Calculus II

Calculus II (257 - 15531)
Study: Bachelor in Biomedical Engineering
Semester 2/Spring Semester
1st Year Course/Lower Division

Please note: this course is cross-listed under the majority of engineering departments. Students should select the course from the department that best fits their area of study.

Students are Expected to have completed:
Calculus I, Linear Algebra

Compentences and Skills that will be Acquired and Learning Results:

The student will be able to formulate, solve and understand mathematically the problems arising in Biomedical Engineering. To do so it is necessary to be familiar with the n-dimensional Euclidean space, making a special emphasis in dimensions 2 and 3, visualizing the more important subsets. He/she must be able to manage (scalar and vector) functions of several variables, as well as their continuity, differentiability, and integrability properties. The student must solve optimization problems with and without restrictions and will apply the main theorems of integration of scalar and vector functions to compute, in particular, lengths, areas and volumes, moments of inertia, and heat flow.

Description of Contents: Course Description

1. Differential Calculus in several variables
1.1. R^n as an Euclidean space; topology
1.2. Scalar and vector functions of n variables
1.3. Limits and continuity
1.4. Differentiability

2. Local properties of functions
2.1. Higher-order derivatives
2.2. Differential operations on scalar and vector functions: geometrical properties
2.3. Free and constrained optimization

3. Integral Calculus on R^n
3.1. Double and triple integrals
3.2. Changes of variables and applications

4. Integrals over curves and surfaces
4.1. Line and path integrals
4.2. Surface integrals
4.3. Integral theorems of vector analysis in R^2 and R^3

Learning Activities and Methodology:

The learning methodology will include:
- Attendance to master classes, in which core knowledge will be presented that the students must acquire. The recommended bibliography will facilitate the students' work
- Resolution of exercises by the student that will serve as a self-evaluation method and to acquire the necessary skills
- Exercise classes, in which problems proposed to the students are discussed
- Fortnightly tests
- Final Exam
- Tutorial sessions
- The instructors may propose additional homework and activities

Assessment System:

- Fortnightly tests (average over top 4 scores) 40%
- Final exam (60%)

Basic Bibliography:

J. E. Marsden and A. J. Tromba. Vector Calculus, 6th. edition. W. H. Freeman. 2012
M. D. Weir, J. Hass, and G. B. Thomas. Thomas¿ Calculus, Multivariable. Addison-Wesley. 2010

Additional Bibliography:

J. Stewart. Calculus. Cengage. 2008
M. Besada, F. J. García, M. A. Mirás, and C. Vázquez. Cálculo de varias variables. Cuestiones y ejercicios resueltos. Garceta. 2011
M. J. Strauss, G. L. Bradley, and K. J. Smith. Multivariable Calculus. Prentice Hall. 2002
P. Pedregal Tercero. Cálculo Vectorial, un enfoque práctico. Septem Ediciones. 2001
R. Larson and B. H. Edwards. Calculus II, 9th. edition. Cengage. 2009
S. Salas, E. Hille, and G. Etgen. Calculus. One and several variables. Wiley. 2007
T. M. Apostol. Calculus. Wiley. 1975