bernoulli equations. 1.3. Second-order equations with constant coefficients. 1.4. Applications. 2. Functions from Rn to R? 2.1. Real-valued functions of n real variables and vector-valued functions. 2.2. Limits and continuity. 3. Differential Calculus in Rn 3.1. Partial derivatives. Directional derivatives. Gradient. 3.2. Tangent plane. 3.3. Differentiability. 3.4. Derivative of the composite function. 3.5. Higher-order derivatives. Schwarz’s theorem. 3.6. Implicit function theorem. 3.7. Local and absolute extrema. 3.8. Constrained extrema: method of Lagrange multipliers. 4. Integral Calculus in Rn 4.1. Double and triple Riemann integrals: definition and examples. 4.2. Properties of integrable functions. 4.3. Change of coordinates. 4.4. Applications.
Main Bibliography	[1]Calculus- A Complete Course, Robert Adams, Christopher Essex, 10ª edição, 2022, Pearson [2] Cálculo, vol. II, James Stewart, 2006, Pioneira Thomson Learning [3] Cálculo, vol. 2, Howard Anton, Irl Bivens, Stephen Davis, 8ª edição, 2007, Bookman [4] Análise Real, vol.2 - Funções de n Variáveis, Elon Lages Lima, Coleção Matemática Universitária, IMPA (Brasil), 2007. [5] Análise Real, vol.3 - Análise Vetorial, Elon Lages Lima, Coleção Matemática Universitária, IMPA (Brasil), 2007. [6] Vector Calculus, J. Marsden, A. Tromba, 2003, Freeman and Company. [7] Cálculo, vol. II, T. Apostol,1994, Reverté
Language	Portuguese. Tutorial support is available in English.