Johnny5

Nice picture of the path, not exactly a circle, not an ellipse, but a closed orbit nonetheless.

To really say you have answered the question, your approach has to predict circular motion. In fact, really it should predict the actual motion. You have to model everything, decide what are the important variables, and ultimately predict a roughly circular path. It's not an easy problem. Regards PS: I can't resist trying to answer it. Here is another link to something which says that the boomerang precesses. I am trying to find a mathematical derivation which predicts circular motion. Here is an exact quote from the source above: Here is another site which discusses lift. Here is someone who is getting to the heart of it All the math needed, for a first approximation to understanding the motion of a boomerang, and thus the answer to your question, is at the site there. Read from "What is a couple?" to equation 8. Start out with this: t = I a The LHS is the greek letter tau, and stands for "torque" or "turning force." On the RHS the letter I stands for moment of inertia, and the letter alpha stands for angular acceleration. I am trying to find a site which gives a clear presentation.

I went and got my textbook "Elementary Classical Analysis" by Jerrold E. Marsden and Micheal J. Hoffman I'm not sure if that's the same Hoffman who wrote my linear algebra text. At any rate he lists the field axioms as: commutativity of addition/multiplication associativity of addition/multiplication additive/multiplicative identities additive/multiplicative inverses distributive axiom non-triviality axiom He also lists six order axioms

No i don't have access to that book unfortunately. Hmm. Well I have a list, why don't you criticise mine. Let me see... Let a,b,c denote arbitrary Real numbers. Closure under addition and multiplication: a+b is a real number a*b is a real number Commutativity of addition and multiplication: a+b=b+a a*b=b*a Associativity of addition and multiplication: a+(b+c)=(a+b)+c a*(b*c)=(a*b)*c Multiplicative and additive identity elements: There is at least one real number called zero, denoted by 0, such that: 0+a=a There is at least one real number called one, denoted by 1, such that: 1*a=a Multiplicative and additive inverses: Given any real number a, there is at least one real number -a, called the addive inverse of a, also called "negative a" such that: a+(-a)=0 Given any real number a, if not(a=0) then there is at least one real number 1/a, called the multiplicative inverse of a, also called the reciprocal of a, such that: a*(1/a)=1 Distributive axiom: a*(b+c)=a*b+a*c Axiom of addition: If a=b then a+c=b+c Axiom of multiplication: If a=b then a*c=b*c Non-triviality axiom: not(0=1) Order axioms For any real number x: not (x<x) For any real numbers, x,y,z: if x<y and y<z then x<z For any real number x, there is at least one real number y, such that x<y For any real number x, there is at least one real number y, such that y<x And I use some definitions. Is the list above complete, is it minimal? Is there a preferrable set of field axioms? I'm not clueless as to the answers to my own questions, but I am interested in seeing what others call "the field axioms".

Why don't you just send me all of your field axioms. Yours personally. That will save us both a great deal of time. Thank you Dapthar

Ok, any others you left out?

Ok thanks.

http://mathworld.wolfram.com/FieldAxioms.html And one more thing... The above list of axioms at wolfram is not minimal. There is nothing logically wrong with giving a list which isn't minimal, its just that once the observation is made, the list of axioms can be reduced. From a memory standpoint it is desirable to have the axioms minimized, so from a human standpoint such a thing is desirable, yet there is nothing inconsistent which will happen by listing too many axioms. here is what I mean though: They give the following statements both as axioms: Distributivity (addition) a(b+c) = ab+ac Distributivity (multiplication) (a+b)c=ac+bc And of course they give commutativity of multiplication: ab=ba Now, what they do not do is use the universal quantifier to say what they really mean. And there is a way around that though, which I will use below: Let a,b,c denote arbitrary real numbers. Axiom:a*b=b*a Axiom: a*(b+c)=a*b+a*c Axiom: (a+b)*c = a*c+b*c However, the third axiom above is a theorem of a system with the first two axioms. Theorem: For any real numbers x,y,z: (x+y)*z=x*z+y*z Let a,b,c denote arbitrary real numbers. By the closure axiom, b+c is also a real number. Therefore: a*(b+c)=(b+c)*a by commutativity of multiplication using real number a, and real number (b+c). Now using the distributivity axiom it follows that the following statement is true in the axiomatic system under discussion: a*(b+c)=a*b+a*c Therefore, by the properties of equality the following is true: (b+c)*a = a*b+a*c And the numbers a,b,c were arbitrary real numbers, hence the statement above is true for any real numbers a,b,c. That is: "x in R, "y in R, "z in R: (x+y)*z = z*x+z*y Which was to be proven. QED Also note that I used the closure axiom too.

Field axioms for real numbers Here you see the non-triviality axiom included.

Well what is a field supposed to be in the very first place? You are turning truth into something relative if you do that. not(0=1) is some kind of absolute fact. To say that the additive identity equals the multiplicative identity, changes the meaning of what 0,1 denote. You cannot de-stabilize what is true, and has a truth value which cannot vary in time for any reason. Yes, I've made my point. Regards

Currently, I am trying to make my way through Euclid's book seven, which is on number theory. I am having trouble with his very first theorem, can anyone here help? I think the reason I am having trouble with it, is that the English translation isn't as good as it could be. Here is a link to proposition 1 of book seven: Euclid's elements, Book 7, Proposition 1 The translation that you see at the website, is not the same as is in Heath. I think Dr. Joyce tried to translate it more clearly on his own. Anyway, I was reading through a graduate text on linear algebra, and around the time they begin to discuss rings, there is a discussion on "relative primeness" and fields. So I want to be able to grasp all of this ultimately, and so I think there is a link of the modern information, to Euclid here. So if anyone can help me decipher this first theorem, that will help me get going on the rest. Thanks PS: I was doing this yesterday, and a few things were confusing to me. First, I do not currently understand what is meant by "one number measures another number" does euclid mean "divide evenly" I think one good example is all it will take. Let the given numbers be 49,50. He starts off with: Actually, I am in the process of answering my own question here. This is the Euclidean algorithm that Dave went over, a few weeks ago. I still remember the algorithm. For some reason Joyce is calling it "antenaresis," don't know where he came up with that. Joyce clearly says that in modern terminology we say 'divides' not 'measures'. And the notation is a|b for a divides b Joyce goes on to say: So start with 7,50. 50-7=43 43-7=36 36-7=29 29-7=22 22-7=15 15-7=8 8-7=1

I am reading this now at Wolfram, on the commutator Wolfram on commutator They mention tensors if you look down a little. I was studying GR about a month ago, and tensors came up. Is the commutator algebra here, related to tensors? Specifially I am wondering about equation 9 at the wolfram site.

This is a good topic for a thread.

Revprez says the above in post 2, and in post 7 Tom says

Raul, There are three spatial dimensions, no more no less. Regards

Another electrodynamics question. You have an infinitely long cylinder made of linear ferromagnetic material of relative permeability Km, and you place it into an initially uniform B field. Let the cylinder axis coincide with the z axis, and let the B field point in the x direction, or y direction. CURL H = 0 What is the B field inside and outside the cylinder? I want to use cylindrical coordinates, define a scalar magnetic potential inside and outside the cylinder, and get the B field this way if possible, but I am not sure if it can be done this way. If there is a better way I am open to that. Thank you