Archive for the 00 Miscellanea Category

Dummy Index

Posted in 00 Miscellanea on October 23, 2008 by ateixeira

This is a concept that appears in Mathematics and it is a pretty useful one. Basically it means that the symbol we are using doesn’t really matter. What matters is the context it’s being used in. For instance in
the following summations \displaystyle\sum_{i=0}^{5}1, \displaystyle\sum_{j=0}^{5}1 the summing index is said to be a dummy one. The result of the sum is the same regardless of whether one uses an i, a k, or whatever other symbol.

Now, are all indexes dummy indexes? No, they are not. For instance in \displaystyle\sum_{j=0}^{l}1 the l index isn’t a dummy one, since the value of the sum changes as one changes the value of  l.

That being said let’s look at a simple proof where some confusion may arise in some people,not really used to think in these terms. Lets us prove that an odd function that is being integrated in symmetric limits yields a 0 integral.

A function,  f(x), is said to be odd if f(-x)=-f(x), and we are trying to calculate \displaystyle\int_{-a}^{a}f\left(x \right)dx.

We know that

\displaystyle\int_{-a}^{a}f\left(x \right)dx=\displaystyle\int_{-a}^{0}f\left(x \right)dx+\displaystyle\int_{0}^{a}f(x)dx
Let’s take a look at the first integral in the right hand side of the equality:
\displaystyle\int_{-a}^{0}f\left(x \right)dx

Let t=-x, hence dt=-dx. Moreover

\displaystyle\int_{-a}^{0}f\left(x \right)dx=\displaystyle\int_{a}^{0}f\left(-t \right)\left(-dt \right)=-\displaystyle\int_{a}^{0}f\left(-t \right)dt=\displaystyle\int_{0}^{a}f\left(-t \right)dt
Remembering the definition of an odd function and the fact that we assumed that f was such a function we have: f(-t)=-f(t) and so
\displaystyle\int_{0}^{a}f\left(-t \right)dt=-\displaystyle\int_{0}^{a}f\left(t \right)dt

Now lets us think for a while. The t variable in that integral is a dummy one or not? If we, in a given function, exchange the t for any given symbol, will that change the value of the integral? No ,it won’t! So t is in fact a dummy variable. Thus we can, for example, change it for x.

At this point I want you to have some furious outcries in your head. That is, of course, if you aren’t used to this type of reasoning. If you are , everything is perfectly normal. The reason for this possible outcry is the change of variable that we made  first t=-x. First this guy tells us that t=-x, and now he tells us that t=x. So what’s it gonna be Mr. Smarty Pants?! Both choices are mutually exclusive (except in the trivial case were t=x=0) and we need a resolution.

The resolution is the fact that the second equality is just us noticing that for effects of computing the integral what matters is the functional form of the function and the limits of integration. So we just changed the symbol in the functional form (not the function itself) and the limits of integration stayed the same. Taking this into account, the result is
-\displaystyle\int_{0}^{a}f\left(t \right)=-\displaystyle\int_{0}^{a}f\left(x \right)dx

and this leads us to:

\displaystyle\int_{-a}^{a}f\left(x \right)dx=-\displaystyle\int_{0}^{a}f\left(x \right)dx+\displaystyle\int_{0}^{a}f\left(x \right)dx=0
Which is the result we expected all along.

Ps: I’ve assumed the reader is familiar with some calculus and if you don’t get all of this don’t despair. I’ll make this a physics course from the bottom up (I’m just trying some things for now and see if I like what I get. ) and in due time I’ll address calculus and it’s foundations. As a matter of fact calculus and some more basic math will be the first things treated on this blog.

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The Plan

Posted in 00 Announcements, 00 Miscellanea on October 23, 2008 by ateixeira

This blog has a very nice plan to it. The plan’s for me to get my knowledge of physics solidified. Too many things are either fuzzy, forgotten, or poorly understood. There was a time were I had things neatly ordered in my mind. But lack of practice and bad habits turned the situation around. But no more of that! Things are going for a U turn and Elvis is entering the building .

This is the list of how things are going down:

1. Basic Mathematics – Based on my course notes of Analysis I,II,III and IV, Linear Algebra, Probability and Statistics (plus some books)
2. Newtonian Physics – Optics and Acoustics will be considered here in principle
3. Electromagnetism – Special Relativity will be studied here. This is the best way to show Maxwell’s equations covariance.
4. Analytical Mechanics
5. Quantum Mechanics
6. Statistical Mechanics
7. Solid State Physics
8. Mathematical Methods of Physics -Some math already studied in point 1 will be viewed in a deeper light, and on the other hand some more sophisticated mathematical machinery will be used.
9. Particle Physics
10. Gravitation and Cosmology
11. Fluid Mechanics – Not much, since I only have a little book in this topic.
12. Non-Linear Phenomena – I don’t know if this point will actually be done. But this is very pretty topic that wasn’t too touched while I was taking the degree and I’d like to know a little more about it.
13. Quantum Field Theory – Maybe a little too technical even for this blog but we’ll see.

Some overlap is expected and that list is more of a guideline than anything else.

Sometimes I may tackle something a bit earlier that it is supposed to, while other times I may tackle some problem that seems like it’s being treated in a later stage. Well, but that’s just how it is.

Things will definitely be done until point 10. I’m not really sure if the last two points will be done. Eitherway we’ll certainly have enough material to take me something like 2 to 3 years in some hard (and fun) work and I hope people won’t lose their interest.

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