Science Forums 2

Doctordick · 08-03-2008

Don't worry about the slow response. I have had a lot of things to do lately and I haven't had much time for the forum either.

Quote:

Originally Posted by AnssiH

It would probably be possible to implement the post editing so that it would display the resulting LaTex render somewhere in real time as you are typing the code... ...but I guess that would make life too easy.

From the way LaTex interpretation is being done now, a real time render would be impossible as many LaTex controles require the closing control to exist before they can be rendered (for example the variable size parenthesis). One would have to establish a default result for any and all such cases in order to define that real time render. I would not expect such a thing soon.

Quote:

Originally Posted by AnssiH

I am a bit uncertain what you mean by "what is in the parenthesis evaluates to an ordinary algebraic expression and does not end up being a differential operator".

As I said, the expression can be taken as ambiguous as to exactly what the differential operator is to operate on; however, things within parenthesis are always to be totally evaluated first so I think it reasonable to insist that, since it is enclosed in parenthesis, the $\vec{\nabla}_i$ can be seen as "not operating on $\vec{\Psi}_1$ as it is outside the parenthesis. And, yes, if you let $\vec{\nabla}_i$ operate on $\vec{\Psi}_1$ the result would be wrong.

Quote:

Originally Posted by AnssiH

Also I am a bit shaky with the term $dV_2$ appearing in places. I remember asking about its meaning before, and it had something to do with having to consider how large an area of possibilities we are considering... Still I am somewhat puzzled about what is it doing there, where ever I see it appearing...

As I explained to you somewhere earlier, that integral sign, $\int$ , was originally a big “S” standing for a “sum”. The number of terms in the sum was allowed to go to infinity and the terms being summed (which are defined by some function) must individually be brought to zero as their number goes to infinity (otherwise the result would be infinite). For that reason, the terms in the sum are weighted by the “differential” (the variable being integrated over preceded by the letter “d”). The standard way of writing an integral is as follows:

$A=\int_a^bf(x)dx$

An integral over many variables would normally be written.

$A=\int_{a_1}^{b_1}\int_{a_2}^{b_2} \cdots \int_{a_n}^{b_n} f(x_1,x_2, \cdots, x_n)dx_1dx_2 \cdots dx_n$

but that just gets too complex when you are dealing with the number of arguments I am working with so I move to somewhat of a shorthand. I omit the limits on the integration (as the limits in all the cases of interest here are over all possible values: i.e., $a=-\infty$ and $b=+\infty$ ) and, instead of all those “d”s I just use dV where “V” stands for volume. I am integrating over the entire abstract volume expressed by the entire collection of arguments. In the case you are pointing to, $dV_2$ refers to the entire abstract volume expressed by the numerical references called “set #2”. Finally in recognition of the differential abstract volume so represented, I only put one integral sign in the expression (essentially integrating over one variable: that one variable being the abstract volume being referred to by $dV_2$ . It just gets rid of a lot of algebraic expressions which really don't need to be there as they add a lot of repetition without actually adding to the clarity of the intended meaning. (That's my opinion anyway.)

Actually, the standard definition of the integral operation has some strong similarities to parenthesis. The “integral operator” (and it can be seen as a mathematical operator) consists of two symbols, the integral sign, $\int$ (with appropriate limits) and the differential “dx” which indicates the argument to be integrated over. These two symbols are placed respectively to the left and to the right of the function to be integrated. The interpretation is that the implied operation is to be completely performed as a unit, uninfluenced by what is outside those symbols. For example, were I to write $\int dx f(x)$ the standard interpretation would be that the answer is xf(x): i.e., f(x) is not to be integrated over as it is not “inside the integral”.

Oh, just one final note. The normal meaning of an integral sign with no limits is that one means to indicate what is called the “indefinite integral”. The “definite” integral has expressed limits and is defined to be the difference between the indefinite integral evaluated at the upper limit and the lower limit. My integrals are actually definite integrals as the limits are clear (I say they are integrated over all possibilities quite a bit) though, as they are written they could be interpreted to be indefinite integrals. Again, I do this for notational simplicity. Letting the reader know I mean these to be integrated over all possibilities is much more convenient than using LaTex to write in the limits everywhere.

I hope I have cleared a few things up.

Have fun -- Dick

Science Forums 2

Hypography?

Share the love!

Sponsors

Friends