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# smart Indians

Discussion in 'The Lighter Side' started by nipperwolf, Nov 22, 2003.

1. ### nipperwolf

Messages:
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Joined:
Oct 3, 2002
When NASA was preparing for the Apollo
> > > > Project, it took the astronauts
> > > > to a Navajo reservation in Arizona for
> > > > training.
> > > >
> > > > One day, a Navajo grand elder and his son came
> > > > across the space crew walking
> > > > among the rocks.
> > > >
> > > > The elder, who spoke only Navajo, asked a
> > > > question.
> > > >
> > > > His son translated for the NASA people:
> > > > "What are these guys in the
> > > > big suits doing?"
> > > >
> > > > One of the astronauts said that they were
> > > > practicing for a trip to the
> > > > moon.
> > > >
> > > > When his son relayed this comment the
> > > > Navajo elder got all excited and
> > > > asked if it would be possible to give to the
> > > > astronauts a message to deliver
> > > > to the moon.
> > > >
> > > > Recognizing a promotional opportunity
> > > > when he saw one, a NASA official
> > > > accompanying the astronauts said, "Why
> > > > certainly!" and told an underling to
> > > > get a tape recorder.
> > > >
> > > > The Navajo elder's comments into the
> > > > microphone were brief. The NASA
> > > > official asked the son if he would translate
> > > > what his father had said.
> > > >
> > > > The son listened to the recording and
> > > > laughed uproariously. But he
> > > > refused to translate. So the NASA people took
> > > > the tape to a nearby Navajo
> > > > village and played it for other members of the
> > > > tribe. They too laughed long
> > > > and loudly but also refused to translate the
> > > > elder's message to the moon.
> > > >
> > > > Finally, an official government
> > > > translator was summoned. After he
> > > > finally stopped laughing the translator relayed
> > > > the message:
> > > >
> > > > "Watch out for these a**holes. They have
> > > > come to steal your land."
> >


text:
# maxLik v1.4-6

0

0th

Percentile

## Maximum Likelihood Estimation and Related Tools

Functions for Maximum Likelihood (ML) estimation, non-linear optimization, and related tools. It includes a unified way to call different optimizers, and classes and methods to handle the results from the Maximum Likelihood viewpoint. It also includes a number of convenience tools for testing and developing your own models.

## Functions in maxLik

 Name Description bread.maxLik Bread for Sandwich Estimator gradient Extract Gradients Evaluated at each Observation MaxControl-class Class "MaxControl" fnSubset Call fnFull with variable and fixed parameters hessian Hessian matrix logLik.maxLik Return the log likelihood value maxBFGS BFGS, conjugate gradient, SANN and Nelder-Mead Maximization compareDerivatives function to compare analytic and numeric derivatives condiNumber Print matrix condition numbers column-by-column maxLik-internal Internal maxLik Functions activePar free parameters under maximization maxNR Newton- and Quasi-Newton Maximization maxValue Function value at maximum maxSGA Stochastic Gradient Ascent maximType Type of Minimization/Maximization AIC.maxLik Methods for the various standard functions objectiveFn Optimization Objective Function vcov.maxLik Variance Covariance Matrix of maxLik objects returnCode Success or failure of the optimization sumt Equality-constrained optimization maxLik Maximum likelihood estimation maxLik-package Maximum Likelihood Estimation summary.maxim Summary method for maximization nIter Return number of iterations for iterative models nParam.maxim Number of model parameters nObs.maxLik Number of Observations storedValues Return the stored values of optimization numericGradient Functions to Calculate Numeric Derivatives summary.maxLik summary the Maximum-Likelihood estimation No Results!

format: plain


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# Even though the foreign exchange student was from a completely differe

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Even though the foreign exchange student was from a completely differe  [#permalink]

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21 Jan 2013, 05:21
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Even though the foreign exchange student was from a completely different culture, it was not long before she felt comfortable between us.

A. between us

B. between most of us

C. around us

D. among us

E. in this culture where the rest of us are comfortable

Doesn't 'Around us' sound better than 'Among us'?

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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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21 Jan 2013, 07:21
The student after was comfortable $$among$$ the other students (or at least in the classroom). Around is completely inappropriate

See the use of around http://www.perfectyourenglish.com/usage/around.htm
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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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21 Jan 2013, 11:24
Why is A incorrect? They could have a three-way and she is between the narrator and some other guy? D is incorrect because it changes the meaning of the sentence.
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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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22 Jan 2013, 06:50
You shouldn't use between if more than two people or entities are involved and these entities are not treated as distinct entities/individuals. It is ok to say there was a three-way between me and Tom and Dan, but it would be incorrect to say "the three-way caused a rift between us." You need to use among when dealing with a collective noun like class/us, where the separate individuals are not treated distinctly. A better guide dealing with between vs. among is:
http://grammar.quickanddirtytips.com/be ... among.aspx.
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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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23 Jan 2013, 00:29
It is incorrect because it changes the meaning of the sentence. You need to use among when dealing with a collective noun like class/us, where the separate individuals are not treated distinctly.
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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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31 Mar 2013, 09:23
I think the new student is one of the students in that class or school. Thus Among is better than Around...

Correct me if i am wrong....
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Re: Even though the foreign exchange student was from a completely differe  [#permalink]

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30 Oct 2018, 04:34
Even though I have solved this one correctly. I still have doubt on this one.

I think we need among because of US used in the sentence.

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Re: Even though the foreign exchange student was from a completely differe &nbs [#permalink] 30 Oct 2018, 04:34
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Computer science

Computational methods and numerical analysis

The mathematical methods needed for computations in engineering and the sciences must be transformed from the continuous to the discrete in order to be carried out on a computer. For example, the computer integration of a function over an interval is accomplished not by applying integral calculus to the function expressed as a formula but rather by approximating the area under the function graph by a sum of geometric areas obtained from evaluating the function at discrete points. Similarly, the solution of a differential equation is obtained as a sequence of discrete points determined, ... (100 of 12,737 words)

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# minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:

Can we remove the embeddness assumption?

Can we replace $S^3$ by $S^n$ for $n\geq3$, still for surfaces (not hypersurfaces)?

Else what are the most reasonable results of "weak" compactness we can expect. For instance, is that reasonable to expect that if we have $\displaystyle \Sigma_\infty =\lim_{n\rightarrow +\infty} \Sigma_n$ in a reasonable sense, do we have $\displaystyle Area(\Sigma_\infty)= \lim_{n\rightarrow +\infty} Area(\Sigma_n)$?

Without embeddedness, the Choi--Schoen theorem is false.

For example, there is a huge family of rotationally symmetric immersed tori in $\mathbb{S}^3$ (the only embedded one is the Clifford torus, by Brendle's solution to the Lawson conjecture). See Brendle's survey, Theorem 1.4, where he shows that there is an infinite family of immersed minimal tori of the form $$\left(\sqrt{1-r(t)^2} \cos s, \sqrt{1-r(t)^2} \sin s, r(t) \cos t, r(t) \sin t\right) \in \mathbb{S}^3\subset\mathbb{R}^4$$ for $r(t)\in (0,1)$ a smooth function. I'll sketch the construction and argue that compactness fails.

Thanks to the symmetry, the minimal surface equation for $r(t)$ can be integrated once, to become $$\frac{r'(t)^2}{r(t)^4(1-r(t)^2)^2}+\frac{1}{r(t)^2(1-r(t)^2)}=\frac{4}{c^2}$$ for some constant $c\in (0,1]$. The solution $r(t) = \frac{1}{\sqrt{2}}$ (i.e. $c=1$) is the embedded Clifford torus.

You can argue that the maximum and minimum values of $r(t)$ (depending on $c$) are given by $$\overline{x}(c)=\sqrt{\frac{1-\sqrt{1-c^2}}{2}}, \underline{x}(c)=\sqrt{\frac{1+\sqrt{1-c^2}}{2}}.$$ Furthermore, the period of the solution is given by $$T(c) = \int_{\underline{x}(c)}^{\overline{x}(c)}\frac{c}{x\sqrt{1-x^2}\sqrt{4x^2(1-x^2)-c^2}}dx.$$

For this to be an immersed torus, one needs that $\frac{2\pi}{T(c)}$ is rational, so that it closes up. You can check that $T(c)\to\sqrt{2}\pi$ as $c\nearrow 1$ and $T(c)\to\infty$ as $c\searrow 0$.

Now, suppose that $c$ is chosen so that $\frac{2\pi}{T(c)}$ is rational. Choose $t$ so that $r'(t) =0$ and $r(t) = \underline{x}(c)$. Then, as computed in Brendle's survey, we have that the second fundamental form at this point satisfies $$h(\partial_s,\partial_s) = r(t)\sqrt{1-r(t)^2}$$ On the other hand, the metric satisfies $$g(\partial_s,\partial_s) = 1-r(t)^2,$$ so the norm of the second fundamental form at this point is at least $$\frac{r(t)}{\sqrt{1-r(t)^2}}=\sqrt{\frac{1+\sqrt{1-c^2}}{1-\sqrt{1-c^2}}}\to\infty$$ as $c\searrow 0$.

As for minimal surfaces in $\mathbb{S}^n$ for $n>3$, I think that the space of even minimal spheres is quite big. See, e.g. Calabi's paper or e.g. this paper. I'm not completely sure about embeddedness of these examples, or whether or not their second fundamental forms blow up (certainly their areas do).

If you have that $\Sigma_n$ converges to $\Sigma_\infty$ as varifolds (up to a subsequence, this always holds assuming a priori area estimates on $\Sigma_n$), then $area(\Sigma_n) \to area(\Sigma_\infty)$.

• @ Otis, Thank you very much for the counter example. For your last remark, I image that the limit object can be singular?I mean, a priori you can converge to a union of smooth and singular mesure, no? To be more precise, is that know that there is a no neck energy property in the bubbling process?
– Paul
Jun 7, 2016 at 11:11
• Hi Paul, the "varifold statement" is essentially a beefed up version of the fact that Radon measures (think of the Hausdorff measure restricted to $\Sigma_j$ with uniform mass bounds admit a convergent subsequence to some measure (the measure has no structure, a priori). No area can be bubbled off (essentially, because area is a subcritical quantity). Another way to see this is via the monotonicity formula: if a minimal surface concentrates a lot of area in a small region, then its overall area must be really huge. Jun 7, 2016 at 11:26
• @Paul If you treat the minimal surfaces as conformal harmonic mappings, that there is no neck and no energy loss follows from the fact that the mapping are conformal. Jun 7, 2016 at 21:16


text:
# How to order 3 images horizontally?

I use the following code to include three images:

\begin{figure}[h]
\includegraphics{delete_gesture.png}
\caption{Awesome Image}
\label{fig:awesome_image}
\end{figure}

\begin{figure}[h]
\includegraphics{ok_gesture.png}
\caption{Awesome Image}
\label{fig:awesome_image}
\end{figure}

\begin{figure}[h]
\includegraphics{settings_gesture.png}
\caption{Awesome Image}
\label{fig:awesome_image}
\end{figure}

Now the images are ordered vertically. I want them horizontally. I tried to use columns or multicols but couldn't find a solution. Any hint?

put them all together in one figure environment and the three minipages without an empty line

\documentclass[english]{article}
\usepackage[demo]{graphicx}
\usepackage{babel,blindtext}

\begin{document}

\blindtext

\begin{figure}[!htb]
\minipage{0.32\textwidth}
\includegraphics[width=\linewidth]{delete_gesture.png}
\caption{A really Awesome Image}\label{fig:awesome_image1}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=\linewidth]{ok_gesture.png}
\caption{A really Awesome Image}\label{fig:awesome_image2}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=\linewidth]{settings_gesture.png}
\caption{A really Awesome Image}\label{fig:awesome_image3}
\endminipage
\end{figure}

\blindtext

\end{document}

• oh, golly -- please make those captions ragged right! they don't need to be that spacy! – barbara beeton Feb 4 '11 at 18:40
• @Barbara: sorry, but the captions is incomplete: "A really awsame image and caption" ... :-) – user2478 Feb 4 '11 at 20:03
• I'm just trying this out now, with three identical images, and I get a strange problem where the center image is shifted down, due to the length of the caption. Is there a way to prevent this with captions of varying length? – celenius Mar 24 '11 at 0:55
• How ahould the images be adjusted, at top or bottom? – user2478 Mar 24 '11 at 17:51
• The problem is that it doesn't align vertically the images if the captions have different lengths, how to correct this? – Ramy Al Zuhouri Mar 18 '14 at 17:55

Here is a solution with subfigure package. It has 2 rows and 2 columns of images. The widths are chosen such that it fits into a column of a 2-column page. I guess you get the idea.

\begin{figure}[t]
\centering
\subfigure[text]{
\includegraphics[width=.225\textwidth]{file}
}
\subfigure[text]{
\includegraphics[width=.225\textwidth]{file}
}

\subfigure[text]{
\includegraphics[width=.225\textwidth]{file}
}
\subfigure[text]{
\includegraphics[width=.225\textwidth]{file}
}

\caption{blablabla}
\label{fig:whatever}
\end{figure}

As you can see it's pretty simple to have to images/objects next to each other -- just put them into the same line. Or you can use multicol inside a figure.

Please note that subfigure is superseded by subfig which provides \subfloat command instead of \subfigure. More compatibility information is found in the subfig documentation.

• Using subfigures is a good idea. I would prefer the subfig package, since it's newer than subfigure and it's widely considered as its successor. – Stefan Kottwitz Feb 5 '11 at 16:26
• according to wikipedia subfigure is depricated. – irrehaare Nov 10 '12 at 16:26
• The subcaption package is the way to go now! It provides a subfigure environment and the \subcaptionbox command. – ypnos Jul 29 '16 at 22:28

I use the following technique:

\begin{figure}[h]
\fbox{\includegraphics{fig1.pdf}}
\hspace{30px}
\fbox{\includegraphics{fig2.pdf}}
\hspace{30px}
\fbox{\includegraphics{fig3.pdf}}
\caption{this is the caption}
\label{materialflowChart}
\end{figure}

This places a thin line around each image (as it uses framebox). You can use \mbox in the same way if you don't want a frame. The \hspace{} command is a convenient way of controlling the spacing between the two images.

• Some notes: \label always comes after \caption. I think you mean '30pt' not 30px, but \hfill would be better. – Martin Scharrer Feb 4 '11 at 14:31
• @Martin: px is the pdf unit and exactly 1 pixel – user2478 Feb 4 '11 at 14:40
• Thanks Martin, I did not know that about \label - I swapped the order. For some reason 30px works on my computer - I thought the px dimensions could be supplied? Is this not the case? – celenius Feb 4 '11 at 14:41
• @Herbert: thanks, I didn't know that pdftex is adding this unit. I had a short look in the TeXbook and run a test with tex beforehand. Should have used pdftex :-) – Martin Scharrer Feb 4 '11 at 14:43
• @Martin - Just curious, why is \hfill better? – celenius Feb 4 '11 at 14:45


text:
# Worksheet on Area of Regular Figures

We will practice the questions given in the worksheet on area of regular figures.

We know the formula for finding the area of regular figures.

Area of a square = side × side

Area of the rectangle = length × breadth

1. Fill in the blanks:

(i) The area of a square whose side is 1 cm is __________.

(ii) A square is considered as the best unit of __________.

(iii) The measure of a region is called the __________.

(iv) The amount of surface enclosed by a closed plane figure is called its __________.

2. Find the area of the following figures:

3. The side of a square field is 125 m. Find its area and also find the cost of laying grass in it at $0.30 per square metre. Find the area of the figures: 4. Find the area of the following figures: 5. Find the area of the regular figures: Answers for the worksheet on area of regular figures are given below. Answers: 1. (i) 1 cm$$^{2}$$ (ii) area (iii) area (iv) area 2. (i) 9 cm$$^{2}$$ (ii) 650 cm$$^{2}$$ (iii) 10 cm$$^{2}$$ 3.$ 4687.50

4. (i) 24 cm2

(ii) 48 cm2

(iii) 67.5 m2

5. (i) 81.25 m$$^{2}$$

(ii) 27.1 m$$^{2}$$

(iii) 220.5 m$$^{2}$$

Area.

Area of a Rectangle.

Area of a Square.

To find Area of a Rectangle when Length and Breadth are of Different Units.

To find Length or Breadth when Area of a Rectangle is given.

Areas of Irregular Figures.

To find Cost of Painting or Tilling when Area and Cost per Unit is given.

To find the Number of Bricks or Tiles when Area of Path and Brick is given.

Worksheet on Area.

Practice Test on Area.


text:
# Tag Info

158

This is really a footnote to Adobe's answer. Light cannot escape from an event horizon. But how can you check that light can never escape? You can watch the surface for some time $T$, but all you have proved is that light can't escape in the time $T$. This is what we mean by an apparent horizon, i.e. it is a surface from which light can't escape within a ...

47

The expression for the power emitted as Hawking radiation is $$P = \frac{\hbar c^6}{15360 \pi G^2 M^2} = 3.6\times10^{32} M^{-2}\ \text{W} = -c^2 \frac{dM}{dt},$$ where the term on the far right hand side expresses the rate at which the black hole mass decreases due to the emission of Hawking radiation. You can see that what happens is that the power ...

31

At first many people didn't care much for black holes. But later people showed that they were pretty unavoidable features of the theory of general relativity and that theory made other quite precise predictions that were tested and found good. So when you are told that black holes are required if you have GR and GR looks like the best game in town then it ...

23

There are two ways to approach your question. The first is to explain what Brian Greene means, and the second is to point out that the "particles being swallowed" explanation is a metaphor and isn't actually how the calculation is done. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in! When a ...

21

There are a number of equivalent ways to think about Hawking radiation. One is pair creation, as endolith mentions, where the infalling particle has negative total energy and so reduces the mass of the black hole. Another way, perhaps more useful here, involves Compton wavelength. If the wavelength of a particle (not just photons, by the way) is greater than ...

21

The reason for many contradictory statements regarding the nature of virtual particles is that they are often invoked for heuristical explanations of phenomena that arise within the framework of quantum field theory. One then tries to justify those explanations by attributing certain properties to virtual particles they do not actually possess. What ...

19

...why do we trust black hole physics? ... (physics which is derived by combining quantum mechanics and GR such as Hawking Radiation, things relating to the Information Paradox, etc. ) Formally, there isn't quite a reason to because we've not observed these things yet. But that's also perfectly okay as well because that is how science sometimes works: we ...

17

A few things: 1)Just because an observer crossing the event horizon doesn't necessarily feel ill effects AT THE TIME OF CROSSING the horizon, it doesn't mean that they won't inevitably end up at the singularity, where there will be plenty of ill effects--all timelike curves that cross the horizon end up at the singularity in a finite amount of proper time. ...

16

Craig Feinstein asked: Does Stephen Hawking believe that General Relativity is wrong? Here is my answer (I will shift my answer there if some one reopen that question): Stephen Hawking did NOT say that black holes do not exist. Hawking used to think black holes are oblivious. Now he admits (like some other people do) black holes have perfect memory, just ...

12

I don't know about extracting mass-energy from inside the black hole; it seems pretty inefficient to me. Hawking radiation isn't that powerful (as well as still being highly hypothetical), but building a Dyson sphere around a star is better. Remember, black holes, are, well, black. If there was enough Hawking radiation, they would cease to be black and would ...

12

To answer this we need to talk a bit about how particles are described in quantum field theory. For every type of particle there is an associated quantum field. So for the electron there is an electron field, for the photon there is a photon field, and so on. These quantum fields occupy all of spacetime i.e. they exist everywhere in space and everywhere in ...

11

To add to Rory's answer- The ability to radiate particles in a random, statistical way, is in a deep sense identical to an object having the property we know as "temperature." So, black holes have a temperature. It has a particular formula that is inversely proportional to the mass of the black hole. If you set that temperature equal to the current ...

10

The conditions for the existence of the Hawking effect are described in classical terms, i.e you need 1) A Lorentz signature metric 2) A horizon (given, for example, by space flowing into a BH faster than the speed of light, or fluid flowing downstream faster than the speed of sound) 3) Surface gravity at the horizon Those conditions are then applied to ...

9

Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why? Very roughly speaking and in as simple terms as possible, inside the black hole, gravity is so intense that the time coordinate and one of the spatial coordinates (the radial coordinate) swap "roles". That's one way to see why you ...

9

Any non-mathematical answer is obviously going to be an oversimplification, but as long as you're happy with that here is my oversimplification. The temperature of a big black hole is lower than a small black hole because the curvature at the event horizon decreases with radius. It's the curvature, i.e. the bending, that determines the temperature - more ...

8

With this answer, I am going to list some of my notes from the paper that @zephyr posted as a comment, http://arxiv.org/abs/0908.1803v1. Using Hawking radiation as a means of mass-to-energy conversion seems, in a word, absurd. The paper, however, addressed exactly that for use in powering a manned spaceship to the stars. Hawking Radiation as a thrust ...

8

As you said, the case of black holes is conceptually totally analogous to the burning books. In principle, the process is reversible, but the probability of the CPT-conjugated process (more accurate a symmetry than just time reversal) is different from the original one because $$\frac{Prob(A\to B)}{Prob(B^{CPT}\to A^{CPT})} \approx \exp(S_B-S_A ).$$ This is ...

8

Yes, black holes are supposedly near-perfect black bodies. They emit thermal radiation called Hawking radiation, which, however, does not originate from beyond the event horizon, but is a consequence of the interaction of the strong gravitational field outside the horizon with the vacuum. The process is sometimes described as the production of 'virtual' ...

8

Yes and no. Electrons - and all other elementary particles - may be viewed as microstates of very tiny black holes. As one considers increasingly heavy elementary particles (e.g. those in the Hagedorn spectrum of string theory), they increasingly morph into black hole microstates. When the elementary particle masses sufficiently surpass the Planck scale, ...

8

Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays. A complete description of this dissolution requires a model of quantum gravity, however, as it occurs when the black hole ...

8

Hawking radiation is such a miniscule effect we can be sure we'll never detect it for a real astrophysical black hole. The Wikipedia article gives some numbers: For a black hole of the mass of the Sun, the power emitted in Hawking radiation amounts to $9\times10^{-29}\ \mathrm{W}$. Even if all this energy were converted to visible-light photons ($4\times10^{... 8 Any black body in space radiates and ends up very cold, might even crystallize. The law of increasing entropy holds for closed systems, in this case the whole system: "all the radiation that left the black body + the black body itself" microstates. In the sense that a black hole behaves as a black body the same holds true, it cannot be considered a ... 7 Hawking radiation is a very robust prediction. It comes simply from applying quantum field theory in the curved space-time near the event horizon. It's also part of the synthesis called "black hole thermodynamics", for which string theory provides an explanation in terms of the statistical mechanics of microstates. In the S-matrix of quantum gravity, if ... 7 I assume you're asking how a black hole can evaporate due to Hawking radiation. The answer is that the Hawking radiation does not come from the event horizon, but instead comes from a region just outside the event horizon so time has not stopped at its position. If you were to watch a black hole form then evaporate, you would never see an event horizon form.... 7 In the case of Hawking radiation, the direct answer is "no, there is no direct test, nor can we imagine one with anything like current technology." But it is not some wild speculation made in vacuum. The extremely closely related Unruh effect can be derived from basic quantum field theory on a curved spacetime, and many QFT and GR texts have at least an ... 7 The black hole initially lost the gravitational energy that was needed to create the pair. The pair-creation model is a bad description of Hawking radiation, which for macroscopic black holes is really photons. The second particle that gets created above the event horizon doesn't have nearly enough energy to escape. It does, however, produce photons above ... 7 The paper pointed out by Daniel's comment gave me a starting point to find more literature on this topic and do further research. After a while, it became clear to me that my question is actually an unsettled (research) question. Therefore, a definitive answer cannot really be expected. Nonetheless, I think it's valuable to provide something of an answer. ... 7 Black holes are basically neutron stars with such a gravitational force that even light cannot escape from it. A black hole is a mathematical solution. A neutron star over the critical mass gets so dense that it forms larger and larger time dilation relative to the outside universe thus we get to see what happens on short time scales. But what causes ... 7 Perhaps the most direct example in particle physics is$J/\Psi$(or any other meson not including up and down quarks) production. The meson has a valence content of$c\bar{c}\$, so it represent a pair of particles knocked on-shell from the nucleon sea. The reaction is not exactly analogous because it requires a rather large input of kinetic energy (as does ...

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• Notes

Find (a) the curl and (b) the divergence of the vector field.
$\mathbf{F}(x, y, z)=x y^{2} z^{2} \mathbf{i}+x^{2} y z^{2} \mathbf{j}+x^{2} y^{2} z \mathbf{k}$

$F(x, y, z)=x y^{2} z^{2} \hat{i}+x^{2} y z^{2} \hat{j}+x^{2} y^{2} z \hat{k}$

$\left|\begin{array}{ccc}{\hat{i}} & {\hat {j}} & {\hat{k}} \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ {p} & {Q} & {R}\end{array}\right|$

vector $=p \hat{i}+Q \hat{j}+R \hat{k}$

$\nabla{x} F=\left[\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right] \hat{i}+\left[\frac{\partial R}{\partial x}- \frac{\partial p}{\partial z} \right] \hat{j}+\left[\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right] \hat{k} \rightarrow$ vector

$\nabla x F=\left [2 y x^{2} z-2 z x^{2} y \right] \hat{i}+\left[2 x y^{2} z-2 z y^{2} x\right] \hat{j}+\left[2 x y z^{2}-2 y x^{2} z^{2}\right] \hat{k}$

$\nabla x F=0 \hat{i}+0 \hat{j}+0 \hat{k}=0$

$\nabla \cdot F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$ Scalar

$\nabla \cdot F=y^{2} z^{2}+x^{2} z^{2}+x^{2} y^{2}$


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# Homework Help: Damped oscillator consecutive amplitude ratio

1. Dec 11, 2011

### Uniquebum

1. The problem statement, all variables and given/known data
Undamped oscillator's period $T_0 = 12s$. Damped oscillator's angular frequency $\omega_1 = \omega_0 * 97\%$ where $\omega_0$ is the angular frequency of the undamped oscillator's. What is the ratio of consecutive maximum amplitudes?

2. Relevant equations
Equation of damped oscillator's motion:
$x = e^{-\alpha t}A_0sin(\omega_1 t + \phi)$
where $\alpha = \frac{b}{2m}$ where $b =$damping constant.

3. The attempt at a solution
Firstly, were' talking about maximums so we can disregard the sin() function.
Calculating $\omega_1 = \omega_0 * 0.97 = \frac{2\pi}{T_0}0.97$.
Thus for the damped oscillator $T_1 = \frac{T_0}{0.97}$

Then we could write something as follows:
$\frac{x_0}{x_1} = \frac{e^{-\alpha t_0}A_0}{e^{-\alpha t_1}A_0}$
but we have no clue of alpha nor about x_0 and x_1... Any help appreciated.

2. Dec 11, 2011

### Stonebridge

3. Dec 11, 2011

### Uniquebum

I ended up using the formula

$\zeta = \sqrt{1-(\frac{\omega_1}{\omega_0})^2}$
And got approx 0.243 out of it. In my answer spreadsheet they claim the answer to be 0.21 however. Now i'm wondering whether i got it right or not... heh :) Thanks for the help either way.


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# James Clerk Maxwell

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James Clerk Maxwell (Edinburgh, June 13, 1831 – Cambridge, November 5, 1879) was a Scottish physicist best known for his formulation of electromagnetic theory and the statistical theory of gases. He also provided quantifiable evidence for the three-colour theory of light and the correct theory of the rings of Saturn. He is regarded by most modern physicists as the scientist of the 19th century who had the greatest influence on 20th-century physics, and is ranked with Galileo Galilei, Isaac Newton, and Albert Einstein, and the main creators of quantum mechanics, Werner Heisenberg, Erwin Schrödinger, and Paul A. M. Dirac.

The Maxwell equations, the Maxwell-Boltzmann distribution, and the unit of magnetic flux, the maxwell, were named after him, as was a crater on the moon.

(PD) Image: John R. Brews
Timeline for key scientists in classical mechanics

## Biography

### Family

James Clerk Maxwell came from two prominent and affluent Scottish families, the Maxwells and the Clerks, both of lower nobility and heavily interrelated.[1]

James's father, John, was a younger paternal grandchild of Sir George Clerk, and accordingly was simply named John Clerk, while his elder brother had inherited the title and name of their grandfather, and thus was also called Sir George Clerk. John legally added the name Maxwell later, after he had inherited from his grandmother Dorothea (née Lady Clerk Maxwell) the Middlebie estate in the Galloway region, in south-west Scotland. The estate is 10 kilometres north of Castle Douglas, the market town, and about 70 kilometres west of the town of Middlebie, Dumfriesshire.

After marrying Frances Cay in 1826, John Clerk Maxwell moved with his bride from Edinburgh to the Middlebie estate. There they built a country house named Glenlair. The house was in a rather isolated spot, even by Scottish standards, the closest cities being Glasgow (110 kilometres north, a full day's journey at the time) and Edinburgh, which was two whole days of travel away.

### Early youth

James was born in Edinburgh, where his parents had gone to ensure proper medical attention at his birth. He was the first son of his mother, who earlier had lost a daughter of a few months old. Mrs Clerk Maxwell was at the relatively advanced age of 40 when she gave birth to James. Soon after the birth the family went back to the Glenlair house. James, who remained their only child, was brought up in the Galloway region, where he played with the local children and acquired a broad Scottish accent, despite his upper-class ancestry.

Maxwell's mother died in 1839 from abdominal cancer, the same disease to which Maxwell would succumb at exactly the same age, 48 years. James grew up alone with his father, with whom he had a happy and close relationship. At a young age James received private lessons from a dull and uninspired tutor, who claimed that James was slow at learning, though in fact he was very inquisitive and had a phenomenal memory. At the age of 10 James was sent to the Edinburgh Academy to receive a proper education. During term time, he was living either with his mother's unmarried sister Jane Cay, or with his father's sister Isabella, the widow of James Wedderburn. Both aunts lived in Edinburgh.

At the academy he became a lifelong friend of Peter Guthrie Tait, who would become also a distinguished mathematical physicist and one of the advocates of Hamilton's quaternion theory. In the beginning James had a difficult time at school because of his Galloway accent and because he wore strange but practical clothes, designed by his father.

Maxwell was showing unusual mathematical ability at school, and at the age of 15 invented, by analogy with the construction of an ellipse, a way of drawing ovals using a piece of string. This work was published in the Proceedings of the Royal Society of Edinburgh, and although not an important paper, is remarkable for such a young author.

### Student days

At the age of 16 Maxwell entered Edinburgh University. He was not unusually young to enter a Scottish university; at that time, these were hybrids between modern secondary schools and universities. Maxwell studied mathematics, philosophy, and physics and left after three years without graduating to go up to Cambridge, at first to Peterhouse College, which housed many Scotsmen, and after one term to Trinity College.

By the time he entered Cambridge, Maxwell was very knowledgeable about English literature and was a reasonably good poet. And, of course, he had a vast knowledge of mathematics and physics. In his first year at Cambridge he had to undertake systematic studies, as any student, although he had already published a few papers, including a valuable one on elasticity. His tutor, the mathematician William Hopkins, noted his strong geometrical insight but also that in analysis his powers were somewhat less. Maxwell followed the lectures by G. G. Stokes and encountered William Thomson again, whom he had first met while at Edinburgh University.

In 1854 Maxwell was second wrangler (second place at the competitive exam, the "mathematical tripos", behind Edward John Routh) and shared the Smith's Prize (a prestigious competitive award for an essay that incorporates original research) with Routh. The following year, Maxwell was elected to a fellowship at his college, Trinity, and started serious research on two of his favourite topics, colour theory and electromagnetism.

### Aberdeen

Clerk Maxwell kept his fellowship at Trinity for only a short time. Because his father's health was deteriorating, he sought a professorship in Scotland. He hoped that the long vacations of Scottish universities would permit him to spend many months each year at his beloved Glenlair with his father. In addition, some of Maxwell's letters to his father seem to indicate that the prospect of becoming a Cambridge don did not appeal to him; he recognized the narrowing tendencies of college life. He became Professor of Natural Philosophy at Marischal College, Aberdeen, but before the appointment was announced his father died, in early April 1856. In June 1858, Maxwell married Katherine Mary Dewar, the daughter of the principal of Marischal College and seven years James' senior. The marriage remained childless.

Maxwell's most important work in Aberdeen, besides the continuation of his studies on colour, was a theoretical study on the nature of Saturn's rings. He proved that the rings must consist of myriad small particles, or moonlets, each in their own orbit around the planet, and that the ring could neither be solid nor fluid. Maxwell was awarded the £130 Adams Prize in 1859 for his essay On the Stability of Saturn's Rings.

While in Aberdeen, Maxwell was nominated twice for a Royal Medal of the Royal Society for his work on the perception of colour, and was twice passed over. But a nomination in May 1860 by G. G. Stokes and William Hallowes Miller (Cambridge Professor of Mineralogy) for the Rumford Medal was successful.

In the fall of 1859, the Chair of Natural Philosophy at Edinburgh became vacant when Maxwell's old teacher and friend James D. Forbes moved to a less demanding post in St Andrews. Maxwell and several of his contemporaries applied for this post, including Peter Guthrie Tait and Edward John Routh. Tait was chosen — he was preferred over Maxwell because it was felt that he was the better teacher.

In 1860 King's College and Marischal College, both in Aberdeen, merged forming the nucleus of what would become Aberdeen University. Maxwell was made redundant by the merger, even though he was the son-in-law of one of the principals and had won the very prestigious Rumford Medal and the Adams Prize.

After having lost his position in Aberdeen and being jobless, Maxwell headed to Glenlair with his wife. He didn't stay unemployed for long and soon found another job. In July 1860, Maxwell was appointed to the vacant Professorship of Natural Philosophy in King's College, London. Almost immediately after the appointment and before even having moved to London, Maxwell suffered an attack of smallpox. Because smallpox is very contagious and life-threatening, his servants were afraid to enter the room where Maxwell lay sick, but his wife nursed him lovingly and he soon recovered and started his work at King's college.

### London

The work at King's College was more exacting than that in Aberdeen. There were nine months of lecturing in the year, and evening lectures to artisans, and so on, were recognised as a part of the professor's regular duties. Maxwell retained the post until the spring of 1865, when he was succeeded by Professor W. G. Adams, but continued lecturing to the artisans during the following winter. In London, Maxwell did most of his great work on electromagnetism, but he also worked on his colour theory and kinetic theory of gases in a laboratory he established in the top floor of his London home. He earned a reputation, at least among his neighbors, as an eccentric scientist. His colour experiments had him gazing into a long, coffin-like, lightbox. His gas experiments were conducted with steam during the dead of summer. His wife, as de-facto lab assistant, kept the stove stoked and water boiling through many hot summer afternoons. Other experiments required cooler temperatures provided by large amounts of ice, probably also hauled to the attic by his wife.[2] These experiments earned him great respect among his peers, as evidenced in June 1861, when Maxwell became, at the age of just 29, a fellow of the Royal Society of London, which entitled him to the postnominal FRS.

### Glenlair

Courtesy Cavendish Laboratory
Front Glenlair House

In 1866 Professor and Mrs. Maxwell returned to Glenlair because of Maxwell's ill health. While out riding — Maxwell was an enthusiastic horseman — he had grazed his head on a tree branch and the wound had led to a severe attack of erysipelas, an inflammatory disease characterized by severe headaches and vomiting. There is reason to suspect that Maxwell's earlier bout of smallpox had made him sensitive to this disease.

While at Glenlair, he began writing his celebrated two-volume treatise on electricity[3]. It is of interest to note that while Maxwell's original paper of 1865 did not use quaternions — all equations were written out in components — in this work they are used extensively. In his treatise Maxwell refers to a book on quaternions by his friend P. G. Tait.[4] There is evidence[5] that Maxwell familiarized himself with Hamilton's quaternions as late as the early 1870s. Later, Maxwell's equations were recast by Oliver Heaviside into the vector format that is still in use today.

During his Glenlair period that lasted until 1871, Maxwell served as examiner and moderator at Cambridge in 1866, 1867, 1869, and 1870, instituting important and useful reforms in the tripos.

Maxwell was a deeply pious man. Lewis Campbell and William Garnett record that visitors to Glenlair between 1866 and 1870 were struck with the manner in which the daily prayers were conducted by the master of the house. The prayers, which seemed extempore, were most impressive and full of meaning.[6]

Courtesy of the Master and Fellows of Trinity College, University of Cambridge
Dr and Mrs James Clerk Maxwell 1869

### Back to Cambridge

In 1870 Cambridge University received a splendid gift from William Cavendish, the seventh Duke of Devonshire, who had been second wrangler in 1829 and was in 1870 Chancellor of the university. The duke, directly related to Henry Cavendish, gave the university £6300 for the construction of a physics laboratory. In 1869 a physical sciences syndicate had recommended the foundation of a physical laboratory and the estabishment of a chair in experimental physics. Until then there was no general laboratory in Cambridge; the physics fellows experimented at home or in their colleges. As always, there were insufficient funds to implement the committee's recommendation, which led the duke to intervene and make his splendid offer. The laboratory was to be called the Cavendish laboratory and the new chair (£500 per annum) was to be the Cavendish Professorship of Experimental Physics.

William Thomson and Hermann von Helmholtz were approached for the job, but both declined. Maxwell, at present considered the greatest of the three, was elected in 1871. He thus became the first Cavendish Professor.

Maxwell devoted great energy to the building and equipment of the new laboratory. It was ready for occupation and functioning in 1874 (and would stay in the same place for a full century; the Cavendish moved to a new location in West Cambridge in 1974). Maxwell had started lecturing in October 1871, and in his long inaugural lecture he expressed in detail his philosophy of teaching physics and the function of the laboratory for teaching and research.

During his last Cambridge period, Maxwell undertook the publication of Henry Cavendish's electrical papers. The manuscripts had been gathering dust for about a century. It is believed that they would have influenced greatly the development of electric science had they been known earlier. Cavendish's eccentricity kept them from being published.

In the summer of 1879, when he was 48 years old, Maxwell showed signs of the same abdominal cancer that had killed his mother when he was eight. He died on November 5, 1879, the year that his great admirer Albert Einstein was born.

## Scientific work

### Colour Theory

Statue of Maxwell, George Street Edinburgh.

Clerk Maxwell started studies on colour vision while still at Edinburgh under the guidance of Professor James David Forbes. Fundamental progress on colour theory — a subdiscipline of physiological optics — had been made earlier by Thomas Young, who had arrived at the conception of three fundamental colours: red, green, and blue.[7] Maxwell continued Young's investigations by studying colour perception with a spinning top that allowed quantitative measurement of the colours being mixed. Maxwell's work contributed importantly to the unravelling of the mysteries surrounding colour vision. As a spin-off of his researches, he was able to show the first ever colour slide, which he projected at the Royal Institution on May 17, 1861.[8] A hundred years later, it was found[9] that Maxwell was lucky that his strategy in designing the slide worked, because ultraviolet acted as a proxy for red in the recording stage, a fact of which Maxwell was unaware. In any case, his slide projection greatly impressed his audience. The work of Young and Maxwell is known today to any computer user who is trying to create his own colours by the RGB colour scheme.[10]

### Saturn's rings

The planet Saturn has a flat circular ring, first observed (but not understood) by Galileo Galilei in 1610. Dutch physicist Christiaan Huygens recognized it as a ring in 1655 and 20 years later Giovanni Domenico Cassini saw that there were actually two rings. In 1850 Harvard astronomer George Phillips Bond discovered a dark obscure ring interior to these two rings. On a visit to Europe in 1851 Bond spoke about his discovery in Cambridge, where he aroused the interest of James Challis, the Plumian Professor of Astronomy. The mechanical stability of the ring was hardly understood at that time. The great Laplace had studied it and found that a solid uniform ring would be dynamically unstable and could not revolve permanently around the planet.

Challis set the stability of Saturn's rings as the subject of the 1857 Adams Prize essay. This prize had been established in 1848 by a few members of St John's college of Cambridge University in honour of John Couch Adams's prediction of the existence of the planet Neptune. Only Cambridge graduates could enter the competition. Clerk Maxwell took up the challenge and submitted his essay on December 16, 1856; his was the only submission.

In the first part of his essay he showed that the rings cannot be homogeneously solid, because, in accordance with Laplace's earlier finding, they would be unstable. The gravitational gradient arising from the main body of the planet would generate stresses that normal matter could not withstand.

In the second part of his prize essay Maxwell assumed that the rings were fluid (but not at rest because they would then collapse into Saturn). This assumption fared no better under realistic conditions because of the development of destabilizing waves. Therefore he was led to assume that the only system of rings that can exist is one composed of an indefinite number of unconnected solid particles, revolving around the planet with different velocities according to their respective distances. He thus concluded that the rings consisted of what were effectively clouds of small solid satellites. This is the correct description, as confirmed by the Voyager 2 space probe in 1981.

Maxwell was awarded the Adams prize on May 30, 1857. Two years later he summarized his more than 80 pages essay in a short article.[11]

### Molecular velocity distribution

Kinetic gas theory was established by Daniel Bernoulli in 1738 in a chapter of his book Hydrodynamica. Bernoulli assumed an elastic fluid (i.e., a gas) to consist of "very minute corpuscles, which are driven hither and thither with a very rapid motion".[12] Bernoulli was the first to recognize that pressure is caused by collisions of the particles with the walls of the vessel containing the gas, and that the speed of the particles increases with increasing temperature. In the nineteenth century the kinetic theory of gases was pursued, among others, by James Prescott Joule in England and Rudolf Clausius in Germany. Especially the latter worker had a determining influence on Maxwell.

Sometime early 1859 Maxwell read a translation of a paper by Clausius, in which the mean path length—the distance covered by molecules beween successive collisions—played an important role.[13] It is likely that Clausius' paper drew Maxwell's attention because he had just finished his work on Saturn's rings in which he had considered the motions and path lengths of myriads of small particles. Maxwell became interested in viscosity, diffusion rates, and other transport properties of gases.

In his earlier investigations, Clausius assumed that all molecules move with the same speed. He used the mean value of the square of the velocity of molecules as the speed of the molecule in the gas. Before Maxwell, Clausius never thought of velocity distributions, which express the percentages of molecules that have velocities differing from the average velocity. Maxwell, who saw clearly that such statistical distributions could be physically important, wrote a memoir for the British Association in 1859, in which he gave the velocity distribution of the molecules in a gas. He also calculated the viscosity coefficient of a gas in terms of mean free paths, and wrote about diffusion, and the conduction of heat.[14]

In his kinetic work, Maxwell relied heavily on statistical ideas. Maxwell's interest in probability theory dated back to 1850 when he read an essay by John Herschel on the work by the Belgian social scientist Adolphe Quetelet, Theory of Probabilities. Later, in 1857, he read the first volume of Henry Thomas Buckle History of Civilization in England, in which Quetelet's statistical methods are applied to history. It is likely that Maxwell's interests in contemporary debates on statistics led him to applications of probability methods in his 1860 papers on the dynamical theory of gases. The introduction of statistical factors into the treatment of appropriate physical problems must be regarded as one of Maxwell's foremost contributions to the advance of physics.

In his 1860 papers Maxwell used simplified models of a perfect gas in which the principal interaction between molecules occurred during collisions. On the assumption that the three orthogonal components, vx, vy, and vz, of the velocity distribution are independent, he demonstrated that the distribution of total velocities (speeds), v2vx2 + vy2 + vz2, among the molecules in the assembly would be governed by a function of the form

${\displaystyle f(v)=A\,v^{2}e^{-v^{2}/B},}$

in which A is related to the density of atoms and B is related to the average kinetic energy of the atoms, or, equivalently, the temperature. This function, which is mathematically similar to those that occur in many simple statistical problems, is commonly designated the Maxwell distribution law, or, alternately, the Maxwell-Boltzmann law, in recognition of the work of Ludwig Boltzmann (1844-1906), who greatly extended research on statistical assemblies.

When Maxwell used his distribution law to calculate the viscosity of his ideal gas, he was surprised to find that the result was independent of pressure, or density. The same was found to be true of thermal conductivity. Careful measurements of the viscosity and thermal conductivity of several gases, carried out at home with the aid of his wife, confirmed the theoretical results for appropriate lower ranges of pressure. This success demonstrated that the theory could be used for other calculations. It also provided a basis for estimating atomic size and an accurate determination of Avogadro's number.

Clausius pointed out some errors in Maxwell's 1860 treatment of diffusion and conduction of heat, which Maxwell corrected in 1866.[15]. In the same work Maxwell modified his theory in consequence of the results of his experiments on the viscosity of air at different temperatures. Further he introduced an intermolecular repulsive force depending on intermolecular distance R as R−5.

### Maxwell's demon

In 1868 Maxwell mentioned in a letter to his friend Tait a finite being ("Maxwell's demon")[16] that could violate the second law of thermodynamics. This law had been formulated in the 1850s by William Thomson and Rudolf Clausius. Clausius stated that it is impossible that heat flows spontaneously up a temperature slope, i.e., heat does not go naturally from a low temperature bath to one of higher temperature; work is required to achieve such a transport of energy.

Maxwell thought up a vessel, filled with an ideal gas, that is divided into two partitions A and B by a division in which there is a small hole. The hole is guarded by a being, Maxwell's demon, who can see the individual molecules and who opens and closes the hole, so as to allow only the swifter molecules to pass from A to B and only the slower ones to pass from B to A. He or she will, thus, without performing any work raise the temperature of B and lower that of A. The internal energy of an ideal gas being proportional to temperature, the demon decreases the energy of A and increases that of B. In other words, the demon causes heat to flow from the low temperature partition (A) to the high temperature partition (B) without expenditure of work, in contradiction to Clausius' expression of the second law. Since this process is the opposite of a natural (spontaneous) process in which entropy increases, it is clear that the entropy of the total system A + B decreases by the action of Maxwell's demon.[17]

Much has been written about a solution of this paradox. A simple-minded answer is that, according to quantum mechanics, individual molecules are indistinguishable, so that the demon cannot see individual molecules. A different solution was discussed by Leo Szilard in 1929,[18] who argued that the demon is part of the total system and that the demon's entropy increases by receiving and storing information. The demon's increase in entropy must be added to the total entropy and thus cancels the decrease in entropy of the gas in the two chambers. For refinements of Szilard's argument, other solutions, and the relation of entropy to information theory, see Ref.[19]; a web source, describing the paradox and its solutions, is in Ref.[20].

### Electricity and magnetism

James Clerk Maxwell's derivation of the classical field equations of electromagnetism represents his greatest triumph. This accomplishment places him in the rank of the very great physicists, such as Galileo, Newton, and Einstein.

#### On Faraday's Lines of Force

As an undergraduate in Cambridge University, Maxwell read Michael Faraday's Experimental Researches in Electricity (1839-1855), in which Faraday explained magnetism by force lines traversing space, instead of by actions at a distance. Thomson had cast some of Faraday's ideas into mathematical form and guided by his work Maxwell went a step further. In his first paper on electromagnetism (1855), entitled On Faraday's Lines of Force,[21] one of his aims was to overturn the idea of action at a distance.[22]

Maxwell presented a — what he called — geometrical model of lines of force in space. This representation rested on equipotential surfaces (scalar functions) to which the force lines (gradient vectors) are orthogonal and give paths of steepest descent from one equipotential surface to the next. Maxwell was guided by the physical analogy of incompressible fluids; not that Maxwell believed electricity to be a fluid, but he recognized the mathematical analogy of field lines to the velocity vectors of flowing fluids.

#### On Physical Lines of Force

Five years later, in 1861, Maxwell returned to the subject and wrote a series of four papers entitled On Physical Lines of Force.[23] In this work he introduced a mechanical ether model, based on "molecular vortices" and "idle wheel particles", that is rather foreign to the modern reader. However, he inferred two momentous consequences: the usual conduction current has to be augmented with a displacement current and the ether sustains transversal vibrations propagating with a velocity c, now known as the speed of light.

To put the second consequence into perspective it must be recalled that in 1856 c had been measured by Wilhelm Eduard Weber and Rudolf Kohlrausch as the factor connecting two different definitions of electric charge (see statcoulomb and abcoulomb): by Coulomb's law and by Ampère's equation. Almost immediately Gustav Kirchhoff noticed that their value of c was surprisingly close to the speed of light determined by Hippolyte Louis Fizeau in 1849. And now Maxwell had shown that electromagnetic waves propagate with exactly the same speed.

In the introduction of part III of his On Physical Lines of Force Maxwell puts forward the supposition that there is only one medium, one ether, carrying both light and electromagnetic phenomena. On p. 22 of the same paper, he writes the following paragraph which is one of the landmarks in the history of the exact sciences:

The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.

#### A Dynamical Theory of the Electromagnetic Field

In 1865 Maxwell wrote his most important paper "A Dynamical Theory of the Electromagnetic Field,[24] in which he introduces a set of equations, now known as the Maxwell equations. This paper is very accessible, also by the modern reader, because he abandoned his mechanical model of vortices and idle wheels, and only applies concepts that are still being taught in modern university courses. He presents twenty equations, divided in eight groups labeled (A) to (H), with twenty variables. Since he did not yet know vector or quaternion notation the different components of his vector fields are counted as individual parameters and the equations are written out in components. For a representation of his equations in modern vector form and comments on them, see the Addendum.

#### The Treatise

Maxwell's electrical work was ultimately summed up in his Treatise on Electricity and Magnetism[3]. The first edition appeared in 1873, the second in 1881, after the author's death, and was edited by W. D. Niven, although it had been revised in part by Maxwell. The third edition, 1891, was edited by Joseph John Thomson. The book is not a systematic treatise in the ordinary sense, and it would be hard to go through it from cover to cover. It contains lots of interesting material and insights, but this has to be dug out the hard way. Maxwell's equations appear only in Chapter IX of part 4, well into the second half of the second volume. Maxwell uses both component and quaternion notation. The book appeared to contemporary physicists, and even to the immediate following generations, as a great but inaccessible monument.

## References and notes

1. For instance, James's great-grandmother from father's side was Dorothea, Lady Clerk Maxwell. She was the daughter of William and Agnes (née Maxwell) Clerk Maxwell. Dorothea married Sir George Clerk and — as usual in those days — carried the married name Clerk, instead of her father's name Clerk Maxwell.
2. Lewis Campbell and William Garnett (1882) The Life of James Clerk Maxwell MacMillan, London Online
3. J. Clerk Maxwell (1873) A Treatise on Electricity and Magnetism. Clarendon Press, Oxford.
4. Tait PG (1985) Elementary Treatise on Quaternions Oxford
5. Crowe MJ A History of Vector Analysis Dover, New York p. 129
6. Lewis Campbell and William Garnett (1882) The Life of James Clerk Maxwell, MacMillan, London
7. It is interesingt to note in this context that the competing colour theory (Zur Farbenlehre) of Young's famous German contemporary, Goethe, has had no impact on scientific colour theory.
8. J. Clerk Maxwell(1861) On the theory of three primary colours Proc Royal Institution Great Britain 3:370-4
9. Evans RM (1961) Maxwell’s colour photograph Scientific American 205:118–27
10. Dougal RC et al. (2006) Then and now: James Clerk Maxwell and colour Optics Laser Technol38:210–8. An in-depth review of Maxwell's work on colour theory with a link to present day applications of colour science. http://dx.doi.org/10.1016/j.optlastec.2005.06.036
11. Maxwell JC (1859) On the stability of the motion of Saturn's rings Monthly Notices, Astronomical Soc London 19:297-304
12. Newman JR The World of Mathematics, vol. II, p. 774, excerpt from Daniel Bernoulli's book Hydrodynamica (1738)
13. Clausius R (1850) On the mean length of the paths described by the separate molecules of gaseous bodies on the occurrence of molecular motion Phil Magazine, Ser. 4 17:81-91 (1859)
14. Maxwell JC (1860) Illustrations of the dynamical theory of gases I. On the motions and collisons of perfectly elastic spheres Phil. Magazine, Ser. 4 19:19-32; II. On the process of diffusion of two or more kinds of moving particles among one another Phil. Magazine, Ser. 4 20:21-37
15. J. Clerk Maxwell, On the dynamical Theory of Gases, read May 31, 1866; published in Philosophical Transactions of the Royal Society of London, vol. 157, pp. 49-88 (1867)
16. J. Clerk Maxwell, Theory of Heat, Longmans, Green, London (1871); reprinted Dover, New York (2001). Maxwell wrote about a "finite being", the name "demon" was coined by W. Thomson in 1874.
17. One can confirm this easily for a monoatomic ideal gas by direct calculation. The energy of one mole of such gas is 3RT/2, with R the molar gas constant and T the absolute temperature. The entropy of one mole is S(T) = R ln(VT 3/2) + S(0). Assume that the amount of gas in A and B stays constant (one mole each) during the demon's operation and that both partitions have the same volume V. By conservation of energy
${\displaystyle \scriptstyle \Delta T^{A}+\Delta T^{B}=\,0.}$
Assume further that the initial temperature of A and B is equal to T. Under these conditions the entropy change, caused by the demon, is
${\displaystyle \scriptstyle \Delta S^{A}+\Delta S^{B}={\frac {3}{2}}\,R\,\ln {\Big [}1-\left({\frac {\Delta T^{A}}{T}}\right)^{2}{\Big ]}.}$
Because the temperature decrease ΔTA is limited by TA = 0 and TA = T, it follows that the argument of the logarithm lies between zero and unity, and hence the logarithm is negative and so is the entropy change of A + B.
18. L. Szilard, Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. [On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings]. Zeitschrift für Physik, vol. 53, pp. 840-856 (1929) doi
19. Maxwell's Demon 2, Entropy, Classical and Quantum Information, Computing edited by Harvey S. Leff and Andrew F. Rex, Institute of Physics Publishing, Bristol, 2nd edition, 2003
20. Maxwell's demon - blog post on the history of thermodynamics, with detailed treatment of the demon
21. J. Clerk Maxwell, On Faraday's Lines of Force, Trans. Cambridge Philosophical Society, vol. 10, pp. 27-83 (1855)
22. P. G. Tait in a posthumous article on Maxwell in the 1911 edition of the Encyclopaedia Britannica. Online
23. J. Clerk Maxwell, On Physical Lines of Force, Philosophical Magazine 4th series, Four Parts, I: vol. 11, pp. 161-175; II: vol. 11, pp. 281-291, 338-347; III: vol. 13, pp. 12-23; IV: vol. 13, pp. 85-95 (1861, 1862).online
24. J. Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field" Phil. Trans. Roy. Soc 155:459 - 512 (1865) online)


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# 1.2 Unit circle: sine and cosine functions  (Page 7/12)

 Page 7 / 12

Find the coordinates of the point on the unit circle at an angle of $\text{\hspace{0.17em}}\frac{5\pi }{3}.\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\text{\hspace{0.17em}}$

Access these online resources for additional instruction and practice with sine and cosine functions.

## Key equations

 Cosine $\mathrm{cos}\text{\hspace{0.17em}}t=x$ Sine $\mathrm{sin}\text{\hspace{0.17em}}t=y$ Pythagorean Identity ${\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1$

## Key concepts

• Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
• Using the unit circle, the sine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ whereas the cosine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the x -value of the endpoint. See [link] .
• The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See [link] .
• When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See [link] .
• Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See [link] .
• The domain of the sine and cosine functions is all real numbers.
• The range of both the sine and cosine functions is $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$
• The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
• The signs of the sine and cosine are determined from the x - and y -values in the quadrant of the original angle.
• An angle’s reference angle is the size angle, $\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis. See [link] .
• Reference angles can be used to find the sine and cosine of the original angle. See [link] .
• Reference angles can also be used to find the coordinates of a point on a circle. See [link] .

## Verbal

Describe the unit circle.

The unit circle is a circle of radius 1 centered at the origin.

What do the x- and y- coordinates of the points on the unit circle represent?

Discuss the difference between a coterminal angle and a reference angle.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis.

Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

The sine values are equal.

## Algebraic

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $t$ lies.

$\text{sin}\left(t\right)<0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{cos}\left(t\right)<0\text{\hspace{0.17em}}$

$\text{sin}\left(t\right)>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)>0$

I

$\mathrm{sin}\left(t\right)>0$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)<0$

$\mathrm{sin}\left(t\right)<0$ and $\mathrm{cos}\left(t\right)>0$

IV

For the following exercises, find the exact value of each trigonometric function.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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s.
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for screen printed electrodes ?
SUYASH
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of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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Cesar
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Stotaw
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Azam
Hello
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text:
# Renderer¶

The clara.viz.core.Renderer class consumes the DataDefinition object and produces, after rendering, images or video streams. It also exposes ways to query and set rendering properties such as camera settings, lighting properties or material properties (transfer functions).

## Usage¶

For the renderer to work three things needs to be provided

• what to render

• how to render it

• where to send the rendered images

### What to render¶

The DataDefinition class provides an easy way to specify input data used by the renderer. The clara.viz.core.Renderer.set_arrays() function can also be used if absolute flexibilty and control on how data is provided is required.

### How to render¶

The render settings are organized as a dictonary and can be

### Where to send the rendered images¶

The renderer can generate still images or video streams.

To create a still image call the clara.viz.core.Renderer.render_image() function. The output of this call is either raw RGB/depth data or a JPEG encoded image.

For rendering to a video stream call the clara.viz.core.Renderer.create_video_stream(). The call takes a callback function which is called when new video data is produced. The call also returns a clara.viz.core.VideoStream object which can be used to configure the video stream (width, height, frame rate and bit rate) and to start/pause/stop the stream.


text:
## POD Floor and Ceiling

### Some data do not support …

#### … a POD curve that goes to zero on the left, or to one on the right.

That means (but perhaps not as obviously as it should) that the standard POD(a) model with asymptotes at zero and one won’t fit the data.

For situations with a data “floor” (a non-zero lower asymptote) or a data “ceiling” (an upper asymptote less than one) we must use a model having the appropriate characteristics, shown in Figure 1.

### POD floor and ceiling

Notice in Figure 2 that there is a handful of very small cracks that are reported as “hits,” the consequence of a corrupting background signal (e.g. “noise”). That means that the lower value of POD cannot be zero. And that means that the standard POD(a) model won’t work. (Well, you can force it to give you an answer, but it will be wrong, as we shall see.)

###### Figure 2 – Data inappropriately described without a POD “floor.”

OK. So who cares about the lower left? We’re interested in the upper right.

Ignoring the non-zero lower asymptote produces an overall poor fit because the model tries to compensate for the poor fit on the lower left with a shallow slope. That, in turn, causes a lousy fit at the upper end as well. This results in a $$a_{90/95}$$, the cracksize with 90% probability of detection and 95% confidence, that is too large, and is incorrectly computed to be 42.

###### Figure 3 – Data is now correctly described using a POD “floor.”

With the POD floor modeled correctly the overall fit is greatly improved and $$a_{90/95}$$ is 28, as seen in figure 4.

### What to Remember:

While the statistical methods in MIL-HDBK-1823A work in the large majority of situations, they do not work in all situations. When the model (any mathematical model, not just a POD model) doesn’t have the same characteristics as the data, then it is the wrong model. Yes, you can torture the software to use it anyway, but your answer will be wrong. No verbal legerdemain can trick Mother Nature, and if you insist on trying (so you can check the box and move on), ultimately she will have her revenge.

The Standard mh1823 POD Software has rudimentary capabilities for detecting a POD ceiling by computing the observed fraction of misses for targets larger than a purported $$a_{90/95}$$ and comparing that empirical POD with POD = 90%. The new 2018 Workshop Addendum can compute a POD Ceiling (or Floor), and confidence bounds for it, plus the Akiake Information Criterion, and Schwarz Criterion for statistical significance.

If your data suggest that their POD minimum value never reaches zero, or their maximum doesn’t approach one, you will need the 2016 POD Workshop Addendum software to model a POD Floor or Ceiling with a 3-parameter model having a lower asymptote (POD > 0) or an upper asymptote (POD < 1).

The advanced POD Workshop Software is an integral part of the 2-day POD Workshop and Short Course.


text:
# [Thermo] - Maxwells Relations? Prove the Validity of the Equations

1. Apr 18, 2008

### tesla22

1. The problem statement, all variables and given/known data

We are working on some problems for class and we are given statements which I accept as valid but don't know how to prove they are valid. I believe I have to utilize the maxwell relations but the terms seem unfamiliar to me.

2. Relevant equations

(1)
Partial
(d^2f / ds^2)_T = T / Kv

(2)
Partial
(d^2h / ds^2)_P = T / C_p

(3)
Partial
(d^2u / ds^2)_v = T / C_v

3. The attempt at a solution

For the first equation, I know that the isothermal compressibiity K = -1/v partial(dv/dP)_T

For the second equation I also know that C_p/T = partial(ds/dT)_P = 1/T * partial(dh/dT)_P and i need to take this knowledge to combine the equations but I don't see how I would be getting the square on the ds out of this. Obviously I'm missing something.

C_v/T should be the same proof with u in place of h holding v constant.

Thanks for any help.

2. Apr 18, 2008

### Mapes

Take a look at my recent post on Maxwell relations. To handle the second derivative, note that $\left(\frac{\partial H}{\partial S}\right)_P$ is also known as $T$. So differentiate $T$ with respect to $S$ again at constant pressure. Does this help?

3. Apr 21, 2008

### tesla22

Thank You, I think the answers were staring me in the face but in a different format and i didn't put 2 and 2 together until your response. I now have a solution where T=du/ds with v constant and T=dh/ds with p constant (correct?) and plugging these in to the equations cv=T(ds/dt)_v and cp=T(dt/ds)_p to get the desired proof. I think its right.

I'm still struggling with the third one however so any assistance would be appreciated. How do you make the proper symbols appear?

4. Apr 21, 2008

### Mapes

The first relation can't be true, the units don't match up.

5. Apr 21, 2008

### tesla22

Yes thats the one i'm struggling with sorry, the first one. The 2nd and 3rd I believe I have proven as said in the previous post.

I completely screwed up the first one, the actual problem is as follows:

(d^2f / dv^2)_T = 1 / Kv

This is the one I am struggling with at the moment. Sorry I was trying too hard to convey the format of the problem that i got the variables wrong somehow.

6. Apr 21, 2008

### Mapes

OK, that makes a lot more sense. Try writing out the differential form of the Helmholtz free energy dF and taking the derivative of that twice with respect to V at constant T. The first derivative equals a well-known parameter.


text:
# Antirhombicosicosahedron

Antirhombicosicosahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymArie
Elements
Components5 cuboctahedra
Faces40 triangles as 20 hexagrams, 30 squares
Edges120
Vertices60
Vertex figureRectangle, edge lengths 1 and 2
Measures (edge length 1)
Volume${\displaystyle \frac{25\sqrt2}{3} ≈ 11.78511}$
Dihedral angle${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 125.26439°}$
Central density5
Related polytopes
ArmySemi-uniform Srid
RegimentArie
DualCompound of five rhombic dodecahedra
ConjugateAntirhombicosicosahedron
Convex coreRhombic triacontahedron
Topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The antirhombicosicosahedron, arie, or compound of five cuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 30 squares, with two of each joining at a vertex.

It can be thought of as a rectification of either the small icosicosahedron or the rhombihedron, or the cantellation of the chiricosahedron.

Its quotient prismatic equivalent is the cuboctahedral pentachoroorthowedge, which is seven-dimensional.

## Vertex coordinates

The vertices of an antirhombicosicosahedron of edge length 1 can be given by all even permutations of:

• ${\displaystyle \left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0\right),}$
• ${\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{\sqrt{10}}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{4},\,±\frac{3\sqrt2-\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{8}\right).}$


text:
How To Simplify Rational Expressions? - Examples With Detailed Solutions

How to simplify rational expressions? Grade 11 examples are presented along with detailed solutions and full explanations. More questions with answers at the bottom of the page are also included.

 How to simplify rational expressions? How to simplify rational expressions? The rules of addition, subtraction, multiplication and division of rational expressions are used to simplify complex expressions. Rational expressions with the same denominator are added or subtracted as follows: $\dfrac{A}{B} \pm \dfrac{C}{B} = \dfrac{A \pm C}{B}$ Rational expressions are multplied as follows: $\dfrac{A}{B} \cdot \dfrac{C}{D} = \dfrac{A \cdot C}{B \cdot D}$ We divide two rational expressions by multiplying the first rational expression by the reciprocal of the second rational expression as follows: $\dfrac{A}{B} \div \dfrac{C}{D} = \dfrac{A}{B} \cdot \dfrac{D}{C} = \dfrac{A \cdot D}{B \cdot C} \;\; \text{or} \;\; \dfrac{ \dfrac{A}{B} }{ \dfrac{C}{D}} = \dfrac{A}{B} \cdot \dfrac{D}{C} = \dfrac{A \cdot D}{B \cdot C}$ If you have difficulties in simplifying rational expressions, review the tutorials in How to Add, Subtract and Simplify Rational Expressions and How to multiply, divide and simplify rational expressions and then start the present tutorial. The examples with detailed solutions and explanations in this tutorials will help you overcome any difficulties in simplifying rational expressions on the condition that you understand every step involved in solving these questions and also spend more time practicing if needed. I will present the examples below starting with fractions first and then with rational expressions, with more challenging questions as you walk through the tutorial. You need to understand each step! Example 1: Simplify: $\dfrac{\dfrac{2}{3} + 5}{4} +\dfrac{1}{2}$. Solution: We first simplify the numerator $\dfrac{2}{3} + 5$ of the fraction $\dfrac{\dfrac{2}{3} + 5}{4}$. Convert to common denominator. $\dfrac{\dfrac{2}{3} + 5}{4} +\dfrac{1}{2} = \dfrac{\dfrac{2}{3} + 5 \cdot \dfrac{3}{3}}{4} +\dfrac{1}{2} = \dfrac{\dfrac{17}{3}}{4} +\dfrac{1}{2}$ Change 4 to a fraction 4 / 1 and simplify $\dfrac{\dfrac{17}{3}}{4}$ by dividing the fractions which comes to a multiplication by the reciprocal. $= \dfrac{\dfrac{17}{3}}{\dfrac{4}{1}} +\dfrac{1}{2} = \dfrac{17}{3} \cdot \dfrac{1}{4} +\dfrac{1}{2}$ Multiply the fractions $\dfrac{17}{3} \cdot \dfrac{1}{4}$. $= \dfrac{17 \cdot 1}{3 \cdot 4}+\dfrac{1}{2} = \dfrac{17}{12} +\dfrac{1}{2}$ Convert to common denominator and add. $= \dfrac{17}{12} +\dfrac{1}{2} \cdot \dfrac{6}{6} = \dfrac{17+6}{12} = \dfrac{23}{12}$ Example 2: Simplify: $\dfrac{\dfrac{x+1}{x-2}+\dfrac{x}{x+1}}{\dfrac{1}{x+1}}$. Solution: We first simplify the numerator $\dfrac{x+1}{x-2}+\dfrac{x}{x+1}$ by converting to the same denominator and applying the rule of the sum of two rational expressions whose lowest common denominator is (x - 2)(x + 1). $\dfrac{\dfrac{x+1}{x-2}+\dfrac{x}{x+1}}{\dfrac{1}{x+1}} = \dfrac{\dfrac{x+1}{x-2} \cdot \dfrac{x+1}{x+1} +\dfrac{x}{x+1} \cdot \dfrac{x-2}{x-2} }{\dfrac{1}{x+1}} = \dfrac{ \dfrac{(x+1)^2+x(x-2)}{(x+1)(x-2)} }{\dfrac{1}{x+1}}$ We now apply the rule of division of rational expressions by multiplying by the reciprocal of $\dfrac{x+1}{1}$. $= \dfrac{((x+1)^2+x(x-2))}{(x+1)(x-2)} \cdot \dfrac{x+1}{1}$ Simplify by cancelling common factors. $= \dfrac{((x+1)^2+x(x-2))}{{\colorcancel{red}{(x+1)}}(x-2)} \cdot \dfrac{\colorcancel{red}{(x+1)}}{1} = \dfrac{((x+1)^2+x(x-2))}{(x-2)}$ Expand and simplify. $= \dfrac{2x^2+1}{x-2} \;\; \text{for} \;\; x \ne -1$ Example 3: Simplify: Simplify: $\dfrac{\dfrac{x-1}{3x+2}+3}{\dfrac{x+1}{6x+4}} - 2$. Solution: We simplify $\dfrac{x-1}{3x+2}+3$ by first converting to the same denominator $\dfrac{\dfrac{x-1}{3x+2}+3}{\dfrac{x+1}{6x+4}} - 2 = \dfrac{\dfrac{x-1}{3x+2}+3 \cdot \dfrac{3x+2}{3x+2}}{\dfrac{x+1}{6x+4}} - 2$ then applying the rule of the sum of two rational expressions. $= \dfrac{\dfrac{x - 1 +3 \cdot (3x+2)}{3x+2}}{\dfrac{x+1}{6x+4}} - 2$ Expand and simplify $x - 1 +3 \cdot (3x+2)$ . $= \dfrac{\dfrac{10x + 5}{3x+2}}{\dfrac{x+1}{6x+4}} - 2$ We use the rule of division of rational expressions which reduces to a multiplication by the reciprocal on: $\dfrac{\dfrac{10x + 5}{3x+2}}{\dfrac{x+1}{6x+4}} = \dfrac{10x + 5}{3x+2} \cdot \dfrac{6x+4}{x+1}$. $= \dfrac{10x + 5}{3x+2} \cdot \dfrac{6x+4}{x+1} - 2$ Factor $= \dfrac{5(2x + 1)}{3x+2} \cdot \dfrac {2(3x+2)}{x+1} - 2 =$ and simplify . $= \dfrac{5(2x + 1)}{{\colorcancel{red}{3x+2}}} \cdot \dfrac {2{\colorcancel{red}{(3x+2)}}}{x+1} - 2 = \dfrac{10 (2x+1)}{x+1}-2$ Convert to the same denominator, apply the rule of addition of rational expressions and simplify. $\dfrac{10(2x+1)}{x+1}-2 \cdot \dfrac{x+1}{x+1} = \dfrac{10(2x+1) - 2(x+1)}{x+1} = \dfrac{2(9x+4)}{x+1} \;\; \text{for} \;\; x \ne -2/3$ Example 4: Simplify: $\dfrac{ \dfrac{2-x}{x+2}-\dfrac{4}{x+2}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}}$ Solution: Apply the rule of subtraction of rational expressions to $\dfrac{2-x}{x+2}-\dfrac{4}{x+2}$ and $\dfrac{1}{x+3}-\dfrac{4}{x+3}$ $\dfrac{ \dfrac{2-x}{x+2}-\dfrac{4}{x+2}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}} = \dfrac{ \dfrac{2-x -4}{x+2}}{\dfrac{1-4}{x+3}}$ then simplify. $= \dfrac{ \dfrac{-x - 2}{x+2}}{\dfrac{-3}{x+3}}$ Apply the rule of division of rational expressions and factor $= \dfrac{-x - 2}{x+2} \cdot \dfrac{x+3}{-3} = \dfrac{-(x + 2)}{x+2} \cdot \dfrac{x+3}{-3}$ and simplify $= \dfrac{-{\colorcancel{red}{(x+2)}}}{{\colorcancel{red}{x+2}}} \cdot \dfrac{x+3}{-3} = \dfrac{x+3}{3} \;\; \text{for} \;\; x \ne -2$ Example 5: Simplify: $\dfrac{ \dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}}$ Solution: Apply the rule of multiplication to $\dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3}$ and the rule of subtraction to $\dfrac{1}{x+3}-\dfrac{4}{x+3}$. $\dfrac{ \dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}} = \dfrac{ \dfrac{4(2-x)}{(x+2)(x+3)}}{\dfrac{1 - 4}{x+3}}$ Apply the rule of division: $= \dfrac{4(2-x)}{(x+2)(x+3)} \cdot \dfrac{x+3}{-3}$ Simplify. $= \dfrac{4(2-x)}{(x+2){\colorcancel{red}{(x+3)}}} \cdot \dfrac{{\colorcancel{red}{(x+3)}}}{-3} = -\dfrac{4(2-x)}{3(x+2)} \;\; \text{for} \;\; x \ne -3$ Example 6: Simplify: $\dfrac{1}{1+\dfrac{1}{x+\dfrac{1}{x+1}}}$. Solution: We first convert the terms in $x + \dfrac{1}{x+1}$ to the same denominator. $\dfrac{1}{1+\dfrac{1}{x+\dfrac{1}{x+1}}} = \dfrac{1}{1+\dfrac{1}{x \cdot \dfrac{x+1}{x+1}+\dfrac{1}{x+1}}}$ We now add $x \cdot \dfrac{x+1}{x+1}+\dfrac{1}{x+1}$ and simplify. $= \dfrac{1}{1+\dfrac{1}{\dfrac{x^2+x+1}{x+1}}}$ Note that $\dfrac{1}{\dfrac{x^2+x+1}{x+1}}$ is a reciprocal is equal to $\dfrac{x+1}{x^2+x+1}$. Hence $= \dfrac{1}{1+ \dfrac{x+1}{x^2+x+1}}$ Convert the terms in $1+ \dfrac{x+1}{x^2+x+1}$ to the same denominator. $= \dfrac{1}{1 \cdot \dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1}}$ Simplify and apply the rule of addition to $\dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1}$. $= \dfrac{1}{ \dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1}} = \dfrac{1}{ \dfrac{x^2+x+1+x+1}{x^2+x+1}} = \dfrac{1}{ \dfrac{x^2+2x+2}{x^2+x+1}}$ The last expression is the reciprocal of a rational expression which may be written as. $= \dfrac{x^2+x+1}{x^2+2x+2}$ More Questions: Simplify the following expressions - Answers at the bottom of the page. a) $\dfrac{\dfrac{2}{5} + 7}{\dfrac{4}{3}} +\dfrac{1}{3}$ b) $\dfrac{\dfrac{x-2}{x+3}+\dfrac{x}{x+3}}{\dfrac{1}{2x+6}}$ c) $\dfrac{\dfrac{x-1}{3x-12}+3}{\dfrac{x+1}{x-4}} - \dfrac{2}{3}$ d) $\dfrac{ \dfrac{2-x}{x+1}-\dfrac{4}{x+1}}{\dfrac{1}{x-5}-\dfrac{4}{x-5}}$ e) $\dfrac{ \dfrac{2-x}{x+2} - \dfrac{4}{x+3}}{\dfrac{1}{x+3} \cdot \dfrac{4}{x+3}}$ f) $\dfrac{\dfrac{1}{2}}{1+\dfrac{1}{\dfrac{1}{x-1}-2}}$ Answers to the above questions a) $\dfrac{353}{60}$ b) $4(x-1)$ c) $\dfrac{8x-39}{3(x+1)}$ d)$\dfrac{ (x+2 ) (x-5)}{3(x+1)}$ e) $\dfrac{(-x^2-5x-2)(x+3)}{4(x+2)}$ f)$\dfrac{-2x+3}{2(-x+2)}$

Updated: 20 January 2017 (A Dendane)