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Bignose

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Posts posted by Bignose

  1. It is probably impossible to stress how important the Fundamental Theorems of Calculus really are. Well, hence the word 'Fundamental'. See if ajb's links help your question, or if any of the other first links that come up from searching "fundamental theorems of calculus" in google. There are lots of different way of explaining it on the just the first page of Google results -- one of them will probably jive. In fact, if one of them helps a lot, come back and tell us which one was best for you. I like the wikipedia entry best myself -- the animation of the converging Riemann sums was good as was the more than one proof -- but I'd like to know which one worked best for you and why.

  2. If it is linear, a matrix algebra appraoch should be good -- there are certain restraints on whether the matrix is invertable or not, but in the vast majority of time the matrix is invertable.

     

    If the system is nonlinear, usually an interative method is necessary, like Newton's method. Once you get a handle on Newton's method, then you can explore various improvements -- almost all of the iterative methods are based on linearizing the system and making an approximate improvement toward a solution. See http://en.wikipedia.org/wiki/Newton's_method

     

    As a decent guide, if you are planning on writing computer software to help you solve these systems, computational fluid dynamics (CFD) books are about as good a start as any. A three dimensional viscous flow is a 4 equation non-linear system of equations. (x-velocity, y-velocity, z-velocity, and pressure are all coupled together.) Any good introductory CFD book will have lots of discussions about how to solve coupled systems of equations.

  3. I have no idea what the percentage is, but there has to be a pretty significant percentage of people who are working at the same time they are attending the community college and really need to have a schooling option nearby instead of across the state. The cost of schooling isn't just in the school itself, these students would have to incurr the costs of moving and rent in a different (quite possibly much more expensive) city.

     

    Not to mention at least nearby here, the community college does a fair amount of community service stuff, like running several bands (at least 2 jazz bands, 1 classical band, etc.) and teaching other adult classes like quilting classes.

  4. What are you saying? Are you saying that I'm too stupid to understand it?

     

    In addition to what swansont said, beyond the level of a high school physics class speaks for itself. That means, you will not cover the necessary material to read a journal article that discusses the research. In fact, unless you are an active researcher in the area, virtually no one will have enough background necessary to read a recent paper and get much more out of it than a summary. What this means is that once you take at least a university level class, maybe even a graduate level physics class, then you will be at the level to get a lot more out of the paper, and understand what the researchers are doing.

     

    Next time you think you are being insulted, sit back, relax, and understand that tone does not carry well over the Internet, via the written word. I in no was was trying to tell you that you were stupid, but you said you were taking high school physics and at that level you will not be able to understand the math and discussion in a peer-reviewed scientific journal like Journal of Tribology for example. (where if you really want to look see http://cat.inist.fr/?aModele=afficheN&cpsidt=7227671 ) Look, there are many, many topics discussed on this board that are well beyond my level. What that says about my stupidity I suppose is up to you to decide, but I have never looked at a quantum mechanics book, and only when I studied tensors I saw a tiny bit about relativity. That does not mean I couldn't undersand it, but it would take me quite some time to get to a level where I could read a current QM or special relativity (or many, many other subjects) research paper and get anything whatsoever out of it. I would and do fully accept the fact that someone would and should tell me that QM is above my level right now.

  5. There's no way of knowing coefficient of friction without testing (i.e. pulling or pushing).

     

    That's not quite 100% true, there are researchers who study the molecular interactions between surfaces to try to predcit friction coefficients. Suffice to say, however, this kind of research will be well beyond a high school level physics class. And, the experiments needed to measure the coefficient of friction are rather easy and cheap to set up and perform, so in all practical cases, testing will be the way to find out the answer.

  6. This is not true, at least not in the formal rules of chess (you can invent all house rules you want).

     

    A person can only win if either 1) you put the king in checkmate, or 2) your opponent resigns.

     

    You can force draw from a losing position if 1) you can avoid checkmate for 50 moves (which means a white-black pair) without a pawn's moving (at which point the number resets), or 2) you get put in stalemate (which means there is no move available such that the king will not be putting himself in check). A draw can also be claimed when there is 'insufficient mating material,' generally K+K, K+KB or K+KN. There are a few other obscure 'insufficient mating material' combinations which are probably unnecessary to explain. Lastly, a draw can also be claimed when a position is repeated three times on the board (and by repeated, that takes into account the potentialities to castle and the availability of en passant moves by pawns).

     

    To computerages:

     

    To improve your endgame, you should probably study various endgame strategies including the principals of 'opposition' and how to advance pawns successfully (or prevent their advance), as well as the ways in which rooks (especially) and minor pieces can control parts of the board. A K+KQ endgame should be the most straightforward in which you can force a mate, so if you are struggling with this, you might want to study the strategic strengths and weaknesses of pieces (how to use them to control the board), because a firm grasp on the pieces will make mating with them self-evident.

     

    I'm a subscriber to the site http://www.redhotpawn.com (you don't have to pay to play, but you only get six games at once) which is a correspondence chess website, but there are many other free online chess sites, too.

     

    The more you play, the better you will understand the pieces, but understanding subtle chess strategy is a life-long process (and I'm a neophyte to be sure).

     

    Nemesio

     

     

     

    If you can show the forced win, there is no 50 move rule. But, you have to show that you know how to do it. K+N+B vs. K is a forced win, but in the worst case it could take well over 100 moves to force it. And if you don't know how to do it, the polite thing to do is to offer a draw. There are some other forced wins that take a long time. K+N+N vs. K is a draw, but K+N+N vs. K+P is a forced win, but it takes perfect timing. You have to bottle up the king and still leave him moves to make with the P, so that he cannot stalemate himself, like in K+N+N vs. K.

     

    These are pretty advanced, but endgame study is usually the most beneficial for new players. To put it simply, the choices you make in the middlegame determine what kind of endgame you end up playing, so being more proficient in the endgame can allow you to steer the middlegame to the endgame you want.

  7. It looks like a graphical way of doing the foil method. and then the counting is just doing the addition of the singles, tens, hundreds, etc. It is kind of neat looking, however, though as clarisse said multilying 79x989 would definately be slower than doing it via more traditional methods.

  8. If you're going to define "reason" like that, this exercise is rather pointless.

     

    Cap'ns got a pretty good point here. Otherwise almost all leisure activities are done/were created for not much reason. E.g. most sports, soccer, basketball, football, baseball, golf, etc? For that matter, sports fanaticism (Boston Red Sox fans, I'm looking at you), or even watching a sporting event you yourself are not participating in is done for no real reason either. I can't think of any real reason most board games would have been invented. (Chess is an analogue to war, but is Monopoly designed to teach people to hoard wealth?) I think even farther, crossword puzzles don't accomplish much. This list can go on and on. Most leisure activities had to have been invented 'because someone was bored,' because there is no real reason behind them. I mean, seriously, look at caber tossing -- you know that someone had to have been pretty bored to come up with that.

  9. My bad, I misinterpreted what you meant. I misread it since it is unnecessary to say what the height would be, an "equilateral triangle with side of 18" completely defines the situation. That said, show us what work you have done, and we will do our best to verify its accuracy. We are not going to post our solution just for you to check at home, because, frankly, we cannot just take your word that you have done the work yourself.

  10. Please don't take this as an insult, but Google comes up with over 300,000 pages. Some of the first ones the equation you are looking for:

     

    http://ising.phys.cwru.edu/surfactants/droplets2.html

     

    http://www.itconceptfr.com/BrochureTracker/Documentation_Tracker_VA.htm

     

    and many, many more.

     

    Also, any decent fluids textbook that has a surface tension chapter will have plenty of info about the Young-Laplace equation.

     

    p.s. I've also always heard it called the Young-Laplace equation (not L-Y), though results come up in both orders. Either way, there are several thousand good webpages out there.

  11. For a single sphere travelling in an infinite fluid, the drag coefficient has simple expressions in the high velocity and low velocity regions, but not intermediate. To know what region you are in, you need to know the particle Reynolds number: Rep = Udr/m

     

    where U=particle's velocity

    d = diameter of particle (sphere)

    r = density of the fluid

    m = viscosity of the fluid

    Rep is a dimensionless number (so all the units of U,d,r & m have be converted so that they cancel)

     

    Then, the drag coefficient, CD, can be written as:

     

    For Rep<0.3, CD=24/Rep

    For 0.3<Rep<500, CD=(24/Rep)*(1.0 + 0.15*Rep^0.687)

    For Rep>500, CD=0.44

     

    This is just one of several correlations that exist, most of them varying in their representation of the intermediate region. The low Rep region, known as the Stokes flow region, can be analytically calculated, whereas the high Rep very turbulent region has been confirmed experimentally many, many times.

     

    with the drag coefficient, you can write the drag force:

     

    Fd = (pi/8)*CD*d^2*r*|V-U|*(V-U)

     

    where V is the fluid velocity (Note the absolute values!)

     

    So, you see, when the flow speed is low, because of the Reynolds number dependence in CD, the drag force is proportional to the velocity difference to the first power. But, when the speed is high, because CD becomes just a constant, the drag is proportional to the velocity difference squared.

     

    An iterative procedure can be used: Guess a velocity, use that guess to calculate a CD, then calculate the velocity for that CD. If the calculated velocity and guessed velocity are the same, you are done, but normally, you have to use that newly calculated velocity as the next guessed velocity and perform the cycle a few times for the desired accuracy.

     

    Then, of course, all the other difficulties in calculating drag: non-spherical particles, the presence of other particles, the effects of the walls of the container the fluid is in, just to name a few.

  12. I don't have a problem with conversion now that I know what it is, I am just saying it is a pretty awkward notational system (which is probably why I have never seen it before).

     

    Consider, you want to mix two solutions together x parts M/5 and y parts M/10. x/5 + y/10 = 2x/10 + y/10 = (2x+y)/10.

     

    That was an easy one. What if it was 3 part solutions? M/7, M/13, and M/17. You have to mulitply all those fractions out, or bascially convert it to decimal notation anyway.

     

    And what do you do if the fraction cannot be converted into 1 over an integer? M/6.84 is really just using an awkward notation for notation's sake.

     

    That's why I said give me moles per liter (I should have explicitly said in decimals) or ppm anyday, since you don't have to fool around with these wierd fraction and stuff.

  13. I am reading a fairly old paper today (1930), and they used a notation for concentration that I am unfamiliar with and can't seem to find info in Google about it. Here is the quote from the paper:

     

    "The concentrations of caffeine (Merck U.S.P.) were M/500, M/750, M/1000."

     

    Can someone tell me what M/500 means? Moles? Parts per million?

     

    Thanks in advance.

  14. Yeah, but why try "brute-force" or "guess-and-check" when iterative techniques like Newton's method will (typically) tend toward a solution. That way you aren't just guessing in the dark, your guesses will (most likely) improve with each time. Brute force has got to be the absolute last resort.

  15. 1) stay as far away from the textbook as humanly possible.

    it's a known fact that textbook are hard to extract useful information from.

    also, most students hate textbooks. for a reason.

     

    Known fact, eh? Perhaps you could give me a reference on this known fact? Perhaps you could explain why textbooks keep getting written? Perhaps you could explain why several people, myself included, read textbooks all the time to expand our knowledges? Look, I agree that there are poor texts out there, just like anything else in life, but there are also some really, really great ones. Broad generalizations like this don't get help further discussions. As a student, if you really dislike the text assigned by the professor, you should take it upon yourself to go to the library and look up other books on the same subject. Maybe one of them will click better with you. Chances are awfully low that the book from your class is the only one written on the subject, and if it is a common subject, the library will have dozens of books on the subject.

     

    3) only give out homework if the homework does not have a monotone feeling.

    meaning: no repeating problems unless they are optional, and no forced structure.

     

    This depends an awful lot on what it is. For example, when teaching differentiation, you want to give lots of homework problems calculating the derivative of many, many functions. This is so that the method of calculating the derivative becomes second nature. There are some things that the only way to really know something is to repeat it over and over. Practice makes permanent. I actually feel that the average math class does not give enough homework, because the average student today has very little math intuition -- intuition gained from practice.

     

    That said, there is too much 'busy work' handed out as well. Sometimes it is necessary, but just to assign work simply so that there is work to do is definately a waste.

  16. Yes, you have to look up the Goodness of Fit test, many statistical design of experiments books will have info on it, like that R.A.Fisher book I cited above.

     

    Swansot and I are saying the same thing. I was just trying to show you how meaningless R^2 values become. Because R^2 of the (n+1) oder polynomial will be exactly R^2=1 for n data points (as will all n+m, m>=1, order polynomials), what does R^2 tell you? Basically nothing whatsoever at all, since for every (n+m) order polynomial, it is one. This was a contrived example to show you how meaningless R^2 values really are.

  17. plagiarism is defined to be "the unauthorized use or close imitation of the language and thoughts of another author and the representation of them as one's own original work. "

     

    So, so long as you acknowledge (and cite properly) that the orignal idea came from a different author, you certainly can modify and expand and thoroughly make info your own. You just have to give credit where credit is due.

  18. I think in a decently run university level class the students should be responsible for learning on their own a fair % of the time anyway. The prof only gets 40 some-odd hours of lecutre a semester. Take away exam periods, and you might be looking at 40-35 total hours. You cannot cover all the material that you want or need to cover in just lecture. The students have to be able to do some work on their own. The one teaching methods class I took some time ago suggested that about 25% of self-taught information was a good mix. That is, 25% of the material is assigned in the required reading, and students could ask questions about the reading, but the professor would not set aside a day or 3 to "lecture" on the material. It was up to the student to teach themselves. And, you make sure the students are self-teaching themselves with homework and exam questions.

  19. woelen makes a good point here. There are numerous identified and like he said an infinite amount of functions out there. Just some of them have special names and applications.

     

    You know sin and cos, but there is also gamma function, error function, Bessel functions, Airy functions, hyperbolic sine and cosine, hypergeometric function, and many many more. These functions should be thought of in exactly the same way as sine and cosine, since the functions are well defined and their values tabulated. Before computers became really common people, would have books with nothing but tables of these functions' values.

     

    In general, whenever a mathematician needed a special function to solve problem, they invented it. E was invented, as least partially, to help evaulate integrals of sine. As you keep taking math classes, and solve more and more complicated problems, you will see where these new functions arise just naturally.

  20. coefficient of determination, that is R^2 right? If so, R^2 is the most abused statistic ever.

     

    Let me propose a thought experiment. Take the limit again when you have 20 data points. Fit that with a 21st order polynomial. What is the R^2 in this case? Exactly 1. Does this indicate to you that the 21st order polynomial is the best fit? What about a 22nd order or 23rd or 100th order. Each of those should have an R^2 of exactly 1 also.

     

    R^2 tells you how good of a fit to your guessed function the data fits. And this is all it does. Period. It does not compare for different trial functions. It does let you compare for guesses between the same form of the trial function. I.e. is will change the m & b in the linear model y = mx +b, but it does not tell you if y=mx + b or if y = mx^2 + nx + b. I suspect you know, or you can certainly contrive examples, where over a small range of x, the y=x and the y=x^2 models (and the y=x^3 and so on...) all look pretty similar. Which one is right? Hopefully the physics of the situation will lead you in the right path.

  21. Please don't think of this as ducking the question but any book on Mechanics of Solids will have a derivation.

     

    I can do it, but it is rather lengthy, and (please forgive me if I am making a hugely wrong assumption about a 16 year old here) but the math is probably beyond what you have learned. The derivation all the way from the conservation of momentum involves integrals of vector and tensor fields and 3-D calculus operators like the gradient and mathematical relations that turn volume integrals into surface integrals (The Divergrence Theorem).

  22. 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.

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