Tough Derivative

[schneidrax]

I'm trying to take the derivative of an improper integral by applying Integration by Parts. But my calculations appear incorrect.

I've tried changing the TOLerance and selecting the Singular Endpoint option all to no avail. Perhaps its a problem with Mathcad 2001i.

In any case, can someone offer assistance?

P.S.:--Just as an aid, I've attached a simple example to illustrate Integration by Parts.

[IRstuff]

Are we talking the same refractive index:
http://en.wikipedia.org/wiki/Refractive_index

In optics, the refractive index is inversely proportional to the speed of propagation in the material, so rarely is the index less than 1, and usually greater than 1. Glass is typically around 1.5.


TTFN,
Eden

[schneidrax]

On 9/13/2009 12:48:27 PM, eden_mei wrote:
>Are we talking the same
>refractive index:
>http://en.wikipedia.org/wiki/R
>efractive_index
>
>In optics, the refractive
>index is inversely
>proportional to the speed of
>propagation in the material,
>so rarely is the index less
>than 1, and usually greater
>than 1. Glass is typically
>around 1.5.
>
What you say is certainly true for glass. However, in the ionosphere (a plasma) the radiowave refractive index can be less than 1.

[ptc-1368288]

On 9/13/2009 9:55:13 AM, schneidrax wrote:
>Jean-
>
>Although I do appreciate your
>efforts, I don't understand
>them. However, I can offer
>some guidance:
>
>First, the function that you
>employed for the refractive
>index, n(r) = r, is not a
>reasonable choice for this
>problem which addresses the
>propagation and bending of
>radio rays in a planetary
>atmosphere. The refractive
>index must always be greater
>than zero otherwise radio
>propagation ceases. (It can,
>however, be less than one).
>
>Second, any atmospheric
>refractive index model should
>approach unity as its upper
>limit, since beyond the
>atmosphere (in vacuum) n(8)=1.
>I used this boundary condition
>in applying both Leibniz' Rule
>and Integration by Parts.
>
>Third, propagation takes place
>only above the planet's
>surface. Thus, 0 < a < r1 <<br> >r. The refractive index need
>not be specified within the
>planet where r < a.
>
>Finally, I've been using a
>Venusian model for the
>refractive index [see my
>Section 5]. It would help me
>if you could use that same
>model to illustrate your
>points.
>
>Thanks, as always, for your
>efforts.
_____________________________

I have reached your Venusian model, from which you calculate several derivatives. Then you plug some definitions depending upon the long top part, which long top part results from solving (A) ... then the final plots do not agree. Conclusion: the top part is not correct, then your solving (A) is wrong. I can't comment about (A), couldn't get any source quick.

Past your Venusian model, "my" Mathcad 11.2a crashes on rescrolling up/down. I will not re-open your work sheet on potential damage to my Mathcad. Where does (A) comes from ? You must tell that to collabs, especially if it is derived correctly.

jmG

[schneidrax]

On 9/13/2009 11:01:56 PM, jmG wrote:
>
>I have reached your Venusian model, from
>which you calculate several derivatives.
>Then you plug some definitions depending
>upon the long top part, which long top
>part results from solving (A) ... then
>the final plots do not agree.
>Conclusion: the top part is not correct,
>then your solving (A) is wrong. I can't
>comment about (A), couldn't get any
>source quick.

I still think Equation (B), the Integration by Parts of Equation (A) is correct (see my response to Tom above and the attachment thereto).
>
>Past your Venusian model, "my" Mathcad
>11.2a crashes on rescrolling up/down. I
>will not re-open your work sheet on
>potential damage to my Mathcad.

Perhaps "your" Mathcad 11.2a crashes because of the literal subscripts in Tough_Derivative.mcd.
This was mentioned by Tom in his response. I'll try to eliminate these subscripts later today.

Where
>does (A) comes from ? You must tell that
>to collabs, especially if it is derived
>correctly.

I'm not willing to go back prior to my "Square One". Let's just take it from the Get-Go that (A) is an improper integral for which we are seeking its derivative.

Again, thanks for looking at this.

[ptc-1368288]

The first A is not an integral. No CAS will neither solve symbolic nor numerical. You can convince yourself by these proofs. If you can't plot or otherwise interpret the integrand, zap. You must have got that relation from purely figurative view of the mind like Wikipedia or other papers.



jmG

[ptc-1368288]

Reading my mind, the relation between a function and its 1rst derivative falls in the DE category [ Lagrange ? ). Integrating A is then solving a DE, which you have to express. There is lot of book stuff behind, in book style. Again, integral A has no mathematical meaning, only a way to say, but not a WayToDo.

jmG

[schneidrax]

On 9/12/2009 12:23:33 AM, jmG wrote:
>The first A is not an
>integral. No CAS will neither
>solve symbolic nor numerical.
>You can convince yourself by
>these proofs. If you can't
>plot or otherwise interpret
>the integrand, zap. You must
>have got that relation from
>purely figurative view of the
>mind like Wikipedia or other
>papers.
>
Nope, but I did mangage to get the derivative of (A) in three different ways. (see my earlier response to Tom today).

Although I didn't apply any of your suggestions this time around, I'm always appreciative of your willingness to help. Thank you, Jean.

[ptc-1368288]

But in a word: SUCCESS ! Not so shure

Nope, but I did manage to get the derivative of (A) in three different ways. (see my earlier response to Tom today).

Although I didn't apply any of your suggestions this time around, I'm always appreciative of your willingness to help. Thank you, Jean.
__________________________

Nothing will change collab: the derivative of an integral is the integrand. Your problem seems to root in the interpretation of some book style that you have refused providing. Not knowing anything of that, you should be able to plot one way or another the integrand of Zeta(,).
You know how I create a "complete work sheet" and how I check work sheets passed in the collab. I have recreated an old work sheet "Improper integral" with variable path of integration like in your case seemingly, just plug your integrand Zeta, define everything .

Clean/define/redefine/manage/abstract that commented work sheet, will look again. The source of (A) would have solved the problem long time ago. Not all book styles (DE's, integrals ...) are executable. The Mathcad integrator is executable.

Jean

[ptc-1368288]

On 9/11/2009 12:54:44 PM, schneidrax wrote:
>I'm trying to take the
>derivative of an improper
>integral by applying
>Integration by Parts. But my
>calculations appear incorrect.
>......
>In any case, can someone offer
>assistance?
>______________________________

I have reconsidered your integral. Not an "offer", just eventual assistance. I can't reconciliate with unsourced and unabstracted projects, especially in your proposal where you jump to numerical solution before terminating the literal calculus solution.
Visit the thread above "Integration by parts",
where your integral is in demo, for purpose.
Sorry, can't collaborate directly, because of the above stated reasons.

Jean

[ptc-1368288]

... revisit the work sheet:

http://collab.mathsoft.com/~Mathcad2000/upfile?128711,11

You can see my point(s) about carrying the maths as scalar. The Mathcad quick plot is based on "scalar maths", with lot of the most interesting analysis done conclusively and visually. The Mathcad quick plot subcontract is a very advanced tool that makes Mathcad so powerful. Also, you can appreciate my untractable objection to use units in Mathcad and probably in all CAS in general: because maths are scalar.

Jean

[ptc-1368288]

... if necessary, I will keep updating the work sheet:

http://collab.mathsoft.com/read?128713,11

Jean

[ptc-1368288]

On 9/21/2009 12:32:13 AM, jmG wrote:
>... if necessary, I will keep
>updating the work sheet:
>
>http://collab.mathsoft.com/rea
>d?128713,11
>
>Jean
_______________________________

... just did.



jmG

[ptc-1368288]

On 9/21/2009 1:31:35 AM, jmG wrote:
>On 9/21/2009 12:32:13 AM, jmG wrote:
>>... if necessary, I will keep
>>updating the work sheet:
>>
>>http://collab.mathsoft.com/rea
>>d?128713,11
>>
>>Jean
>_______________________________
>
>... just did.
>
>jmG
================================

... now, in here:

http://collab.mathsoft.com/~Mathcad2000/upfile?128716,11

jmG

[ptc-1368288]

On 9/21/2009 2:03:55 AM, jmG wrote:
>On 9/21/2009 1:31:35 AM, jmG wrote:
>>On 9/21/2009 12:32:13 AM, jmG wrote:
>>>... if necessary, I will keep
>>>updating the work sheet:
>>>
>>>http://collab.mathsoft.com/rea
>>>d?128713,11
>>>
>>>Jean
>>_______________________________
>>
>>... just did.
>>
>>jmG
>================================
>
>... now, in here:
>
>http://collab.mathsoft.com/~Mathcad2000/
>upfile?128716,11
>
>jmG

+++++++++++++++++++++++++++++++++

... ... more (+) as it goes !
Integrand(u,p) = Final(u,p)
Therefore, integrating Final(u,p) is the solution !
Don't ask me to digest before bed time.

[ptc-1368288]

... the last word is that simple:

The function might be integrated by Mathematica only [suppose], and quite a pest of an advanced gear box. Therefore for those less equipped, some smart cat devised an equivalent under the form you had presented as an integrand, that eventually can be integrated by minimal CAS.

Would be damned interesting if 14 users would run the work sheet ?

jmG

[ptc-1368288]

... read more.

jmG