]> Tutorial

Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)

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Next topic: chapter 1 Differentiation, section 1 Real Numbers, page 1 Irrational Numbers

Tutorial - instructions for use

In order to benefit from the present tutorial, one should try all the options described below, and follow the instructions.

There are four links on the top and bottom of each page (present included):

  1. Home takes you to the home page.
  2. Table of Contents assists you in finding the chapter, section and page that you are looking for. The Table of Contents includes also a link to the Table of Figures.
  3. The A-Z Index helps you finding a particular topic.
  4. Help gives you assistance. If you cannot find the appropriate answer to your request, write to the author: dagan@tauex.tau.ac.il .

In addition to the windows opened by the four links listed above, the courseware is based on three types of windows: page, figure and solution.

All the courseware links open in the same window. However by right-clicking on a link, one can choose by convenience, to open it in a new tab or in a different window.


The window you are viewing now, is of the type "page". It contains text, embedded occasionally with mathematical expressions, which are viewed fine by the browsers FireFox and Safari only. Other browsers could encounter difficulties, with the mathematical expressions, but are expected to be able to do well in the future with the help of MathJax, which is under development and is not implemented here.

On the top and bottom of the page window, there are links for the previous and next page that are suggested for the user, who wants to cover all the subjects of this courseware.

Different topics of a page are subtitled by red characters. You are reading the topic "Page".

The main points (expressions or words) of a paragraph are displayed in bold letters. This is very helpful in scanning the text and looking for a particular subject.

The size of text and the mathematical expressions depend on your browser set-up and monitor resolution. For resizing use the zoom-in and zoom-out instructions of your system. The size of text and of the mathematical expressions change proportionally.

The navigation between the page-windows is done by clicking the link of a page (e.g. the link for the previous or the next topic), or by moving back and forth, using the browser's buttons. The use of the links "Table of Contents" or "A-Z Index" is another method of navigation.

Following is an example of a mathematical expression - the Pythagorean theorem for a right-angle triangle, where  c  is the length of the hypotenuse, and the symbols  a  and  b - of the other sides.

c 2 = a 2 + b 2 (

The fenced numbers on the right of a mathematical expression represent in that order: chapter, section, page and mathematical expression.


The abbreviation Fig. before the link's name indicates a figure-window (graphical). It is convenient some times to open the figure in a different window, thus a corresponding page window is available simultaneously. A title appearing in the figure-window represents also a link back to the correspoding page window.

The Fig. windows are based on SVG (Scalable Vector Graphics), that has strong support of the main browsers (FireFox, Safari, Google-Chrome and Opera), excluding the Microsoft ones, which are under development.

They contain graphical information and text. Most of the figures are interactive, and/or some are animated. Yellow buttons, from the control bar on the top of each Fig. window, activate the interactivity and the animations. In some cases the interactivity is enhanced by more buttons, appearing in additional places of the window. However there are few simple sketches, that do not have a control bar and are not animated, but could only be resized by stretching and/or squeezing of the corresponding window.

The text and the graphics change sequentially. The user may progress according to his individual speed by clicking the appropriate buttons. An animation can be speeded up, paused, and resumed at any interval. Some animations can be repeated with different parameters.

After opening any Fig. window, make sure by the zooming facility of your system, that you have the whole window displayed. In addition, the graphics scalability allows (if necessary) to zoom-in, zoom-out, and pan any detail.

The extensive use of colours in the figures, which is natural in graphics, also appears in the text. We have attempted to match the colour of the text with the appropriate detail in the graphics, in order to focus the attention of the user to that particular detail.

All these features are explained and illustrated in the Fig. Pythagoras, where a proof of the popular Pythagorean theorem appears as a tutorial. After opening it, make sure by the zooming facility of your system, that you have the whole Fig. page displayed in the window.

If you are interchanging "pages" with Fig.s very often it is advisable to use the bowser FireFox or Safari, which give best best support for both.

Exercises and Solution

In general the last topic in a page window is subtitled Exercises. It contains exercises to be solved by the user, which are based on the theory and examples in the page. Each exercise is numbered, and the "Exercise number" is a link to the solution-window. The "page number", located at the top of the solution window, provides the link back to the corresponding page-window.

In addition to the detailed solution, there are instructions for the user to evaluate ones score on the basis of ones solution, done independently. Ten points is the maximal score for each exercise. Here is a live example of an exercise, which you should follow, including the solution:

Exercise 1. A right-angle triangle has two identical sides of length s each. How long is the third side?

Experienced User

If you are happy with the knowledge you received from the tutorial, you are now an "experienced user" and can choose any topic from the "Table of Contents". including the tutorial.

Next topic: chapter 1 Differentiation, section 1 Real Numbers, page 1 Irrational Numbers

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