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The present courseware is motivated by a course given at Tel-Aviv University, and intended for first year undergraduate physics students . The material presented here, covers mathematical subjects, many learned rigorously in mathematical courses, but unfortunately are needed for studying physics and engineering in the early stage. The present courseware intends to cover the gap and convey to the students the necessary mathematical tools in due time. Some parts could also be used for teaching other empirical-science students, who need some mathematical knowledge in the early stage of their study.
Mathematics is the most rigorous discipline. In order to understand well the meaning of the mathematical notions, theorems and applications; one needs a very thorough and deep approach. This cannot be achieved completely by a course similar to the present courseware, and an additional thorough mathematical course is necessary, at least "Differential calculus with a single variable", which should be taught simultaneously.
On the other hand, just learning mathematics rigorously does not supply the student with the complete meaning and understanding of any particular mathematical tool for practical science applications. In this respect the present courseware is complemantary to the rigorous mathematical attitude for empirical-science students.
The rich animated and interactive graphics, applied here, represent a powerful tool for understanding the material. They can be projected in class during the lectures, and studied independently at a later date at home, or in a library. These could also benefit pure mathematical teachers and students.
The material covered by the previous version 1.0 of the courseware was "Calculus of a single variable". It is extended in the present version 2.0 by adding "Calculus of many variables"
The question is, how to present and convey the material to the students without the rigorous proofs. Whenever the proofs are short, they should be used. If not, one can give the proof for a simpler case and just state that "it can be shown that..." . Another approach is to relate a particular subject to a very concrete and known practical example and of course to state that "also in the general case it can be shown that...".
Any approach that is not rigorous should be reasonable and based on logic. A theorem should not be presented without clarification and justification. One cannot base anything on intuition. An educated guess is more appropriate, emphasising educated. Science students need to be tought logic and reasoning as part of their education.
One can see how intuition could play a destructive role in Fig. Intuition. As a matter of fact many first year students fail to give a satisfactory answer to the second question asked in this example.
In addition the trend of the examples and exercises should be, as far as possible of applicable nature, rather than being entirely abstract.
The students are required to have learned higher level mathematics in school, although the courseware fills the gap. There are however exceptions: planar geometry and trigonometry are taken for granted, but only at their most basic level. For instance some planar geometry was used in Fig. Pythagoras in relation to the page Tutorial.
Students with a lower background of mathematics can also follow the course, but should make effort and spend more time in the beginning. Afterwards, there should be no difference with the rest of the students.
The list of books for introducing mathematics to the students of physics and engineering is extensive, therefore I mention a few, which I am in particular fond of, but this of course is a very biased sample.
One book that covers more or less, the material of the present courseware, but is richer in physical examples is:
On the other hand, a book that covers much more mathematical material for physics and engineering, can be used as a general reference book, such as:
For a more detailed study of the theory with plenty of exercises, there are many books covering the paricular subjects. The following series of books are very popular, and just select the book corresponding to your subject:
The most important requirement for any science student is a mathematical handbook, where all the necessary formulae and tables are under the fingertips. Here is an example:
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