DrRocket

Choose a basis in which B is diagonal. Read the Wiki article and vview this as a constrained optimization problem. Use Lagrange multipliers.

Here's my take on that article. 1. Although it dates from 1984 I doubt that the situation is any better today. I would guess that it is worse. 2. I am not particularly surprised that students made conceptual errors in applying mechanics to real-world examples. Besides the obvious lack of mastery of the subject there are some issues not identified in the study: 1) The desired responses are based on idealizations of the real-world scenarios. Unless students recognize the expected idealization they may miss the point or give incorrect answers. 2) Some familiarity with the experimental apparatus (e.g. dry ice pucks and air hoses) is necessary to make sense of the problem presented, and non-ideal behavior may introduce confusion. 3. The difficulty in identifying forces in a given situation does not surprise me, if the students were taught mechanics in a physics class. I have seen a Ph.D. student (nuclear physics) have difficulty with this in a very elementary setting. On the other hand engineering students are taught very early on about "free body diagrams" and don't seem to have such problems. (When I showed the technique to the physics Ph.D. student, the response was instant understanding and "That's neat."). 4. The study correctly recognized that "Of particular concern is the apparent failure of universities to help precollege teachers develop a sound conceptual understanding of the material they are expected to teach." I think getting understanding implanted in secondary school teachers is the real solution, but that is easier said than done. 5. Some of the problems presented and the expected reasoning are actually rather sophisticated and require an ability with abstract mathematics that relatively few students really have. The problem asking if two rolling balls ever have the same speed is such a problem -- it is an existence proof in essence (It was slower here there and faster there so it must have been equal somewhere in between.) 6. The problems seem to be world wide. So no particular educational system is indicted, or exonerated. 7. The study paper was by a lady in the U. of Washington Physics department and referenced work by her students for Ph.D. dissertations. If Ph.D. degrees in Physics (rather than psychology or Ed.D degrees in education) were awarded for such work then I am appalled. Aside: Back in my undergraduate days, mechanics was taught to engineers by the Engineering Science Department. They complained that physics students who took their classes "Were great at theory, but couldn't solve the problems." This still baffles me. You simply cannot really understand a theory but not be able to apply it. Equally you cannot apply something unless you understand the underlying theory. What this tells me is that people can somehow "get by" through regurgitation of what some view as "theory" without having any real understanding. I don't know how that is possible except through rote learning and canned testing. This makes me really appreciate Moore method classes and seminar courses where student participation dominates. I hate pure lecture courses.

You still have not explained what it really is that you are trying to do, or the source of the question. It would also help to know something of your background in mathematics. The question of critical points of the rayleigh quotient ties back to your thread on maxima and minima. But this question is a bit advanced for either a calculus class or a class on matrices and linear algebra. Moreover, the Wiki article ought to give you a couple of ideas where to start. This is not your basic undergraduate homework problem.

Yes, you did.

Not surprising. I have been told by students that they had no interest in understanding the theory, but rather just wanted to "plug and chug". I saw a lot more of this attitude in industry and there it is scary because you get people who are good at manipulating sophisticated computer codes, but don't understand what those codes really tell you or what the limitations are. I have seen "equations" derived by PhD's with perfect symbol manipulation -- but having no relationship to the actual question at hand. Right answer. Wrong problem. This is the same student failing but at a higher and more costly level. [quote name=Cap'n Refsmmat'If you'd like to see the examples from the research, here's the citation: McDermott, L. C. (1984). Research on conceptual understanding in mechanics. Physics Today, 37(7), 24. American Institute of Physics. http://physicstoday....d/v37/i7/p24_s1 If you can't get a full-text copy easily, I can send you a PDF. I don't have access. I would appreciate a PDF. Thanks. Students seem to more or less demand such courses. You fix it by ignoring such demands and instead run classes with real content, that challenge the students and by failing those who can't or won't meet reasonable standards. This is not hard to do. Text books already exist. It does require that legislatures and administrations recognize that not everyone will meet those standards and that some children will be left behind. What won't work is demanding that educational effectiveness be measured by average scores on standardized texts. When you do that you get exactly the conduct that you incentivize -- people teach the test. There is a real problem with empasis on things other than understanding. http://www.lehigh.edu/~shw2/ap2001.html Students today have the same basic genetics that students have always had, plus ready access to a vast amount of information. If they are weak on critical thinking and deep understanding it is because such has not been demanded of them. Critical thinking is not "taught", it is developed through exercise, and with that excercise comes understanding. One can demand exercise.

When you take a philosophy class all bets are off. One cannot expect any sort of conclusion from a philosopher. I cannot imagine a more damning statement about AP physics classes or exams in a university physics class. Which is not say that it is not true, but rather that there is way too little emphasis on understanding and way too much on cook book exercises. This is even scarier. What is the origin of these misconceptions ? I don't see how high school physics cannot be a significant source, unless the students did not take high school physics. In the latter case I don't quite see how the basis could be a university level calculus-based physics class unless the students simply don't have the appropriate background. In any case this sounds like a serious deficiency in either preparation or reasoning on the part of the students themselves. They cannot possibly be reading and understanding the text or paying attention in lectures and still maintaining such basic misconceptions. Maybe they believe too much of the nonsense that is available on the internet. This is more of a problem with critical thinking than with any specific subject matter, but I suspect it gets back to expectations from primary and secondary school, and maybe too much junk from poor sources.

We were not talking about truth vs provability. Your assertion -- that a number for which the rationality or irrationality is unproven is actually then neither rational nor irrational -- is just flat wrong, and therefore extremely misleading to a newbie trying to understand mathematics. Moreover truth and provability ARE NOT the same thing. The people who disagree with you are called mathematicians. The distinction between "true" and "provable" is critical to understanding the Godel incompleteness theorems, and some of Paul Cohen's work. So, if you contend differently you probably need to do some more study.

That sounds a bit light for a computer science curriculum. You should probably take a calculus-based physics class. Calculus was invented as part of Newton's development of mechanics. You shuld also take a class in linear algebra or a combination of linear algebra and ordinary differential equations. You might also consider an introductory course in Real Analysis, which is basically "calculus done right". Two excellent texts are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin, both junior-senior undergraduate level books.

Some of the best classes that I have taken were taught by what is known as the "Moore method" or "Texas method" in which the burden was fully placed on the student and the lecturer did not explain anything or answer questions. They merely moderated and sometimes contributed to the critique by the class of work presented by the students. I don't believe that any "tenured professor" is incapable of explaining a concept to a student who is actively trying to learn (as opposed to expecting to be "taught") or of answering questions ("helpfully" is in the beye of the beholder and a perfectly reasonable answer may not be recognized as such by a lazy or poor student). There is WAY to much emphasis placed on "self esteem" by many students, and primary and secondary school teachers. Students may not like blunt answers, but sometimes they need to hear them. I repeat, the burden to learn belongs to the student. A great deal of the problems in education lie in the failure of the student to meet that responsibility. What cannot be excused is failure of the instructor to understand the subject matter, and the teaching of distortions and downright falsehoods. Then something is horribly wrong in their first class in physics, likely in high school. Veclocity and acceleration are so well-defined and fundamental that only someone who fails completely to understand very basic physics and mathematics could create such confusion. We are back to requiring competence in secondary school teachers. There is also the unfortunate reality that only a minority of U.S. students possess both the capability and inclination to understand any but the most elementary mathematics and science. Capability is probably more ubiquitous than interest, but both are necessary.

Which is understandable when you consider that many of their teachers have no deeper understanding. As I said, the problem starts with incompetent secondary school teachers. The calculator is but a symptom, but an important symptom. I rather doubt that they were incompetent in their own field, at least if you are speaking of mathematicians or physicists. Competence in one's field may not be a sufficient condition for a good teacher, but it is a necessary condition. And it is a condition that many secondary teachers fail to meet, even if they have a BS in the discipline. I find complaints about "incompetent lecturers" at the university level suspect. At that level the responsibility of the lecturer is to aid the stufdent in learning, not teach. The failure is quite likely on the part of the student to meet his obligation to learn.

There are a number of problems that contribute to what you observe. 1. Many secondary school teachers, even mathematics and physics teachers are incompetent in science and mathematics. People who are competent in mathematics and science but who do not hold teaching certificates are often not permitted to teach in the public school system, even though they would be qualified to teach the same subjects in a university setting. 2. There is an excess reliance on computers and calculators as "black boxes". This unfortunately extends to some mathematics classes. Students tend therefore to lack fundamental understanding even if they can "find the answer" to textbook problems. Lack of competence on the part of teachers tends to promulgate this problem. 3 The curriculum has been watered down. Real credit at some discerning schools is not given for AP scores below the top level of "5". This untimately comes back to people teaching subjects in which they lack competence. 4. Much AP emphasis, in calculus in particular is on symbol manipulation rather than understanding. The result is that students who learn their calculus at a university tend to have deeper understanding that those who learn it in AP calculus -- this opinion was confirmed to me by an MIT engineering professor. In short, AP classes by themselves are not the answer. If you want to improve the situation the place to start is with the competence of secondary school teachers in mathematics and science. The major research university in my state has a project to do this. Initially it was headed by a well-known mathematician, but I don't know if he felt he made much progress, and it is now going in to a longer term phase. When students can't do simple arithmetic calculations quickly in their head they have no hope of being able to follow a lecturer in a derivation of nearly anything. It is a BIG problem in terms of understanding. Simple example: Students often have trouble with fractional exponents -- because they can't add fractions.

Absolutely not. Any real number is unequivocally either rational or irrational. It may not be known which is the case, but the answer is never "neither". I seem to recall, not positive though, that it is an open problem whether [math] e + \pi[/math] is rational or irrational.

It is not correct that in general (x^tAx)Bx = (x^tBx)Ax for symmetric A and B (even if they are positive-definite). For an ordinary Rayleigh quotient R(M,x) where M is a symmetric matrix, the critical points are eigenvectors of M. This problem is well beyond just matrix algebra. The generalized Rayleigh quotient requires symmetric positive-definite matrices. Are your A and B positive-definite ? What are you really doing ? You might want to read this Wiki article http://en.wikipedia.org/wiki/Rayleigh_quotient

"13 choose 3" is the usual terminology in the mathematics community, and it is reflected in LaTex as well -- {13 \choose 3 }= \frac {13!}{3!10!} gives you [math] {13 \choose 3 }= \frac {13!}{3!10!}[/math] . and in general for "n choose m" you have [math] {n \choose m} = \frac {n!}{m!(n-m)!}[/math]

You have a problem with the term, AxBx. It doesn't make sense since Ax and Bx are both column vectors. Did you mean to write x^tABx = x^tBAx ? If so, think about the transpose of each side and about the dimension of the indicated product.

A black hole is a relativistic effect of an extreme concentration of mass/energy governed by the Einstein field equations of general relativity. A black hole is really a 4-dimensional structure in the Lorentzian spacetime manifold, and curvature of that manifold is both extreme and the major issue. A hurricane is a fluid dynamic manifestation of continuum mechanics as described by the Naviere-Stokes equation, which is non-relativistic and applies in flat 3-space. The large-scale structure to which you refer is essentially 2-dimensional, though the flow involved is 3-dimensional and locally turbulent. Any similarity between a hurricane and a black hole is pure coincidence.

Given your relation [math] A=A_0 e^{ \frac {-0.693t}{T_{1/2}}}[/math] You will need to specify a value for [math]T_{1/2}[/math] and then you can compile a list for [math]A/A_o[/math] for specific values of [math]t[/math] using Excel or just a calculator. What is it that you are ultimately trying to do ? This seems like an odd problem for personal amusement or a commercial application. Is this homework or some sort of class assignment ?

Trust me, there is. But the beauty of it lies in the subtlety of Dirac's ridicule, which may not be evident if you lack significant experience with mathematical proofs and derivations, and a dry sense of humor. Dirac inserted the knife very deftly -- the work of a master. Somewhat more obvious but equally pointed and amusing was Wolfgang Pauli's " That's not right - that's not even wrong" . The best ridicule is done so deftly that a dense subject fails to recognize it for what it is. This, of course, requires an intelligent and perceptive broader audience to appreciate it. On a different note, this is one of the better pieces that I have seen regarding ad hominem arguments. http://plover.net/~b.../adhominem.html

Change from what ?

A fallacy is a breach of formal logic. An assertion is not a logical argument and cannot be a logical fallacy. An assertion can be fallacious (i.e. tending to deceive or mislead) but not a fallacy. In the case that you cite the assertion is an unsubstantiated opinion, intended to convey the impresssion that those who ridicule are likely ignorant and therefore unworthy of consideration. It is in in fact a subtle form of an ad hominem attack. Student (interrupting a lecture in which an equation is being bderived) : "Professor, I don't understand." silence more silence Student: "Professor, aren't you going to answer my question ?" Professor: "That was not a question." The professor in the incident was P.A.M. Dirac. Anyone care to characterize Dirac as ignorant or unworthy of consideration ? "An ad hominem (Latin for "to the man" or "to the person"), short for argumentum ad hominem, is an attempt to negate the truth of a claim by pointing out a negative characteristic or belief of the person supporting it.[1]" http://en.wikipedia.org/wiki/Ad_hominem

A good surgeon will naturally pick the best operation. Mathematicians work with the operation at hand, or invent one that provides insight in the situation that is presented. Quite often there are several operations available simultaneously, and the result is a richer algebraic structure , like a module, vector space or algebra. But in the case of a monoid one considers a very restrictive case, and only one operation is under consideration. Where is it that you are encountering the concept of a monoid ? Quite typically one's first exposure to abstract algebra is via groups. There is much more that can be said about groups. If you have a particular text that you are reading, please tell us what it is. If not, I can recommend Michael Artin's Algebra.

It needs to be clearly stated that an ad hominem argument is one in which the validity of an assertion is questioned on the basis of some negative characterization of the proponent. It is not merely an insult. Thus "you are an idiot and therefore your statement is absurd" is an ad hominem argument, but "your statement is absurd and therefore you are an idiot" is an observation and an insult, but not an ad hominem argument. Exactly the same logical issues arise when expert opinion is used -- only in this case the ad hominem perspective is adopted to lend credibility to an assertion because of the nature of the one who asserts. In the case that you cite, it is kitkat who makes an ad hominem argument. iNow's statement was logically irrelevant, perhaps valid, and depending on kitkat's view of hypocrisy might be taken as pejorative, but not an ad hominem argument. Note that while an ad hominem argument is not valid in the context of formal logic, it is not necessarily poor reasoning in the larger context. We engage in ad hominem reasoning every time that we request a citation from the peer reviewed literature, or cite an expert opinion. In the modern environment in which the ability to publish, easily and widely, is afforded to anyone with a keyboard and internet connection it is in fact necessary to discriminate among those who publish. Life is just too short to spend time dissecting in detail the rantings of the numerous nut cases found today on the internet -- and science forums attract more than their fare share. I know of no instances in which great ideas were lost to ridicule. Pythagoras may have received criticism when irrational numbers were discovered, but his ideas survived the death of the man. Einstein's was work, in the darkest days of the Reich was called "Jewish physics", but Einstein's ideas and reputation seem to have prevailed over those of his detractors. Valid ideas seem to always win out in the end. But let's be realistic. Revolutionary valid new science never has and will not ever arise from the rantings of some delusional amateur in an internet science forum. The first step in doing research and developing radical new valid science is understanding what is already known, the basis of that knowledge and the limits of its applicability. That requires serious intense study. There is a reason why almost all research scientists have Ph.D. degrees. I seem to observe that purveyors of tripe take any criticism of their ideas ortheir grasp of subjective matter as personal attacks. Thus criticism of ideas is interpreted erroneously as an ad hominem attack. I also note a tendancy for some to be more concerned with the "self esteem" of fringe posters than in content or intellectual honesty (note the degree to which positive and negative "reputation points" are cast on the basis ofsuch concerns rather than technical content). This goes to the point where requests for homework help that are in essence outright cheating are tolerated because "the professor should forsee this possibility". Sincere newbies deserve consideration and tolerance. But kid gloves are not for wackos, cheaters or even just lazy students. The railing against ad hominem arguments by the lunatic fringe is simply a tactic used to promote nonsense, and an attempt to have absurdities given apparently serious consideration. In a venue without the mollifying influence provided by an academic environment and peer review, the risk of intellectual chaos promulgated by articulate and prolific lunatics is real (think Farsight or your favorite creationist). It is therefore expedient, and in fact necessary, to learn to discriminate between arguments and sources that are deserving of the expenditure of intellectual capital and tripe that should be dismissed out of hand, not even read fully. This is ad hominem reasoning at its most productive, and it is necessary for those who wish to learn real science in a finite lifetime.

What you have been told is correct, but let's try a slightly different tack. Think of a binary operation as a generalization of multiplication or addition. So, given elements a and b from your set, call it M, a*b is another element and "*" is the binary operation. For a monoid * is required to satisfy the following 2 properties: 1. Associativity --- a*(b*c)=(a*b)*c 2. Identity element --- There is an element I, called an identity, such that I*a=a*I=a for any a in M If in addition "*" satisfies the requirement that 3. Invertibility -- for any a in M there is a b in m such that a*b=b*a=I then the monoid is called a group. A typical example of a group are the nxn invertible matrices for some fixed n with entries taken from thev real (or complex) numbers. Groups are one of the vfundamental structures studies in abstract algebra courses. Monoids have less structure, lacking invertibility. If only property 1 is satisfied then your set M is called a semigroup. So a monoid can also be called (and commonly is) a "semigroup with identity". Quite commonly the bibary operation is written simply as multiplication, so as "ab" rather than "a*b". Note that the following property has NOT been required 4. Commutativity -- a*b=b*a In general, even when written as simple multiplication, the operation is not assumed to be commutative (think of matrix multiplication). When the operation is commutative the algebraic structure is called "commutative" or " Abelian" (after the famous mathematician Niels Henrik Abel). So one can have an Abelian group, an Abelian semigroup, or an Abelian monoid or equivalently a commutative group, a commutative semigroup or commutative monoid.

Theories are accepted, not "believed in", because they possess predictive power that is substantiated by a significant body of verifiable and repeatable experimental or observational data. The major physical theories are the quantum field theories of the electroweak force and the strong interaction (quantum chromodynamics) and general relativity. But since general relativity and quantum theories are inherently incompatible, I know for a fact that they cannot all be absolutely correct, and I strongly that each of them are only very good approximations under the appropriate circumstances. So, you could say that I don't really "believe" in any of them. Nevertheless they are the best available explanations of natural physical phenomena. There is no single well-defined theory of evolution, but rather a broad framework. Nevertheless, that framework is consistent with what is known about chemistry, genetics and the fossil record. It is the best available explanation for speciation. Not that a "theory" is a high level scientific construct. It is more than a hypothesis, and a great deal more than a conjecture or a speculation.

[math]\sum Forces = 0[/math] [math]\sum Moments = 0[/math] After that it is just practice in applying those two equations.