text:
User:Charles Jeffrey Danoff/CELTA log/Week 1

These are the entries in my CELTA Log for week one of my four week course in late 2010.

@ TH

 * Most memorable part of the day was learning its possible to teach someone a language using only that language even if they know nothing. It was wild, but I learned how-to use basic Swahili pretty quickly (don't know how long I'll have retention, but ...)
 * Start w/hello
 * My name is ...
 * Bring out an object (e.g. orange / apple / water)
 * Go over what is this? This is?
 * Yes / No
 * 1, 2, 3
 * My fellow students and teachers are cool and seem genuinely interested in learning how-to teach English better.

HW

 * Working mostly at a coffee shop with a peer and a little at Grand Central Terminal and my Aunt's house, spent roughly 2 hours or-so preparing my lesson plan and a worksheet "Find someone who... re-mix" for my 20 minute lesson tomorrow.
 * Did a solid job of working right after school, could've been better about working more efficiently on the train and at my aunt's, learned I'll have a lot of work to complete after class.

TP 1
My first lesson went alright. Started with one of my favorite jokes from the ITESLJ, which got laughs from the teachers, but not any of the students. Got right into the task next. Only had 20 minutes and 5 Ss, so I didn't spend too long on the instructions, and for the most part the activity went well. Afterwards, tutor suggested I use simpler language and slow down a little.

@ TH
After teaching, watched 5 colleagues give 20 minute lessons. Then got feedback from our tutor. In the afternoon went over learner styles in preparation for our first written assignment, then went over a wide variety of styles to teach vocabulary. I was quite tired, and not ony top of my game as a student throughout most of the day. Small part of my listlessness was what I judged to be material designed more for those who've never taught before, which I found boring.

HW
Went to Barnes and Noble's coffee shop work on the next day's lesson plan. A couple fellow students were there too, which was fun to chat, but a little distracting. Got most of my lesson plan work done there, then finished all but the Vocabluary explanation sheets on the train home, which I just finished now on the inbound train the following morning. It is important to get to know my peers, but this afterschool time needs to be a little more disciplined, because in the ensuing weeks I assume it will get harder.

TP 2
Second lesson went much better, beginning with the students laughing at my joke. I was a little unprepared b4 class, because I wanted to print out my lesson plan, but was unable to, which flustered me a little. Once on stage I also realized I had not memorized enough of my lesson plan to flow naturally through the steps I had written. Out there it was fine, because I just flowed from one step to another. The tutor gave me some suggestions while teaching, including having the students read privately, instead of chorally, and have them work i pairs, when I was trying to haveenas a group. The pairs went well, save one weaker S amongst 2 strong Ss. Eventually, I took him aside and worked with him privately. Feedback I got afterwards was mainly about grading my language, speaking slower and using more ICQ's when I teach.

@ TH
Had more energy for this afternoon, which was nice. Something I've noticed is that despite being out of college over 3+ years, and having taught professionally, as soon as I was back in the student's role, I went back into all of my old routines, like lounging back in my chair and struggling to stay awake. Cognizant of that (and with more rest) I was much more attentive today to my posture and to pay attention. I find some of the material we've covered to be a little remedial, but today had a lot of new stuff that helped keep my attention, and which I think'll be useful in the classroom.

HW
Didn't have anything due the next day, so the work I did after class was prepping for my Friday lesson, giving me a chance to get ahead in my studies, which was welcome. I did the work just in one of the teaching house classrooms, which was nice, because it saved me some money, and while there if I had any questions I could go and talk to my tutor who was in her office. Got most of the work done in a little over an hour.

@ TH
Spent the whole morning watching fellow students teach for 40 minutes each. Learned some things watching them, but learned more about the students who are the same ones I teach. Especially one French gentleman, who is the subject of my first written assignment. We have to do a profile of one of our students as a learner. I was taking notes on him. Afternoon today was fun. Spent the first half talking about passive voice, which was interesting and kinda tough. I guess when you add (be + past participle) to something it becomes passive voice, or as Wikipedia (http://en.wikipedia.org/w/index.php?title=English_passive_voice&oldid=395481025) says "The passive voice is a grammatical construction (a "voice") in which the subject of a sentence or clause denotes the recipient of the action rather than the performer." After that we talked about lesson planning, breaking down the whole CELTA/TH NY process, which they were careful to note, "may be different from that of the school where [we] teach." I guess if I were really clever, I'd use this log to summarize what I learn each day as a quick review of sorts.

HW
Finished my lesson plan for Friday using the printer and copier available in the St. John's building. I then got some prelim work done on my Focus on the Learner assignment.

TP 3
Taught a lesson on laughter. Got the students laughing, and received positive feedback from my colleagues and tutor. Biggest thing I need to work-on is eliciting answers/responses from the ss, b/c I talk too much on stage. Have a touch of dancing bear going right now, need to get them speaking.

@ TH
Afternoon on CCQs and phonology was engaging and helpful.

HW
Did some writing of my Focus on the Learner assignment on the train Friday and Saturday, then finished it Sunday afternoon/evening. Created a how-to document to explain an online exercise for a student's pronunciation, and published it on Scribd here.


text:
Talk:Motivation and emotion/Book/2010/Stress and health

-- Jtneill - Talk - c 06:21, 14 December 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User:Egm6321.f10.team2.reserve/HW7

= Problem 7.4 =

From the lecture slide Mtg 38-3

Given

 * {| style="width:100%" border="0"

$$ \displaystyle

f(\theta )={{T}_{0}}{{\cos }^{6}}\theta $$

(Eq 4.1)
 *
 * }

 * {| style="width:100%" border="0"

$$ \displaystyle

f(\theta )={{T}_{0}}\exp \left[ -{{(\frac{2\theta }{\pi })}^{2}} \right]$$

(Eq 4.2)
 *
 * }

Find
i) Without calculation find property of $$\displaystyle {{A}_{n}}$$

 * $$\displaystyle \left. \begin{align}

& {{A}_{2k}}=0 \\ & {{A}_{2k+1}}=0 \\ \end{align} \right\}k=0,1,2,3...$$

ii) Evaluate non-zero coefficients.

Part a

 * $$\displaystyle f(\mu )={{T}_{0}}{{(1-{{\mu }^{2}})}^{3}}$$

where $$\displaystyle \mu =\sin \theta $$.

When $$\displaystyle n=2k+1$$ and when $$n\displaystyle \ge 7$$, we have,

 * $$\displaystyle \left. \begin{align}

& k=0\Rightarrow n=1 \\ & k=1\Rightarrow n=3 \\ & k=2\Rightarrow n=5 \\ & k=3\Rightarrow n=7 \\ \end{align} \right\}{{A}_{n}}=0$$

Then for evaluating non-zero coefficients.

For simplicity setting $$\displaystyle \mu =x$$ we get $$\displaystyle \mu =x$$.

Therefore we derive coefficients using WolframAlpha ,

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

\begin{align} & {{A}_{0}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{0}}(x)dx}=\frac{1}{2}\int\limits_{-1}^{1}{{{T}_{0}}{{(1-{{x}^{2}})}^{3}}*1*dx=}0.457{{T}_{0}} \\ & {{A}_{2}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{2}}(x)dx}=\frac{5}{2}\int\limits_{-1}^{1}{{{T}_{0}}{{(1-{{x}^{2}})}^{3}}*(\frac{3{{x}^{2}}}{2}-\frac{1}{2})*dx}=-0.762{{T}_{0}} \\ & {{A}_{4}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{4}}(x)dx=}\frac{9}{2}\int\limits_{-1}^{1}{{{T}_{0}}{{(1-{{x}^{2}})}^{3}}*(\frac{35{{x}^{4}}}{8}-\frac{15{{x}^{2}}}{4}+\frac{3}{8})*dx}=0.374{{T}_{0}} \\ & {{A}_{6}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{0}}(x)dx}=\frac{13}{2}\int\limits_{-1}^{1}{{{T}_{0}}{{(1-{{x}^{2}})}^{3}}*(\frac{231{{x}^{6}}}{16}-\frac{315{{x}^{4}}}{16}+\frac{105{{x}^{2}}}{16}-\frac{5}{16})*dx}=0.069{{T}_{0}} \\ \end{align}$$

(Eqs 4.3)
 *
 * }

Part b

 * $$\displaystyle f(\theta )={{T}_{0}}\exp \left[ -{{(\frac{2\theta }{\pi })}^{2}} \right]$$

Then,

 * $$\displaystyle f(\mu )={{T}_{0}}\exp \left[ -{{(\frac{2\mu }{\pi })}^{2}} \right]={{T}_{0}}\exp \left[ -(\frac{4{{\mu }^{2}}}) \right]$$

where $$\displaystyle \mu =\sin \theta $$.

Since f(x) is an even function for k=0,1,2,3… we get $$\displaystyle {{A}_{2k}}$$.

As to evaluating non-zero coefficients we will repeat the same procedure by setting $$\displaystyle \mu =x$$ to get $$\displaystyle f(x)={{T}_{0}}\exp \left[ -(\frac{4{{x}^{2}}}) \right]$$.

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$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

\begin{align} & {{A}_{0}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{0}}(x)dx}=\frac{1}{2}\int\limits_{-1}^{1}{{{T}_{0}}\exp \left[ -(\frac{4{{x}^{2}}}) \right]*1*dx=}1.173{{T}_{0}} \\ & {{A}_{2}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{2}}(x)dx}=\frac{5}{2}\int\limits_{-1}^{1}{{{T}_{0}}\exp \left[ -(\frac{4{{x}^{2}}}) \right]*(\frac{3{{x}^{2}}}{2}-\frac{1}{2})*dx}=-0.639{{T}_{0}} \\ & {{A}_{4}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{4}}(x)dx=}\frac{9}{2}\int\limits_{-1}^{1}{{{T}_{0}}\exp \left[ -(\frac{4{{x}^{2}}}) \right]*(\frac{35{{x}^{4}}}{8}-\frac{15{{x}^{2}}}{4}+\frac{3}{8})*dx}=0.714{{T}_{0}} \\ & {{A}_{6}}=\frac{2n+1}{2}\int\limits_{-1}^{1}{f(x){{P}_{0}}(x)dx}=\frac{13}{2}\int\limits_{-1}^{1}{{{T}_{0}}\exp \left[ -(\frac{4{{x}^{2}}}) \right]*(\frac{231{{x}^{6}}}{16}-\frac{315{{x}^{4}}}{16}+\frac{105{{x}^{2}}}{16}-\frac{5}{16})*dx}=-0.726{{T}_{0}} \\ \end{align}$$

(Eq 4.4)
 *
 * }

= Problem 7.5 Finding second homogenous solution of Legendre equations of order 0 and 1 =

From the lecture slide Mtg39-1

Given
The Legendre equation is,

 * $$\displaystyle (1-{{x}^{2}}){y}''-2x{y}'+n(n+1)y=0$$

with its first homogeneous solution with respect to order 0 and 1,

 * $$\displaystyle \begin{align}

& {{P}_{0}}(x)=1 \\ & {{P}_{1}}(x)=x \\ \end{align}$$

Find
Prove the second homogenous solutions of Legendre equation with respect to order 0 and 1 are,

 * $$\displaystyle \begin{align}

& {{Q}_{0}}(x)=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right)={{\tanh }^{-1}}x \\ & {{Q}_{1}}(x)=\frac{x}{2}\log \left( \frac{1+x}{1-x} \right)-1=x{{\tanh }^{-1}}x-1 \\ \end{align}$$

Solution
During the derivation of Legendre equation we assumed that $$\displaystyle x=\sin \theta $$ meaning $$\displaystyle x$$ has the domain of $$\displaystyle [-1,+1]$$ in which +1 and -1 are its singularity point.

Verification of $$\displaystyle {{Q}_{0}}(x)$$
When the order of Legendre equation equals 0,

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$$ \displaystyle

(1-{{x}^{2}}){y}''-2x{y}'=0$$

(Eq 5.1)
 *
 * }

Eq 5.1 can be immediately reduced to a linear first ODE with varing coefficients by taking $$\displaystyle {y}'=z$$,

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$$ \displaystyle

{z}'+\frac{2x}{{{x}^{2}}-1}z=0$$

(Eq 5.2)
 *
 * }

The integrating factor is,

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$$ \displaystyle

\begin{align} h(x) & =\exp \left[ \int_ – ^{x}{\frac{2x}{{{x}^{2}}-1}dx} \right] \\ & =\exp \left[ \int_ – ^{x}{\frac{1}{1-{{x}^{2}}}d(1-{{x}^{2}})} \right] \\ & =\exp [\log (1-{{x}^{2}})] \\ & =1-{{x}^{2}} \\ \end{align}$$

(Eq 5.3)
 *
 * }

Hence,

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$$ \displaystyle

\begin{align} z & ={{h}^{-1}}(x)\int_ – ^{x}{h(x)*0dx} \\ & =\frac{1-{{x}^{2}}} \\ \end{align}$$

(Eq 5.4)
 *
 * }

Then we can derive $$\displaystyle y$$,

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$$ \displaystyle

\begin{align} y & =\int{zdx}+{{k}_{1}} \\ & ={{k}_{1}}+\int_ – ^{}{\frac{1-{{x}^{2}}}dx} \\ & ={{k}_{1}}+\frac{2}\int_ – ^{}{\left( \frac{1}{1+x}+\frac{1}{1-x} \right)dx} \\ & ={{k}_{1}}+{{k}_{2}}\frac{1}{2}\log \left( \frac{1+x}{1-x} \right) \\ \end{align}$$

(Eq 5.5)
 *
 * }

Since $$\displaystyle {{P}_{0}}(x)$$ and $$\displaystyle {{Q}_{0}}(x)$$ are two homogeneous solutions of Eq 5.1, we can say,

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$$ \displaystyle

{{Q}_{0}}(x)=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right)$$

(Eq 5.6)
 *
 * }

On the other hand,

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$$ \displaystyle

\begin{align} & \tanh x=\frac{{{e}^{2x}}-1}{{{e}^{2x}}+1} \\ & \Rightarrow x=\frac{1}{2}\log \left( \frac{1+\tanh x}{1-\tanh x} \right) \\ & \Rightarrow {{\tanh }^{-1}}x=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right) \\ \end{align}$$

(Eq 5.7)
 *
 * }

Thus $$\displaystyle {{Q}_{0}}(x)$$ can also be written as

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

{{Q}_{0}}(x)=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right)={{\tanh }^{-1}}x$$

(Eq 5.8)
 *
 * }

Verification of $$\displaystyle {{Q}_{1}}(x)$$
When the order of Legendre is set to 1,

 * $$\displaystyle (1-{{x}^{2}}){y}''-2x{y}'+2y=0$$

or

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$$ \displaystyle

{y}''+\frac{2x}{({{x}^{2}}-1)}{y}'+\frac{2}{(1-{{x}^{2}})}y=0$$

(Eq 5.9)
 *
 * }

By direct method in Mtg29-3 and Mtg30-1, we derive,

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$$ \displaystyle

\begin{align} & h(x)={{u}_{1}}^{2}(x)\exp \left[ \int_ – ^ – {{{a}_{1}}(x)dx} \right] \\ & ={{[{{P}_{1}}(x)]}^{2}}\exp \left[ \int_ – ^ – {\frac{2x}{{{x}^{2}}-1}dx} \right] \\ & ={{x}^{2}}\exp \left[ \int_ – ^ – {\frac{1}{1-{{x}^{2}}}d(1-{{x}^{2}})} \right] \\ & ={{x}^{2}}(1-{{x}^{2}}) \\ \end{align}$$

(Eq 5.10)
 *
 * }

and then,

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$$ \displaystyle

\begin{align} {{Q}_{1}}(x) & ={{u}_{2}}(x) \\ & ={{u}_{1}}(x)\int_ – ^ – {\frac{1}{h(x)}dx} \\ & =x\int{\frac{1}{{{x}^{2}}(1-{{x}^{2}})}dx} \\ & =x\int{\frac{1}+\frac{1}{(1-{{x}^{2}})}dx} \\ & =x\int{\frac{1}{(1-{{x}^{2}})}dx+}x\int{\frac{1}dx} \\ & =\frac{x}{2}\log \left( \frac{1+x}{1-x} \right)-1 \\ \end{align}$$

(Eq 5.11)
 *
 * }

By Eq 5.7,

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$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

{{Q}_{1}}(x)=\frac{x}{2}\log \left( \frac{1+x}{1-x} \right)-1=x{{\tanh }^{-1}}x-1$$

(Eq 5.12)
 *
 * }

= Problem 7.6 The oddness of $$\displaystyle {{Q}_{n}}(x)$$ with respect to $$\displaystyle n$$ =

From the lecture slide Mtg 39-1

Given
In general the second catogary of Legendre functions have such expression,

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$$ \displaystyle

{{Q}_{n}}(x)={{P}_{n}}(x){{\tanh }^{-1}}(x)-2\sum\nolimits_{j=1,3,5\cdots }^{J}{\frac{2n-2j+1}{(2n-j+1)j}{{P}_{n-j}}(x)}$$

(Eq 6.1)
 *

 * }

where $$\displaystyle J=1+2\left[ \frac{n-1}{2} \right]$$.

Find
Prove that $$\displaystyle {{Q}_{n}}(x)$$ has oddness depending on the value $$\displaystyle n$$.

When $$\displaystyle n$$ is even
From problem 7.4 we have known that if $$\displaystyle n$$ is even then $$\displaystyle {{P}_{n}}(x)$$ is even and vice versa.

On the other hand $$\displaystyle {{\tanh }^{-1}}(x)$$ is always odd, then the first tern of Eq 6.1 is odd.

Since $$\displaystyle J=1+2\left[ \frac{n-1}{2} \right]$$, it's clear that $$\displaystyle J$$ is always odd.

Also $$\displaystyle j$$ is always odd making every $$\displaystyle n-j$$ in the summation odd.

Therefore every $$\displaystyle {{P}_{n-j}}(x)$$ in the summation is odd.

Drawn together, when $$\displaystyle n$$ is even, $$\displaystyle {{Q}_{n}}(x)$$ consists of several odd terms. By problem 7.2 we know that $$\displaystyle {{Q}_{n}}(x)$$ is eventually odd.

When $$\displaystyle n$$ is odd
Going the same process like above, we know that:

$$\displaystyle {{P}_{n}}(x)$$ is odd;

$$\displaystyle {{\tanh }^{-1}}(x)$$ is always odd;

the first tern of Eq 6.1 is even;

$$\displaystyle J$$ is always odd;

every $$\displaystyle n-j$$ in the summation even;

every $$\displaystyle {{P}_{n-j}}(x)$$ in the summation is even;

$$\displaystyle {{Q}_{n}}(x)$$ consists of several even terms thus it's even.

Conclusion
The oddness of $$\displaystyle {{Q}_{n}}(x)$$ depend on the value of $$\displaystyle n$$, and $$\displaystyle {{P}_{n}}(x)$$ and $$\displaystyle {{Q}_{n}}(x)$$ have opposote oddness.

= Problem 7.7 Distance Between Two Spheres =

From the lecture slide Mtg 40-3

Given
$$\displaystyle P\left( {{r}_{P}},{{\varphi }_{P}},{{\theta }_{P}} \right)$$ and $$\displaystyle Q\left( {{r}_{Q}},{{\varphi }_{Q}},{{\theta }_{Q}} \right)$$ for spherical coordinate system.

In Astronomical Convention we have,

 * $$\displaystyle \begin{align}

& x_{P}^{1}=r_{P}^ – \cos {{\theta }_{P}}\cos {{\varphi }_{P}} \\ & x_{P}^{2}=r_{P}^ – \cos {{\theta }_{P}}\sin {{\varphi }_{P}} \\ & x_{P}^{3}=r_{P}^ – \sin {{\theta }_{P}} \\ \end{align}$$

 * $$\displaystyle \begin{align}

& x_{Q}^{1}=r_{Q}^ – \cos {{\theta }_{Q}}\cos {{\varphi }_{Q}} \\ & x_{Q}^{2}=r_{Q}^ – \cos {{\theta }_{Q}}\sin {{\varphi }_{Q}} \\ & x_{Q}^{3}=r_{Q}^ – \sin {{\theta }_{Q}} \\ \end{align}$$

 * $$\displaystyle \cos \gamma =\cos {{\theta }_{Q}}\cos {{\theta }_{P}}\cos ({{\varphi }_{Q}}-{{\varphi }_{P}})+\sin {{\theta }_{Q}}\sin {{\theta }_{P}}$$

 * $$\displaystyle {{\left( r_{PQ}^ – \right)}^{2}}={{\left\| \overrightarrow{PQ} \right\|}^{2}}=\sum\limits_{i=1}^{3}={{\left( x_{Q}^{1}-x_{P}^{1} \right)}^{2}}+{{\left( x_{Q}^{2}-x_{P}^{2} \right)}^{2}}+{{\left( x_{Q}^{3}-x_{P}^{3} \right)}^{2}}$$

Find
Show that,

 * $$\displaystyle {{\left( r_{PQ}^ – \right)}^{2}}={{\left( r_{Q}^ –  \right)}^{2}}\left[ {{\left( \frac \right)}^{2}}+1-2\left( \frac \right)\cos \gamma  \right]$$

Solution
Since,

 * {| style="width:100%" border="0"

$$ \displaystyle

{{\left( r_{PQ}^ – \right)}^{2}}={{\left( x_{Q}^{1}-x_{P}^{1} \right)}^{2}}+{{\left( x_{Q}^{2}-x_{P}^{2} \right)}^{2}}+{{\left( x_{Q}^{3}-x_{P}^{3} \right)}^{2}}$$

(Eq 7.1)
 *

 * }

so we need to calculate each terms.

First Term:

 * $$\displaystyle {{\left( x_{Q}^{1}-x_{P}^{1} \right)}^{2}}={{\left( r_{Q}^ – \cos {{\theta }_{Q}}\cos {{\varphi }_{Q}}-r_{P}^ – \cos {{\theta }_{P}}\cos {{\varphi }_{P}} \right)}^{2}}$$

 * $$\displaystyle {{\left( x_{Q}^{1}-x_{P}^{1} \right)}^{2}}=r_{Q}^{2}{{\cos }^{2}}{{\theta }_{Q}}{{\cos }^{2}}{{\varphi }_{Q}}+r_{P}^{2}{{\cos }^{2}}{{\theta }_{P}}{{\cos }^{2}}{{\varphi }_{P}}-2r_{Q}^ – r_{P}^ – \cos {{\theta }_{Q}}\cos {{\varphi }_{Q}}\cos {{\theta }_{P}}\cos {{\varphi }_{P}}$$

Second Term:

 * $$\displaystyle {{\left( x_{Q}^{2}-x_{P}^{2} \right)}^{2}}={{\left( r_{Q}^ – \cos {{\theta }_{Q}}\sin {{\varphi }_{Q}}-r_{P}^ – \cos {{\theta }_{P}}\sin {{\varphi }_{P}} \right)}^{2}}$$

 * $$\displaystyle {{\left( x_{Q}^{2}-x_{P}^{2} \right)}^{2}}=r_{Q}^{2}{{\cos }^{2}}{{\theta }_{Q}}{{\sin }^{2}}{{\varphi }_{Q}}+r_{P}^{2}{{\cos }^{2}}{{\theta }_{P}}{{\sin }^{2}}{{\varphi }_{P}}-2r_{Q}^ – r_{P}^ – \cos {{\theta }_{Q}}\sin {{\varphi }_{Q}}\cos {{\theta }_{P}}\sin {{\varphi }_{P}}$$

Third Term:

 * $$\displaystyle {{\left( x_{Q}^{3}-x_{P}^{3} \right)}^{2}}={{\left( r_{Q}^ – \sin {{\theta }_{Q}}-r_{P}^ – \sin {{\theta }_{P}} \right)}^{2}}$$

 * $$\displaystyle {{\left( x_{Q}^{3}-x_{P}^{3} \right)}^{2}}=r_{Q}^{2}{{\sin }^{2}}{{\theta }_{Q}}+r_{P}^{2}{{\sin }^{2}}{{\theta }_{P}}-2\sin {{\theta }_{Q}}\sin {{\theta }_{P}}$$

Add all three terms together, then we get

 * {| style="width:100%" border="0"

$$ \displaystyle

\begin{align} {{\left( r_{PQ}^ – \right)}^{2}}= & r_{Q}^{2}\left[ {{\cos }^{2}}{{\theta }_{Q}}\left( {{\cos }^{2}}{{\varphi }_{Q}}+{{\sin }^{2}}{{\varphi }_{Q}} \right)+{{\sin }^{2}}{{\theta }_{Q}} \right]+r_{P}^{2}\left[ {{\cos }^{2}}{{\theta }_{P}}\left( {{\cos }^{2}}{{\varphi }_{P}}+{{\sin }^{2}}{{\varphi }_{P}} \right)+{{\sin }^{2}}{{\theta }_{P}} \right] \\ & -2r_{Q}^ – r_{P}^ – \left[ \cos {{\theta }_{Q}}\cos {{\theta }_{P}}\left( \cos {{\varphi }_{Q}}\cos {{\varphi }_{P}}+\sin {{\varphi }_{Q}}\sin {{\varphi }_{P}} \right)+\sin {{\theta }_{Q}}\sin {{\theta }_{P}} \right] \\ \end{align}$$

(Eq 7.2)
 *

 * }

From trigonometric equations, we know that

 * $$\displaystyle \left( {{\cos }^{2}}{{\varphi }_{Q}}+{{\sin }^{2}}{{\varphi }_{Q}} \right)=1$$

 * $$\displaystyle \left( {{\cos }^{2}}{{\varphi }_{P}}+{{\sin }^{2}}{{\varphi }_{P}} \right)=1$$

 * $$\displaystyle \left[ {{\cos }^{2}}{{\theta }_{Q}}+{{\sin }^{2}}{{\theta }_{Q}} \right]=1$$

 * $$\displaystyle \left[ {{\cos }^{2}}{{\theta }_{P}}+{{\sin }^{2}}{{\theta }_{P}} \right]=1$$

 * $$\displaystyle \left( \cos {{\varphi }_{Q}}\cos {{\varphi }_{P}}+\sin {{\varphi }_{Q}}\sin {{\varphi }_{P}} \right)=\cos \left( {{\varphi }_{Q}}-{{\varphi }_{P}} \right)$$

So putting equations above in the $$\displaystyle {{\left( r_{PQ}^ – \right)}^{2}}$$, we obtain

 * {| style="width:100%" border="0"

$$ \displaystyle

\begin{align} {{\left( r_{PQ}^ – \right)}^{2}}= & r_{Q}^{2}+r_{P}^{2}-2r_{Q}^ – r_{P}^ – \left[ \cos {{\theta }_{Q}}\cos {{\theta }_{P}}\left( \cos ({{\varphi }_{Q}}-{{\varphi }_{P}}) \right)+\sin {{\theta }_{Q}}\sin {{\theta }_{P}} \right] \\ & =r_{Q}^{2}+r_{P}^{2}-2r_{Q}^ – r_{P}^ – \cos \gamma \\ \end{align}$$

(Eq 7.3)
 *

 * }

If we take common parenthesis of $$\displaystyle {{\left( r_{Q}^ – \right)}^{2}}$$, it will be shown

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

{{\left( r_{PQ}^ – \right)}^{2}}={{\left( r_{Q}^ –  \right)}^{2}}\left[ {{\left( \frac \right)}^{2}}+1-2\left( \frac \right)\cos \gamma  \right]$$

(Eq 7.4)
 *

 * }

= Problem 7.9 Deriving Legendre polynominals using recurrence relation =

From the lecture slide Mtg 41-2

Given
The second recurrence relation of Legendre polynominals is,

 * {| style="width:100%" border="0"

$$ \displaystyle

(n+1){{P}_{n+1}}-(2n+1)x{{P}_{n}}+n{{P}_{n-1}}=0$$

(Eq 9.1)
 *

 * }

And Legendre polynominal of order 0 and 1 are,

 * $$\displaystyle \begin{align}

& {{P}_{0}}(x)=1 \\ & {{P}_{1}}(x)=x \\ \end{align}$$

Find
Legendre polynominals of order 2,3,4,5,6.

Solution
From Eq 9.1 we have,

 * {| style="width:100%" border="0"

$$ \displaystyle

{{P}_{n+1}}=\frac{2n+1}{n+1}x{{P}_{n}}-\frac{n}{n+1}{{P}_{n-1}}$$

(Eq 9.2)
 *

 * }

Take $$\displaystyle n$$ as $$\displaystyle 2,3,4,5,6$$ we have,

 * {| style="width:100%" border="0"

$$ \displaystyle

\begin{align} & {{P}_{2}}=\frac{3}{2}x{{P}_{1}}-\frac{1}{2}{{P}_{0}} \\ & {{P}_{3}}=\frac{5}{3}x{{P}_{2}}-\frac{2}{3}{{P}_{1}} \\ & {{P}_{4}}=\frac{7}{4}x{{P}_{3}}-\frac{3}{4}{{P}_{2}} \\ & {{P}_{5}}=\frac{9}{5}x{{P}_{4}}-\frac{4}{5}{{P}_{3}} \\ & {{P}_{6}}=\frac{11}{6}x{{P}_{5}}-\frac{5}{6}{{P}_{4}} \\ \end{align}$$

(Eqs 9.3)
 *

 * }

Solve Eqs 9.3 by sequence we have,

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:90%; padding:10px; border:2px solid #8888aa" |
 * style="width:90%; padding:10px; border:2px solid #8888aa" |

\begin{align} & {{P}_{2}}=\frac{3}{2}x*x-\frac{1}{2}*1=\frac{3}{2}{{x}^{2}}-\frac{1}{2} \\ & {{P}_{3}}=\frac{5}{3}x\left( \frac{3}{2}{{x}^{2}}-\frac{1}{2} \right)-\frac{2}{3}x=\frac{5}{2}{{x}^{3}}-\frac{3}{2}x \\ & {{P}_{4}}=\frac{7}{4}x\left( \frac{5}{2}{{x}^{3}}-\frac{3}{2}x \right)-\frac{3}{4}\left( \frac{3}{2}{{x}^{2}}-\frac{1}{2} \right)=\frac{35}{8}{{x}^{4}}-\frac{15}{4}{{x}^{2}}+\frac{3}{8} \\ & {{P}_{5}}=\frac{9}{5}x\left( \frac{35}{8}{{x}^{4}}-\frac{15}{4}{{x}^{2}}+\frac{3}{8} \right)-\frac{4}{5}\left( \frac{5}{2}{{x}^{3}}-\frac{3}{2}x \right)=\frac{63}{8}{{x}^{5}}-\frac{35}{4}{{x}^{3}}+\frac{15}{8}x \\ & {{P}_{6}}=\frac{11}{6}x\left( \frac{63}{8}{{x}^{5}}-\frac{35}{4}{{x}^{3}}+\frac{15}{8}x \right)-\frac{5}{6}\left( \frac{35}{8}{{x}^{4}}-\frac{15}{4}{{x}^{2}}+\frac{3}{8} \right)=\frac{231}{16}{{x}^{6}}-\frac{315}{16}{{x}^{4}}+\frac{105}{16}{{x}^{2}}-5 \\ \end{align}$$

(Eq 9.4)
 *

 * }

= References =

title: User:Egm6321.f10.team2.reserve/HW7


text:
User talk:Pibeagles

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title: User talk:Pibeagles


text:
User:Egm6321.f10.team6.Kim.MK/hw7

= Show the character of Even and Odd Functions (Problem 7.2)=

From class note lecture 38_1

Problem Statement
Let

Required
Show that: 1. if the $$ \begin{align} \{g_i\} \end{align}$$ is odd, then $$ \begin{align} f \end{align}$$ is odd.

2. if the $$ \begin{align} \{g_i\} \end{align}$$ is even, then $$ \begin{align} f \end{align}$$ is even.

==Solution ==

1. The definition of an odd function is that $$\begin{align} f(x) = -f(-x) \end{align}$$.

If $$\begin{align}\sum_{i=0}^{n} g_i(x)\end{align}$$ is odd function,

 Odd functions, $$\begin{align} y = x^{2n+1}\end{align}$$

$$\begin{align} y= sin (nx) \end{align}$$

2. The definition of an even function is that $$\displaystyle f(x) = f(-x)$$.

If $$\begin{align}\sum_{i=0}^{n} g_i(x)\end{align}$$ is even function,  Even functions, $$\begin{align} y = x^{2n}\end{align}$$

$$\begin{align} y= cos (nx) \end{align}$$

Problem Statement
We have recurrence relations that

Binomial Theorem :

Thus,

Required
Continue power series expansion to find $$\begin{align}{P_3, P_4 , P_5 , P_6 } \end{align}$$, and compare the result to that obtained by a) (7) & (8) class note lecture 36_2

b) the result of HW 7.9

a)
We need the first 7 terms of the expansion

Using (10.5), we get

Therefore, we have confirmed two results are equivalent.

b)
Problem 9 result

title: User:Egm6321.f10.team6.Kim.MK/hw7


text:
User:Fedosin

Academic degrees
Specialist degree, Radiophysics, Perm State University, 1978.

Perm State National Research University is a main University in the middle of Russia, which is known by its fundamental educations.

Professional experience
September 1978 – June 1986

Scientific Researcher of Natural Science Institute (ENI) at Perm State University. Natural Science Institute (ENI) is a multifunctional center of science and research at Perm State University; it has more then 10 laboratories (physics, chemistry, biology, physical chemistry, radiobiology and so on). At ENI, I worked in the laboratory of organic semiconductors. The chemists of the laboratory synthesized some new semiconductors and physicists found their properties. My first task was constructing of special techniques for measurement of Hall effects in organic semiconducting crystals. The accuracy of the equipment after many improvements was about 10 nV, so I used it for my research of Hall effects and conductivity of crystals and powder of the new materials.

May 2000 – September 2003

Scientific Researcher (Laboratory of Radiospectroscopy at Perm State University). During my work in the Laboratory I was a team member in two projects: «Technology for generation of diamonds in graphite by electromagnetic forces», and «Apparatus for demonstration of ball lightning». Both projects were on the base of research grants of Department of science Administration of Perm Region.

Memberships/Affiliations (current)
The editor of Perm scientific site, guest editor of Special Issue Time, Space and Matter: open questions and new perspectives for International Journal of Physics.

Awards
Student research work at Physical Technical Institute by A.F. Ioffe (USSR Academy of Sciences, Sankt-Peterburg), 1977 – 1978. Award of Perm State University competition, 1987 (Powerful source of controlling electric current). Honorary Citizen of Louisville, USA (since 1995). Author of three patents. Who's Who in the World nomination.

Research interests

 * Philosophical analysis.

Study of physical phenomena and then using the results of operations should include the philosophical component. This ensures maximum return on applied research methods and provides an additional synergistic effect. Development of new ideas is impossible without a deeper philosophy.

 * Relativity theory.

The primary goal of any physical theory is the description of phenomena in the framework of a system based on the knowledge of these phenomena in other similar systems. In common relativity describes the transformation of relations of physical quantities from one system to another. This is the basis for modeling of phenomena, the application of similarity theory, networking and laws. Analysis of the relativity of the device work provides important directions of research in technology.

 * Fundamental forces.

It is well known that on a macro scale is the main force of gravitation, the transition to the atoms and molecules is dominated by electromagnetic forces, and at the level of elementary particles are nuclear forces. Description of these forces is based on mathematical formulas derived from the symmetry of interactions. But it is equally important to build physical models of the substantial interactions, which allows understanding the true causes of forces. This may be of significant assistance in applied research, from the ordering of electrons in superconductivity, and ending with the ordering of molecules in liquid crystals.

 * Infinite hierarchical nesting of matter.

In this theory, matter is divided into different levels so that between them it is possible find the relation of similarity. This allows assessment of the values of physical quantities in various systems, using only a few basic characteristics of these systems. This approach complements the method of describing systems based on the use of natural physical units.

Books
The theory of similarity between the atomic and stellar systems is described in the first book. The book presents the theory of Infinite Hierarchical Nesting of Matter and the Lorentz-invariant theory of gravitation (LITG); there is also an additional article about LITG.

A relation is discovered between the mass and binding energy of space objects, corresponding to the Einstein formula (equivalence of mass and energy); discreteness of stellar parameters and quantization of parameters of cosmic systems are revealed; stellar Planck, Dirac, Boltzmann and other stellar constants are determined; combined SPФ symmetry with respect to similarity of physical processes at different scale levels of matter is introduced. For the first time a mathematically precise derivation of the Newton formula of emergence of the gravitation force is provided based on the concept of gravitons; the energy density and penetrability of gravitons in the matter are found. This approach is further developed in article within the framework of Le Sage’s theory of gravitation. On the basis of LITG equations it becomes possible experimental determination of the speed of gravity, the study of gravitational torsion field, the gravitational induction, the gravitational shielding and the gravitoelectromagnetism. The analysis of thermodynamics using the theory of relativity is done, the effective reaction force in the principle of Le Chatelier - Brown is determined, a new fourth energy definition of entropy is presented.

The second book describes the contemporary problems of physics. The most important results are: creation of the extended special theory of relativity in new axiomatics; the analysis of relationship between LITG and the general theory of relativity; the study of the role of relativity in physical theories (see also ); calculation of the angular momentum and radius of proton, representation of the original electron-ionic model of ball lightning (see also the article ).

The philosophy of carriers is described in third book. In this book the following basic problems are solved. Firstly, new philosophical logic called “syncretics” is developed. The discovery was made – all philosophical categories together constitute a mathematical group and have the group properties. As a result, instead of the three previously known philosophical laws (unity and struggle of opposites, transformation of quantity into quality, negation of negation) it becomes possible to formulate new philosophical laws. Among these new laws we could mention the law of conservation and change in the system’s organization, the law of the existence flow, the law of multiplication of structures and many others (a total of 27 new laws). Secondly, a new philosophical system is created – the philosophy of carriers. The basis of this philosophy are five postulates of which all the theory is deduced (similarly the Euclidean geometry has 5 postulates). Thirdly, based on the field equations a theorem is proved that relates the energy fluxes of various kinds in the space volume with the laws of conservation of momentum, energy and angular momentum.

The origin of life was further developed in the fourth monograph. On the basis of a large amount of factual material has been proven that the masses and lengths of living beings faithfully replicate the masses and lengths of the non-living carriars. Thus, the same similarity relations were true not only between different spatial levels of organization of matter, but also between the levels of living carriers of their masses in the hundreds of tons (whales) and up to a mass comparable to the mass of individual atoms (prions). As a result the philosophical conclusion follows about parallel co-existence in the nature of live and lifeless as two philosophical opposites, are jointly organizing the whole nature around them. With respect to the Earth it is proved by the fact that it is biological evolution over billions of years radically transformed the geochemistry of the ocean, the earth's surface and the atmosphere. As a result, the question of the origin of life (from nonliving matter?) is converted to a question about the ways in which life on the lowest scale level has the opportunity to grow and move to the higher scale levels of matter.

One of the results of fifth book is creation of the covariant theory of gravitation (CTG), which is based on the extended special theory of relativity, Lorentz-invariant theory of gravitation, Maxwell-like gravitational equations, metric theory of relativity (MTR) and Le Sage's theory of gravitation. And MTR is formulated on the axiomatic basis, developing the ideas of special and general theories of relativity, which allows us to distinguish between the general relativity and gravitation theory as such. With the help of the language of vectors and tensors electrodynamics and gravitation theory (in the form of LITG and CTG) are axiomatized. Besides the field equations of Maxwell type, the similar equations for the matter are derived, which allowed axiomatizing the theory of matter as well. Using the operator of proper-time-derivative the expression for the four-force density is found and it is shown that the equation of motion of general relativity theory is a special case of the equation of motion of CTG.

Based on the idea of strong gravitation, torsion field and electromagnetic forces in the gravitational model of strong interaction, the nuclear forces responsible for the integrity of nuclei and elementary particles themselves are explained. Besides, the strong gravitational constant is introduced and the stability of electron, proton and atoms in the strong gravitational field and in the proper electric field is calculated. On the basis of the theory of infinite nesting of matter the substantial electron model is constructed and the cause of its spin is derived. The energy definition is given in the tensor form and the meaning of entropy is established. The model of emergence and maintenance of inversely varying magnetic field in space objects is described.

The origin and the internal structure of ball lightning and bead lightning is explained in the same way. A model of appearance and maintaining of inversely varying magnetic field in a rather massive space objects like the Earth and the Sun is described. The arguments against the existence of black holes are presented.

In August 2021, the book “Horizons in World Physics”, Volume 306, was published, in which the scalar and vector potentials, electric and magnetic fields inside and outside a charged cylinder, a spherical system of particles, and also in a relativistic uniform system at rest and in a state of rotation were calculated. When turning from a classical to a relativistic uniform system, a method is used to calculate the internal field potentials with the help of the gauge function that satisfies the Laplace equation. The external electric potential and the field strength are found by separating variables using Legendre polynomials. Separation of variables for calculating the external vector potential and the magnetic field leads to the need to introduce new polynomials proportional to the sine of the spherical zenith angle. The first seven such polynomials are calculated, which are sufficient to find the vector potential in the quadrupole approximation.

Comments
The theory of infinite nesting of matter implies the existence in the Universe of new particles – nuons, the number of which is comparable with the total number of nucleons. At the stellar level the analogues of nuons are the white dwarfs, the number of which exceeds the number of neutron stars. Introduction of nuons allows explaining the redshift and background radiation effects, solving the problem of invisible dark matter and dark energy, and understanding the effect of attenuation of radiation from distant supernovae as a result of photons’ scattering on the nuons.

The purpose of the comments to fifth book is to clarify the following issues:

 * 1) What determines the mass to charge ratio of proton and electron?
 * 2) What is the relationship between the weak, electromagnetic, strong and gravitational interactions; between neutrinos, photons and gravitons?
 * 3) Can the gravitational torsion field be the cause of formation of stationary planetary orbits in the processes of accumulation of matter at the early stages of emerging of protoplanetary disks around stars?
 * 4) What parameters define the relationship between mass and radius of nucleon?
 * 5) At what minimum distances equilibrium of nucleons is achieved under the influence of strong gravitational field and torsion field in the deuteron and in atoms?
 * 6) What transformations of elementary particles’ matter is the emergence of the states of vector W- and Z-bosons, t-quark, tau lepton in accelerator experiments associated with?
 * 7) How can the structure of hadrons and their interaction with each other be explained in the model of quark quasiparticles and based on the fundamental forces without using the theory of quarks?
 * 8) Can we consider that the law of matter amount conservation does not contradict the law of conservation of energy-momentum of the special theory of relativity?
 * 9) How can the neutron substantial model and proton substantial model be grounded?
 * 10) Is there a relationship between the electric and magnetic proton polarizabilities which is not associated with the idea of quarks?
 * 11) How do strong gravitation, torsion fields and electromagnetic fields balance and mutually complement each other in the atoms?
 * 12) What are the dimensions and shape of the electron cloud in the simplest atoms?
 * 13) What is the cause of stationary states and quantization of the energy levels, angular momenta and magnetic moments of electrons in atoms?
 * 14) How can the relations for mass, radius, angular velocity of rotation, rest energy and magnetic moment in the self-consistent model of proton with non-uniform distribution of mass and charge be correlated with each other?
 * 15) Does the law of equipartition of energy fluxes between the matter and field hold during the nucleon rotation?
 * 16) What is the most general expression for the force in physics?
 * 17) Why do muons act like electrons in the atom?
 * 18) Can strong gravitation be used to explain the cold fusion of atomic nuclei?
 * 19) How can the general theory of relativity (GTR) be axiomatized so as to divide the system of axioms into two parts, one of which corresponds to general relativity of phenomena in different frames of reference, and the other corresponds to the theory of the gravitational field as such? Which parts of GTR can be derived from the covariant theory of gravitation ?
 * 20) What shape do the Liénard–Wiechert gravitational potentials have for a material point which at the initial time is not located at the origin of coordinates? How can we using the Liénard–Wiechert potentials of individual points and the superposition principle of potentials accurately calculate the relativistic potentials inside and outside the massive ball?

Other publications
Fundamental long-range fields, which include gravitational and electromagnetic fields, are associated with any objects in the Universe and are considered as the consequence of Le Sage's theory of gravitation. Reduction of gravitational forces and electromagnetic forces to the action of graviton fluxes (particles with the properties of photons and neutrinos) and to the fluxes of smallest charged particles (praons), filling the electrogravitational vacuum, allows us to define more exactly the concept of body mass as a measure of its inertia in the fluxes of gravitons and charged particles (inertial mass is manifested through resistance to any external force that creates acceleration). If in some reference frame these fluxes are compensated, then the body is either moving inertially and has inertial mass at a given constant velocity, or it is motionless and has a rest mass. The relationship between the energy of fundamental fields and the corresponding body mass (as well as the gravitational mass) is described in the article. According to the article, it is the charge component of the force field of the electrogravitational vacuum in the form of fluxes of charged particles within the framework of Le Sage's theory of gravitation, which is mainly responsible for both electromagnetic and gravitational interactions, as well as for the action of other fields inside bodies.

In case of a weak field it is proved that the invariant energy of a massive body is not just the sum of non-gravitational and gravitational energies of the body according to the general theory of relativity, but also is equal to the difference between the matter energy, as the sum of masses of individual baryons of this matter, and the sum of other energies, including the energy of fields and the internal kinetic energy of the matter. For the total energy of the massive body and its mass the expression is derived, which includes only the energy of fundamental fields and the energy of strong gravitation.

As a consequence, the content of the principle of mass and energy equivalence is specified. In the theory of infinite nesting of matter the objects of higher levels of matter consist of the objects of lower levels of matter. If we take a set of objects of a certain level of matter and make up more massive objects of them, then the heavier the object will be, the more its mass will differ from the sum of masses of the original objects. According to the standard opinion, decrease in mass should take place mainly due to contribution of the negative mass-energy of the gravitational field, holding together the matter of massive objects. However, from a philosophical point of view, another possibility can be admitted – the relative mass of objects can increase as we move to higher levels of matter. Such a possibility is admitted in an article, which examines the relationship between mass and energy in a variety of cases, including heating of bodies, fusion of atomic nuclei, as well as analyzes the findings of the general theory of relativity and covariant theory of gravitation. The next article, in which the covariant theory of gravitation is derived from the principle of least action, again confirms the increase in mass of bodies in comparison with the calibration mass-energy of particles of the system due to the contribution of the gravitational energy. In the article the meaning of the cosmological constant and the form of gravitational stress-energy tensor are found. The problem of the relationship between the masses of the entire system and its constituent parts is solved in the articles, where five types of masses of the physical system are defined and it is shown that the mass density of objects at higher levels of matter decreases in comparison with the mass density of bodies that make up these objects.

In article, taking into account the Lagrangian function and the principle of least action, the analysis of the covariant theory of gravitation leads to the Euler-Lagrange equations and to the Hamiltonian function. The latter is expressed in terms of the three-dimensional generalized momentum in an explicit form, as well is determined in terms of the 4-velocity, scalar potentials and strengths of gravitational and electromagnetic fields, taking into account the metric. Such notions are introduced into consideration as 4-dimensional generalized velocity and Hamiltonian 4-vector, and the problem of body mass is again considered. In order to describe the properties of mass three different masses are introduced: one of them is related to the rest energy by Einstein's formula, another is the observable mass, and the third mass is determined from the condition of absence of the energy of macroscopic fields in the matter. It is shown that the action function has a physical meaning as the function that describes the change of intrinsic properties, such as the rate of proper time flow and the increase rate of the phase angle of periodic processes.

Article makes important changes to the theory of gravitation. Based on the fundamental principles, 4-potentials of the acceleration field and pressure field are introduced into the theory. On the basis of these 4-potentials such tensors are constructed in a relativistically covariant way, as the acceleration tensor and pressure field tensor, as well as the acceleration stress-energy tensor and pressure stress-energy tensor. So, the tensors are found that previously were derived phenomenologically and only approximately described the energy-momentum of matter and pressure. The equations of the acceleration and pressure fields are similar in form to the Maxwell equations. Adding the 4-potentials of gravitational and electromagnetic fields and tensor invariants of these fields to the Lagrangian allows us to find the Hamiltonian, i.e. the relativistic energy of the system of multitude of particles and fields. Interpretation of the cosmological constant is given as the energy density of the particles at rest at infinity located at a distance from each other. This allows us to uniquely express the energy and momentum of the system, and to ultimately simplify the equation for the metric. The solution of this equation is obtained and the components of the metric tensor are calculated for the case of a massive and electrically charged body, outside and inside the body.

The article contains the relationship between the field coefficients and dependences of the scalar curvature and the cosmological constant in the matter as functions of the parameters of typical particles and field potentials. Besides, comparison of the cosmological constants inside a proton, a neutron star and in the observable Universe allows us to explain the problem of the cosmological constant arising in the Lambda-CDM model.

The 4/3 problem, according to which the field mass found through the field energy is not equal to the field mass determined through the field momentum, is solved in article and more precisely in the article. It is shown, that in a moving body the excess mass-energy of the gravitational and electromagnetic fields is compensated by the lack of mass-energy of the acceleration and pressure fields. Moreover, for a fixed and a moving body in the form of a sphere the total energy and momentum of all the four fields inside of the body are equal to zero.

In order to describe the dissipation of the kinetic energy of matter fluxes in a viscous medium, the 4-potential of the dissipation field, dissipation field tensor and dissipation stress-energy tensor are considered in the field theory. Further application of the principle of least action allows us to find the dissipation field equations and the equations of motion of matter particles, which are equivalent to the Navier-Stokes equations. Thus, these equations are derived in a covariant form, which also allows us to determine the metric inside the viscous medium and its energy.

Unification of the equations of various fields acting in the matter can be carried out using the concept of general field and the standard procedure for finding the stress-energy tensor and equations of vector fields of any type. As a result we find out that the electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field and other vector fields, are the components of a single general field. Each particular field is relatively independent of the other fields in equilibrium state, when the process of energy exchange between the fields and particles is completed. In equilibrium, particular fields are described by equations that have the same form for all the fields, including the equation of motion.

One result of the concept of the general field is the model of gravitational equilibrium, which allows estimating the parameters of cosmic bodies using simple formulas. These formulas are based only on the field equations, unlike formulas of polytropic model originating from the estimated equation of state in polytropic process. Another result is clarification of the virial theorem — in the model under consideration the energy, associated with the forces acting on the particles, is approximately 5/3 times greater than the kinetic energy of the particles, whereas in the classical approach this ratio is equal to 2. An analysis of the integral theorem of generalized virial allows us to find formula for root-mean-square speed of typical particles of the system, without using the notion of temperature. The relation between the theorem and the cosmological constant, characterizing the physical system under consideration, is shown. The difference is explained between the kinetic energy and the energy of motion, the value of which is equal to half the sum of the Lagrangian and the Hamiltonian.

Also the relativistic uniform system is described in the simplest way in the concept of the general field. The potentials of the acceleration field and the pressure field are calculated in the article for a rotating uniform system and a relativistic relationship connecting the pressure, particle velocity and density is found. In the limit of low velocities, this relationship turns into the standard formula of the kinetic theory of gases. Electromagnetic potentials and fields of a rotating uniform system are calculated in the articles.

The idea of strong gravitation is studied in many articles. Further development of this idea is carried out in article, where it is used to calculate the angular momentum and the radius of the proton. The exact calculation of the proton radius becomes possible in the self-consistent model, where the explanation of emergence of De Broglie wavelength is also presented, as a consequence of the Lorentz transformations applied to the standing waves inside the elementary particles.

In the article it is shown that, according to the metric theory of relativity, the description of events can be made using not only electromagnetic but also gravitational wave, which follows from the equation of similarity of these fields. The article presents the concept of the scale dimension as a generalization of the four-dimensional spacetime to the fifth dimension.

In the book and in the article the electrokinetic model of generation of magnetic fields in cosmic objects is introduced based on the assumption of separation of the charges in matter of cosmic bodies under pressure and high temperature. The difference of equations of motion in the covariant theory of gravitation and in the general theory of relativity is used for explanation of Pioneer anomaly in the article.

Assuming that the photon consists of charged particles of vacuum field (praons), the substantial model of the photon is built. According to the theory of infinite nesting of matter, flows of praons generate electromagnetic forces between the charges and can explain the Coulomb's law, and the charge and mass of praons can be calculated using the coefficients of similarity. Praons are part of nucleons and leptons matter as well as nucleons are the basis of neutron star matter and ordinary stars and planets. Motion of praons inside of a photon leads to the dipole magnetic moment and a nonzero rest mass of the photon as a sum of the masses of all praons constituting the photon.

For vector fields, covariant expressions for the energy, momentum and angular momentum are derived, the angular momentum pseudotensor and the radius-vector of the system’s center of momentum are determined, the integral vector is calculated and the impossibility of interpreting it as a four-momentum of the system is shown, in contrast to how it is done in general theory of relativity. The formula for calculating the relativistic angular momentum is derived in the article.

Field energy theorem for vector fields was proved in the article. Unlike the virial theorem, the field energy theorem is applied not to the particles themselves, but to their fields. With the help of this theorem, the concepts of the kinetic and potential energies of the fields are introduced and the relations between them are found. Various forms of the covariant equation of motion of particles of matter are analyzed in the article. Using the example of electromagnetic field, which is a vector field, the field integral equations are derived in covariant form. A new theorem on the magnetic field of rotating charged bodies is proved in the article. . A description of all known covariant equations for vector fields is presented in the article equation of vector field.

Based on the similarity of properties of photons and money, and on the formula for the density of distribution of photon gas by energies, the corresponding mathematical formula for distribution of annual income per capita is obtained in the article.

The study of systems
A systematic approach is essential in the development of the systems theory and systems science in general. In the course of study of space systems in the theory of Infinite Hierarchical Nesting of Matter, the properties of the hydrogen system were determined and the idea of quantization of parameters of cosmic systems was developed. On the basis of the theorem of SPФ symmetry and the similarity of matter levels in the study of scale dimension, the following characteristics were derived, inherent in the space systems, from the smallest particles to metagalaxies and larger objects:
 * 1) The interpenetration of systems into each other, considered up to infinity.
 * 2) The distribution of cosmic objects by the matter levels, which are the steps in the infinite hierarchy of space systems, based on a geometric progression.
 * 3) The similarity of systems, including the similarity of shapes, sizes, masses, speeds of processes, equations of motion.
 * 4) The interaction of systems with each other as a mode of existence.
 * 5) Direct and indirect replication and reproduction of systems.
 * 6) The systems’ generation of material emissions in the form of fluxes of particles and field quanta, which result in total in formation of fundamental forces, acting on the systems at other levels.
 * 7) Distribution of the systems with living objects among the space systems according to the same laws that are characteristic of the systems with non-living matter (nesting of the living in the living; correspondence of the sizes and masses of living beings to the sizes and masses of space objects; location of living beings at the matter  levels according to geometric progression; recurrence of forms and modes of existence of the living at the similar scales levels; reproduction in the living objects of those fluxes and forces that form the living objects at a new level).

In the philosophy of carriers, presented in 2003, many philosophical laws were formulated as the laws that are valid for all types of systems. These include:
 * 1) The law of development of the system’s opposites.
 * 2) The law of unity and struggle of the system’s opposites.
 * 3) The law of conservation and variation of the system’s organization.
 * 4) The law of the extremum of the system’s organization.
 * 5) The law of similarity of carriers at different scale levels.
 * 6) The law of connection of the organization’s extrema and the motion fluxes.
 * 7) The law of connection of the organization and the existence flow.
 * 8) The law of conservation and variation of carriers.
 * 9) The law of development of carriers (systems).
 * 10) The law of mutually complementary components of the system.
 * 11) The law of reproduction of structures.
 * 12) The law of the essence’s expression.

In the course of formulating the laws, the philosophical definitions of closed system, self-contained system, isolated system and open system were given.

Syncretics is regarded as a general methodology of the systems study in philosophy and systems science (systemology). Syncretics is a multi-valued philosophical logic, generalizing the metaphysical and dialectical logics. This allows us with the help of syncretics to formulate the rules of operations, to find the required combinations of categories in principles and laws, and to apply a systematic approach to any science and field of knowledge. Syncretics and philosophy of carriers substantiate the theory of infinite nesting of matter from a philosophical standpoint.

Topics for future research
For researchers with the skills of numerical simulation of physical phenomena, the following topics are:
 * 1) Checking of the equations of acceleration field of stars in galaxies, based on the rotation curves of stars. Determination of constant $$~ \eta $$ of acceleration field.
 * 2) Checking of the gravitational field equations in covariant theory of gravitation, for the stars in galaxies and cosmology.
 * 3) The visualization of calculation results of Pioneer anomaly, and comparing of pictures of motion in time for different orbits in covariant theory of gravitation and in general relativity.
 * 4) The application of equations of general field for description of internal structure of stars and planets.
 * 5) The simulation of gravitational Newton interaction on the base of theory of graviton field. Supposing a sphere with numerous balls inside it, and a ball outside the sphere we can calculate the gravitational interaction of the ball with each ball inside the sphere and common force. The purpose is to check dependence of the force on the distance.

According to the gravitational mass of a body decreases with increasing of electric charge of the body, as a result of electric mass-energy contribution. This conclusion may be checked in an experiment.

title: User:Fedosin


text:
Portal talk:Arts/Learn/Selection

Hi, I am a professional commercial photographer, I wanted to give my input on this entry:

"Take your camera, be it a DSLR, a point and shoot camera, or your cell phone. Take ten photos a day for a week. Every day, select just one to keep. Every week, select one of those "finalist" photos to keep.

''This will help develop your selective eye for judging quality works. Drawing or any other medium can be reasonably substituted for photography.''"

I have to say; this is useless, if you take a photo and it happens to be good great, but this teaches you nothing about the process. You will never be able to duplicate the elements of the photo unless you know what went in to it. And that can not be done with a point and shoot or a cellphone, or a DSLR on auto.

This promotes leaving the quality of your shots to chance, and understanding nothing of what makes the photo good: What settings in relation to what lighting, in relation to what subject.

It works fine for learning composition, but the same can be said for looking at advertisements in a magazine.

I think some mention of changing and using manual settings, observing their effects, etc, has to be included for this advice to be at all productive or helpful.

Its all very nice, and sweet, but it doesn't teach much of anything.

title: Portal talk:Arts/Learn/Selection


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Talk:Distinction between Economic Growth and Economic Development

I think you scholars need to put more differences between economic growth and development.


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text: Motivation and emotion/Textbook/Motivation/Arousal/Are you stressed?


text: User:DILEEPSAGAR

title: User:DILEEPSAGAR


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Talk:Motivation and emotion/Textbook/Motivation/Arousal

-- Jtneill - Talk - c 01:22, 14 December 2010 (UTC)


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Ideas in Geometry/Symmetry Groups

Learning about Symmetry Groups

Symmetries

There are two different major types of symmetries that can be found in math: rotational and reflective.

Rotational Symmetry describes the different rotations that a shape can make, while still looking exactly the same.

There are four major facts that can be useful with rotational symmetry:

-	Rotations can be described in either fractions or degrees

-	Rotations can be either clockwise or counterclockwise

-	Many symmetrical shapes will have a rotation for every side of their shape

-	Every shape has a rotation that brings the shape all the way back around, which can be referred to as a full rotation or null rotation

The pictures below illustrate a few examples of rotational symmetry (arrows illustrate the sides that can be rotated):

Reflective Symmetry describes the different ways that a shape can be reflected / folded across itself, without any of the shape hanging over the edge.

There are some basic facts that can be useful pertaining to reflective symmetry:

-	It is possible for a shape to have no reflections

-	The lines on the drawings are referred to as lines of reflection

The pictures below illustrate a few examples of rotational symmetry (with lines of symmetry included on the drawings):

Now that we have an understanding of the basics of symmetry, there are a few more topics to cover to fully understand symmetry groups.

MODS

When telling time on a clock, we know that there are only 12 hours, and after 12, we must go back to 1, correct? This is referred to as mod 12.

The basic characteristics of mods can be found in the curriculum for Math 119, which is where the following information comes from:

For n a positive whole number and 0≤a,b≤n - 1, we define the sum of a and b mod n in the same way:

-If a + b < n, then the sum of a and b mod n is a+b.

- if a+b≥n, then the sum of a and b mod n is a+b-n (the n must be subtracted to wrap back around to the beginning again)

When trying this to another example, say mod 5, we follow the same procedures. Lets look at some example problems:

a) 2+4 = 1 (because 2+3 = 5 and since there is 1 remaining, we bring it back around to 1)

b) 3+1 = 4 (since this number does not pass 5 and have to come back around, therefore the final answer is 4)

c) 4+3 = 2 (since it only takes 1 to get to 5, we then have 2 left over, which makes the final answer 2)

Taking It One Step Further

The Following information is borrowed from Math 119's Ideas in Geometry Textbook:

A group is a set with an operation * such that:

-For all x, y in the group, x * y is also in the group.

-There is a special member of the group I called the identity, such that for any x in the group we have x * I = I * x = x

-For any x in the group there is a member if the group x^-1 called x's inverse such that x * x^-1 = x^-1 * x = I

-The operation * is associative. That is, x * y * z = x * (y * z) = (x * y) * z

Thus, in the example of the set {0, 1, 2, 3, 4}, with addition mod 5, it is considered a group.

Conclusion

There are much deeper levels that can be dove into before fully understanding the subject of symmetry groups, but this should give most a basic understanding of the subject matter, which will allow for more extensive thought down the line. The basic conclusion that we can come to is that through the use of the symmetries at the beginning (reflective & rotational), which make up the symmetry group for that particular group, along with the mods, we can gain a general understanding of what Symmetry Groups truly are. It may not be a full understanding, but it is more than we knew before.

title: Ideas in Geometry/Symmetry Groups


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Ideas in Geometry/Center of Triangles

By: Charmaine Ramos (cramos2)

title: Ideas in Geometry/Center of Triangles


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Ideas in Geometry/Spherical Geometry

Basic Points
It is important to recognize and understand these key concepts to fully expand upon properties of spherical geometry.

A point: a point on a sphere is the same as a point on a plane. A line: a line on a sphere is called an arc due to the shape of a sphere. It is also the shortest distance between two points on the sphere . If an arc is extended, it will form a great circle. A great circle, however is the end of the lines path. A great circle: a great circle refers to arcs that are extended to form a circle between two points. Because the arcs are the shortest distance between the two points, the circle being formed will revolve around the entire sphere, as if its radius where the very center inside of the sphere. This circle always slices a sphere in half. There are infinitely many great circles on a sphere. Antipodal points: these points are exactly opposite of each other on the sphere, such as the poles. All great circles run through antipodal points. A circle: Every point has to be the same distance from the center, referencing its radius.

Parallel Lines
In spherical geometry Parallel lines DO NOT EXIST.

In Euclidean geometry a postulate exists stating that through a point, there exists only 1 parallel to a given line.

However, it is important that we remember the definition of a line in spherical geometry. Because a line is defined as a great circle, we can understand that each line goes through two antipodal points, slicing the sphere in half (shown in the first picture). If two lines (great circles) were to be drawn, they will always cross paths as they cross over to the opposite hemisphere (shown through lines A and B in picture 2).

Circle C appears to be parallel to line A. However, Circle C cannot be considered a line, since a line must be an arc that can be extended to run through two antipodal points. Since Circle C does not, it is not a parallel line to A. Instead we can call circle C simply a line of latitude.

Therefore, Parallel lines do not exist since any great circle (line) through a point must intersect our original great circle.

Angles and Triangles
Angles and their properties exist the same in spherical geometry as they do in Euclidean Geometry. An angle in spherical geometry is simply formed by two great circles. Thus, in picture 2 up above, there are angles formed where lines A and B intersect.

A triangle however, is different. In Euclidean Geometry, the sum of the interior angles of a triangle must equal up to 180°, since lines on a plane are very constricted. In spherical geometry, a triangle is formed by three arcs of great circles intersecting. These three arcs can form triangles with interior angle sums of much larger than 180 degrees. For example figure below shows three great circles. One is the line that lies along the equator, and the two other lines are perpendicular to the equator, but they both run through the same antipodal point at the top of the sphere. Therefore, we know that all three lines intersect to form a triangle and we also know that the two lines intersecting the equator already create an angle of 90° each. Thus, we can determine that the interior angle sum of the triangle is already larger than 180°.

In fact, the angle sum can be anywhere from 180° to 540°. We can see how the size of the angles increase and decrease if we examine smaller and larger triangles. The larger the triangle becomes on the sphere, the more Euclidean properties it loses. Thus, the angle sum will become larger.

Picture References
http://en.wikiversity.org/wiki/File:Intersecting_great_circles.png http://commons.wikimedia.org/wiki/File:Linalg_great_circle.png http://www.abc.net.au/science/morebigquestions/sphere.htm http://euler.slu.edu/escher/index.php/File:Great-circles.png

title: Ideas in Geometry/Spherical Geometry


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User:MannaRaps

This user enjoys learning and rapping.

title: User:MannaRaps


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User talk:MannaRaps

 Hello MannaRaps, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 23:27, 6 December 2010 (UTC)

 Woof! My name is Jack Russell. I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

Motivation and emotion
Welcome - you might be the first online participant! (The rest are on-campus students). Maybe share a bit about your learning goals and I can suggest some ideas about how that could work. Sincerely, James. -- Jtneill - Talk - c 23:27, 6 December 2010 (UTC)

title: User talk:MannaRaps


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User talk:Egm6321.f10.team5.steinberg/hw7

Part 1
The Legendre polynomials $$ P_n \! $$ are even when n is even and odd when n is odd (see Problem 7.3 above). That is to say that

$$ P_{2k} \! $$ is EVEN

and

$$ P_{2k+1} \! $$ is ODD for $$ k = 0, 1, 2, 3 ... $$

Mathematically, the product of 2 odd functions is an even function, the product of 2 even functions is an even function and the product of an even function with an odd function is an odd function.

If the integrand of $$ A_n $$ is odd, the integral will be zero since our limits of integration equally straddle $$ \mu = 0 $$. However, if the integrand is even, $$ A_n $$ will be non-zero. We have already examined the even and odd properties of the Legendre polynomials $$ P_n $$. Thus, we must now look at $$ f(\mu) $$ to determine the even and odd properties of $$ A_n $$.

Part 1a
where we recall that $$ \mu := sin(\theta) \! $$

Equation (4.3) is a 6th order polynomial in $$ \mu $$, therefore it is an even function.

Therefore, with this $$ f(\mu) $$ we may conclude the following:

The reason k stops at 3 is that for $$ n \ge 7 $$, $$ A_n $$ is zero, since all Legendre polynomials at or beyond that order are orthogonal to $$ f $$.

Part 1b
For this part, we will examine $$ A_n $$ as a function of $$ \theta $$ since we cannot easily make f an explicit function of $$ \mu $$.

By merely looking at the behavior of $$ f(\theta) $$ at the limits of integration and at zero, it is apparent that it is an even function.

Therefore, we observe that

Part 2a
We now evaluate the non-zero values of $$ A_n $$ using Wolfram Alpha.

Part 2b
We now evaluate the few non-zero values of $$ A_n $$ using Wolfram Alpha.


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Topic talk:Ideas in geometry

Lesson One: Euclid's Axioms

Euclid was known as the “Father of Geometry.” In his book, The Elements, Euclid begins by stating his assumptions to help determine the method of solving a problem These assumptions were known as the five axioms. An axiom is a statement that is accepted without proof. In order they are: 1. A line can be drawn from a point to any other point. 2. A finite line can be extended indefinitely. 3. A circle can be drawn, given a center and a radius. 4. All right angles are ninety degrees. 5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate) To explain, axioms 1-3 establish lines and circles as the basic constructs of Euclidean geometry. The fourth axiom establishes a measure for angles and invariability of figures. The fifth axiom basically means that given a point and a line, there does not exist a line through that point parallel to the given line. The following picture shows a diagram for each axiom.

--Christine Yu 01:43, 7 December 2010 (UTC)

title: Topic talk:Ideas in geometry


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Euclidean geometry/Euclid's axioms

Lesson One: Euclid's Axioms

Euclid was known as the “Father of Geometry.” In his book, The Elements, Euclid begins by stating his assumptions to help determine the method of solving a problem. These assumptions were known as the five axioms. An axiom is a statement that is accepted without proof. In order they are:

 * 1. A line can be drawn from a point to any other point.
 * 2. A finite line can be extended indefinitely.
 * 3. A circle can be drawn, given a center and a radius.
 * 4. All right angles are ninety degrees.
 * 5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate)

To explain, axioms 1-3 establish lines and circles as the basic constructs of Euclidean geometry. The fourth axiom establishes a measure for angles and invariability of figures. The fifth axiom basically means that given a point and a line, there is only one line through that point parallel to the given line.

Quiz
 {All right angles are ninety degrees. + True - False
 * type=""}

{A line can be drawn from a point to any other point. + True - False
 * type=""}

{Rotation of a shape preserves distance. + True - False
 * type=""}

title: Euclidean geometry/Euclid's axioms


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User talk:Xenab

 Hello Xenab, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 02:05, 7 December 2010 (UTC)

title: User talk:Xenab