Cálculo Diferencial e Integral II

Course text

The text was mostly completed in September 2015. Details have been changed and some additional material has been included since then. You are advised to use the $\infty$ semester to have access to the most up to date version.

Support material, in particular exercises, have been extended right up to the end of the 2020/21 academic year.

Work on offering an English translation of the course text has started on October 24, 2021.


To print a page you have to consider:

  • Waiting until all formulas are displayed.
  • Expanding any sections marked Further details or similar.

For students

The course is divided in pages, each page being accessible from the menu.

Let us underline some points on how things are organized:

  • One page may match a lecture or not. Each page tries to deal with one topic and that usually falls between one and two lectures.
  • There will always be material mentioned in the text and not in class and vice-versa.
  • The author of the text does not claim what is the adequate level of detail for each student or how deep is their interest in Mathematics. The text provides collapsed sections that may be omitted on first reading. It is hoped that some students will be curious.
  • This web site includes what at some point was the state of the rart concerning presenting mathematical formulas and graphics on the web. This may cause some instability. If you wish to report technical or mathematical anomalies email
  • Dealing with formulas, some of them rather long, in order to make them readable on a smartphone, is not trivial. Nevertheless it should be relatively easy if you you rotate the screen. A contextual menu option may be available to collapse portions of formulas.
  • The text was not written having a printed version as an objective. In particular, in a printed version, hyperlink functionality will be lost, and most equations, sections and results will not be numbered to give the traditional printed page way of finding references.
  • Some animations produced with sage and three.js (previously jmol) have been included.
  • The original language for the course text is Portuguese as spoken in Portugal and written pre-1990. It is currently being translated into English.

To start reading the text itself click the $\mathbb{R}^n$ menu item.

For mathematicians


The author believes the Mathematics textbook, as practiced in the 20th century, is dying, at least for university Calculus textbooks. Most of these include a large number of pages, implying confusion between what is important and what is secondary, a large number of repetitive exercises and lack of originality, the latter a consequence of the stabilized character of first year university calculus. Hence, the number of printed calculus textbooks can only be justified by economical or curricular reasons.

In spite of the warning in the last paragraph about lack of originality also being applicable to this text, its structure has a few unusual characteristics:

  • The study of integration in the $\mathbb{R}^n$ context using cartesian coordinate systems starts as soon as possible, even before studying continuity and differentiability. The purpose of this is to make the student acquaint himself with geometric reasoning in $\mathbb{R}^n$ with $n\gt 2$. We claim that working with limits of integration in $\mathbb{R}^3$ with cartesian coordinates has an important part to play in the necessary leap from one dimensional contexts. Obviously the change of variables formula in integrals is postponed to a more traditional moment later on.
  • Establishing which bounded functions are integrable in a bounded set in the sense of Riemann is also postponed, but not ignored, until we are able to deal with uniform continuity through the Heine-Cantor theorem. We characterize uniform continuity using oscilation of a function in a set in order to emphasize the the link to Lebesgue’s integrability criterium.
  • The inadequacy of the Riemann integral to deal with simple examples of apparently reasonable surface integrals are not ignored but are not solved.

The publishing of a mathematical text used to be a warranty concerning its quality, in spite of an increasing number of counterexamples. This site is an attempt to show that it is possible to avoid compromising the quality one should expect from a text intended to support a university course, both in a strict mathematical sense but also in terms of web technology or graphics. This is done being conscious of the falacy that the proliferation of web supported mathematical and physical content allows students to bypass Mathematics and Physics lectures which in turn allows diminishing the amount of course time assigned to these areas. Unfortunately the curricula introduced in 2006 at Instituto Superior Técnico were already polluted by such nonsense and the 2020/21 curricula continued this unfortunate trend.


The Course Text is made available according to the Creative Commons - NonCommercial-Attribution 4.0 International. The author considers that this license allows the usual printed redistribution by Student Unions.


This work would not have been possible without several open source software projects mentioned in and, additionally, sage, three.js and jmol. In particular the author  was impressed by on how easy and fast it was to find help in case of problems.

Communications concerning all types of anomalies coming directly or indirectly from students and colleagues contributed decisively to increase the quality of this work. Thanks!

Unfortunate circunstances

These pages reproduce in real time summaries introduced in the school fénix system in spite of the awful technical quality of that system RSS feeds. I recommend reading instead of the practice of introducing semantically irrelevant content in fields created for other purposes.


To report anomalies in the site concerning mathematics, english, formatting or programming, please use email.

Known problems

  • In smartphones in portrait orientation quite a number of formulas will not fit to the width of the page and others will have bad breaks. Rotating to landscape orientation will solve most of these. If this does not work for a particular formula please contact me by email. Some assistive options that automatically collapse or expand parts of equations may be available. The latter functionality should be considered as experimental.
  • This site is not tested in Apple operating systems and, in particular, in the Safari browser. Hence reports of anomalies specific to this context will be very much welcome but need to be as complete as possible. Install Firefox or Chrome if the problem turns out to be Safari specific.

Recent changes

MathJax was updated to version 3.*.* and animations produced with sage ≤ 8.7 and jsmol started being produced with sage ≥ 9.0 and three.js. Unfortunately three.js has some limitations in a few of these.

$3d$ animation example obtained with Sage 9 (beta) and three.js for the paraboloid $z=x^2+y^2$.

In October 2021

Instituto Superior Técnico applied a curricular restructuring where a course such as CDI-II ceased to exist, its topics having been split between two courses, one of which maintaing the course title.

Due to this rather confusing state of affairs and the inevitable decrease in rigor in the sequence of Calculus courses taught at Instituto Superior Técnico, a consequence of diminishing time assigned to its classes, this site has been removed from the domain and started being hosted in a domain registered by the author in Digital Ocean servers. This way one hopes that students will not be confused by its content with respect to current curricula while maintaining the ability to use this material lo learn Mathematics.

This version last edited: João Palhoto Matos, 09/12/2022 19:56:24.