Bignose

Group grading only works if each member of the group gets to submit a closed grade for each group member. That way, if you are in a group where you did 90% of the work, you will note that only 10% of the work was done by the others in a closed, sealed, private report between you and the professor only. If it open, you could get pressured to say everyone worked on it, but if closed you can honestly report that the others did nothing. In terms of what makes a good teacher, I think it is one who makes sure that students with all different learning styles are included. There are some students who like to see the derivation, there are some that like to see examples, there are some who like to see the end product. For example, consider a civil engineering class that is learning how to construct a dam. One the one hand, the mathematically inclined students will enjoy seeing where and how to apply the appropriate equations, whereas the hands-on students would love to see a presentation from an engineer that helped coordinate a dam construction job 10 years ago, or better yet a field trip out to a construction site. A good teacher knows how to involve all the different kinds of learners. It is an unfortunate truth that too many professors concentrate on the kind of learners that they themselves are. The reason is simple, that is the way the professor knows it, and he/she really connects with the students that learn it the same way. But, the really good ones go out of their way to make sure all types are accommodated. It may be very uncomfortable for the professor the first few times going outside of their preferred method, but the really great ones do. This is really a summary of Kolb's learning cycle. ( see https://engineering.purdue.edu/ChE/News_and_Events/Publications/teaching_engineering/chapter15.pdf for example ) Kolb theorized that the way to make sure that a fact is 100% in the student's brain is to perform all the different ways of knowing something. I.e. do both the hands on approach and the derivation and apply the equation. But, since each student has a preferred way of learning, they enter the cycle at different points. The real key is for the teacher to facilitate every single student to go completely around the entire cycle, maybe multiple times, so that every student no matter where they entered the cycle has learned the same principle from multiple points of view -- their preferred method and other alternative methods.

This may be a bit off topic, but Hooke's law can be derived. It is a conservation of momentum equation for a very specific case. Conservation laws have become laws since as far as every case that has ever been studied, the appropriate quantites (mass, momentum, energy) have obeyed the conservation law. Starting with the conservation principle, you can get pretty close to most of the fluid mechanics and solid mechanics equations. Most of the time, a constitutive realtionship needs to be assumed (like defining/assuming what the viscosity of a fluid or the elasticity of a solid is.) but with the conservation and constitutive equations, the equations that describe nature are derived. I guess in these cases, if you don't consider the conservation laws as a basis for the derivations, then they cannot be completely derived, but starting with conservation and using appropriate constitutive realtions quite a lot can be done. All that said, there are statistical methods that give the probabilities of linear fit versus quadratic fit. Look into almost any statistical design book. R.A.Fisher's Statistical Methods, Experimental Design, and Scientific Inference is very well thought of. Especially see the section "Tests of Goodness of Fit"

Well, per the rules, nobody is going to do this for you. What equations do you have at hand that may apply? What relationships do you know between free energy, enthalpy, and entropy? What assumptions do you think can be made about this problem? Basically, show us what work you have done up to this point and where you are having trouble.

You have to very careful here to make sure your words say exactly what you are trying to say. There is a phrase in here that you said was impossible: "...vector A_x is simply the magnitude..." A vector cannot be the same things as a magnitude. One is a vector quantity, and the other is a scalar. They cannot be equal. The magnitude of a vector is a scalar. When writing equations, a vector can only be equal to another vector. The components A_x, A_y, etc. are scalars, but when you write them together with basis vectors, you are representing a vector then. What the phrase should have been was "component A_x of the vector A is simply the magnitude..." Also, i cannot be positive or negative. It is a basis vector that defines in what direction the x coordinte points. If you want a vector that points in the other direction, you put a negative component in front of it. Also, the cosine terms comes from projection operators. They arise when given a vector's magnitude, how much goes along the x-coordinate. JustStuit's example should not have had a subscript x next to the A, just the magnitude of A. That is probably a lesson or two away, and really doesn't have anything to do with just using notation to describe a vector.

Well, I didnt calculate an answer, so I don't know if it is right or wrong You can put your answer back into the PDE and see if the two sides are equal or not. You can always check, and probably should whenever possible. There are many different ways to solve this kind of problem. I know from your posts over the last few months you've been going through orthogonal functions, and these problems from 0 to infinity do lend themselves to being solved via the method of Laplace transforms. Some of the other ones can be solved via the method of Fourier transforms. But, whatever way you solve it, you should get the same answer, provided the problem fulfills the uniqueness theorems (which pretty much any homework problem will). It may be expressed a little differently depending on the method you used, but the answer from any valid method you performed correctly will be unique. So, whether you use separation of variables or any other method, it should be the same answer. BTW, almost any book on PDEs you get from the library will have lots of info on separation of variables, if you want to read up on it. Like I said, it is the first method I always try, since it only takes a few seconds and you can see quickly whether the method will work or not.

Sarah, do you know what the method of separation of variables is? (I mentioned it in your wave equation post.) Let me show you First, we let u(x,t) equal the product of two functions, one only a function of x and the other only a function of t Let u(x,t)=X(x)T(t) Now, plug this into your equation (I'm sorry, but latex doesn't appear right on my screen so I'm gonna try to do this with normal text. And d means partial in this case) du/dt becomes d(XT)/dt which is equal to XdT/dt since X is only a function of x which I'm going to shorthand even further to XT' where the prime means differentiation only on that functions variable. kd^2u/dx^2 becomes kTd^2X/dx^2 or kTX'' So, your PDE is now: XT' = kTX'' or rewritten T'/T = kX''/X Now, look at this, the lefthandside is only a function of t, and the righthandside is only a function of x. How can this be? If and only if both the lefthand and righthand sides are equal to a constant, which I am going to denote R. T'/T = kX''/X = R this can be written as two ODEs now. T'/T = R & kX''/X = R This two ODEs can be solved. The boundary conditions can also be converted using the separation of variables, and should be applied to the appropriate ODE. For example, using the BCs of problem b, it becomes: T'/T = R & T(0)=0 kX''/X = R & X(0)=1 & X(+infinity)=0 p.s. I assume that as x goes to infinity, u goes to zero right? Otherwise your problem statement is missing a boundary condition. After solving the ODEs, you put the solution back together, u=X*T and you have an answer. p.p.s. Like I mentioned in your wave equation post, separation of variables is the very first thing I always try on a PDE that is supposed to have an analytical solution. There are few ODEs that have analytical solutions, and even fewer PDEs, so the PDE homework problems that are out there are almost all separable. There are some out there that require other tricks (like Fourier or Laplace or other transforms), but it rarely hurts to try separation first. You can usually see pretty quickly if it is going to work or not.

"solved by" is definately the wrong word, "represented by" would be much more accurate. One thing I would like to say is to not fall in love with i,j,&k, since many other books use other notations. Learn the concept, don't get hung up by the notation (you would not believe how much trouble some students have with this). Really, those basis vectors (the bold letters) are there to help remind you which part of the vector points in each direction. That is, what part of any given vector points in the x direction, the y direction, etc. There are many different way of representing this: (let ~ mean "represented by") v ~ [vx vy vz] ~ vxi + vyj + vzk It is really a notation thing to aid in bookkeeping.

well, u(x,t)=0 cannot be right, since the equation in a becomes 0=x*t Have you tried separation of variables? That method is usually the first bullet out of my gun when given problems like that. Let u(x,t)=X(x)*T(t), then you should be able to write 2 equations, one for X(x) (and only a function of x) and T(t) for only a function of t. Then solve the two ODEs.

3 mins?!? The trend today is to try to get as short and catchy a phrase as possible. It used to be the 3 min soundbite, then the 30 sec soundbite, now the goal is 10 words or less. E.g., how many times have you heard "cut-and-run" the last 6 months? Politics would not be streamlined if the public didn't basically demand it. Here's a scary statistic I just read yesterday. There is a very close Senatorial Race here in Missouri, where the incumbent has been running tv ads for 2 months or so now. The challenger just began over Labor Day. Both of their consultants expect lots more (and more negative ads) before election day. The quote from a political observer group from Vrigina: Over 80 of people get all of their information (about candidates and up-for-vote issues) from television. That only leaves 20 who even consult any other source, like radio, newspapers or magazines, the candidate's own pamphlets, and the Internet. If only 20% of people are even bothered to look for additional information beyond the sound bite, the process will naturally tend toward a streamlined, sound-bitish, partisan, confrontational, negative-ad-producing process. It is really up to the voters to change the system as we now have it.

Good to meet you too, Martin. I usually read through most new threads up in the physics and math sections, including your reviews of Smolin's book. I usually have very little to add to the physics section, though, since I am engineer, and deal with flows in pipes and mm size particles. I am not familiar with string theory since it is not going to help me compute multiphase flows. But, it is always interesting, and especially interesting is the dynamic as the theory progresses and critiques show up. Here is a real good question, has it really taken 30 years for people to come out an say 'Um, string theory has yet to make a verifiable new prediction,' or have those people always been there and are just now getting press? And of course, by 'those people' I mean the well-trained (physics-wise) and respected member of the physics community, not the woo-woos who think that their theory of how love and the color purple (and no math can describe that) keeps the universe together proving, of course, that all we need it love.

Well this is awfully contradictory, since you are trying to discuss a mathematical concept, and you want to try to limit the amount of math we can use. Rather difficult to do.... That said, the very first thing that comes to mind is repetition is necessary. If the second die does the series you suggested every time, then it is obviously deterministic (non-random). But, there is a finite chance that the series did just occur on its own. The next thing is that the experiment needs to be defined much more carefully. Is an experiment 1000 rolls? Then there is a distribution, P(1st roll = n1, 2nd roll = n2, 3rd roll = n3, ...). which is a huge function with 1000 variables. If each roll is independent, then it becomes P1*P2*...*P1000 where (P1 is shorthand for P(1st roll) & etc.). This could also be the case where 1 experiment is 1 roll of the die, and it is repeated 1000 times. Lastly, I'd like to echo a lot of what bascule said, though in a slightly different way. The human brain works to a huge extent by pattern recognition. And, so, even in truly random situation, the brain is going to try to find a pattern in the data, whether one exists or not. Look how many times cause and effect are announced just because two variables have a correlation? E.g. there is a correlation between having a healthy mouth and overall health of the body. The toothpaste companies in their ads try to tell you that a healthy mouth leads to a healthy body. And while, that is probably true to a certain extent, my bet is that the people who take care of their entire body are much more likely to brush and floss regularly, hence the correlation between the two. One of the best ways to untrain the brain to look for patterns is to work through and understand the nuances of probability and statistics. You can quickly learn that coincidences just happen all the time, and that they don't necessarily mean anything. Also, you can learn how assign values of confidence to know how likely an event or hypothesis is.

All physics aside, I really enjoy reading Gregg Easterbrook's work. I certainly don't always agree with him, but for the most part, he explains why he feels the way he does. Instead of just disagreeing with someone because that someone comes from the other political party, like soooooo many people do. The man writes one heck of a football column each week, too. http://sports.espn.go.com/espn/page2/story?page=easterbrook/060912 And his book, The Progress Paradox, really makes you think about how great we really have life today. There is a great line in the book about how today we can purchase cheap wine from the gas station that would be considered the finest quality to medieval kings. It really puts all those people who bemoan how terrible life is today in their place. Gregg is almost what I would consider a modern Renaissance Man, since he keeps current with quite a few different things. Physicists out there may be upset, since some of his writings have been against the use of so much money to build bigger supercolliders when that money could be used for TB shots in Africa or something. I agree with Martin's summary though, "a generalist shooting from the hip, but he gets some things right."

You could go all the way the other way and donate money to the people who want to bring you "the truth." Check out http://www.projectcensored.org/ for example, their 25 top censored news stories of the year are filled with lot of pseudo-scientific stuffs.

Please show some details, words are inexact, but showing some of the PMM equations will let everyone know what you are proposing.

You can even take post #3 farther. "How far away from downtown are you?" "Oh, about 45 mins." Time is clearly not a measure of distance, yet everyone understands exactly what is meant.

If America started selling beer in metric units, all the college kids would figure it out real quick.

At its most basic, the 'matrix math' and the 'regular math' is exactly the same thing. There is a lot of math that is written that just comes down to notation. y1 = a11*x1 + a12*x2 y2 = a21*x1 + a22*x2 is just regular math for y = A*x Well, sure, its easy for 2x2, but writing out 2000x2000? Matrix notation looks better and better. Now, the real power or matrix (and vector/tensor notation) starts to come when you realize the impact of frame-indifference. I don't know GR and QM well, but I can give you an example from fluid mechanics. The fluid does not in any way know if you are trying to describe its motion using Cartesian, cylindrical, spherical or any other coordinate system. But, if it a Newtonian fluid, it has to obey the Navier-Stokes equations http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html, which are really just a description of conservation of momentum. The biggest point is that the Navier-Stokes equations are valid in any coordinate system, and my best guess is that this is exactly the same for QM and SR etc. The photon you are studying should not be aware that you are trying to describe its motion in cylindrical or spherical or any other coordinate system. Nature is indifferent to our description of it. The Navier-Stokes equation has vectors in it (v for the fluid velocity) and tensors (what you call matrices) (T for the stress tensor of the fluid). Eqn (6) on that page looks simple, only 5 terms. But, depending upon the coordinate system, writing out all the terms can take a lot of space, e.g. the equations written out in spherical coordinates takes up almost an entire type-written page, see eqns (23-25) and compare with (6) above. What I am trying to say, is that the tensor/matrix equations typically are the more general equations, with the details left to later. It is often easy to lose understanding if you get caught up with an equation with several terms rather than a few. Finally, at the other extreme, the distinction between functions and matrices become blurred. That is, a vector becomes a finite representation of an infinite dimensional space also known as functions. The clearest example I can think of right now is Fourier expansion where a vector stores the coefficients of the Fourier decomposition. A finite number of sines and cosines of different frequencies summed together can be very close to equal to an arbitrary function, though it would take an infinite number to sum up to be a perfect match, hence the infinite dimensional aspect. You can expand these ideas of function spaces, inner products, self-adjoint operators, etc. But these ideas are pretty far out there and probably not helping answer this question, so I'll stop now.

When in doubt, Mathworld http://mathworld.wolfram.com/FunctionalEquation.html Note in the References three books by Kuczma on functional equations.

Bookworm, the server was down for a time, so people may not have been able to post to help you. That said, this is the homework help session, and one of the major rules here is that the forum member will help, but you have to show us what you have done to date to solve this on your own. What ideas have you tried? Any equations you think are appropriate?

The answer to all three is infinite. While the chances are incredibly small, the chances are non-zero that every single roll (of 1 die, or multiple dice) will be exactly the same. Similarly, the chances are small that the same number will never come back up, but it is non-zero. What you should be asking are, how many throws will be necessary to expect that the same number comes up twice. This is a very important concept in probability, the expected value, your book should have lots of information on it. and, since this is homework help, what have you done towards calculating this on your own? We (forum members) will help where you get stuck, but we won't do your problems for you.

There are physical problems where the solution involves gamma functions, too. Many diffusion problems have gamma function or error function solutions. While it is not introduced at the same time as sine or cosine, etc., I think of them as just another useful, well-studied, tabulated function that just comes up. Bessel functions are another example of functions that just arise naturally when describing some physical situations.

The application disclaimer is very apt. On many engineering scales, that is, plant level, ad hoc or rules of thumb can be more than sufficeint. Many, many plants have been designed on this without the desingers knowning the basics underlying everything. Real good case in point: turbulence. Many of the broader implications of turbulence are known, and almost every flow on a plant scale is turbulent, but no model can capture every detail of the turbuelnce at all speeds to date. Just because the basic details are not known, and many desingers use simplified models of the turbulence, does not mean those designers are any less successful at their jobs. That said, better understand of the physcial insights have led to better and better turblence models over the years.

At the very least, the drop size gives some information about the surface tension involved, yes?