Bignose

Let's look at SandMan's formula here: 100 meters / 28 unknown units = x meters / 52 degrees so, the unknown units on 28 must be degrees, but what is 28 degrees? 90 degrees minus 52 would be 38. So, the 28 is either wrong or just made up. Not only that, but the lengths and angles aren't in perfect proportion, that's why the trig functions exist.

Oh my goodness! You know what other name adds up to 2? Jesus! Jesus is the devil!?! Someone should really tell the Christians.

A proper physics text should be chock full of math. By proper I mean a book designed to teach and/or discuss physcsis, not one of those popular science trying-to-teach-advanced-physics-concepts-using-analogies-and-pictures books. Whether it is a very basic level high school algebraic-based physics, to a typical university level text full of calculus, to the advanced graduate level texts on string theory or general relativity or any of the other thousands of advanced physics topics, they will all have quite a lot of math in them. Just as one suggestion, check out Roger Penrose's The Road to Reality: A Complete Guide to the Laws of the Universe; it is going to have more math describing physics in it than you've probably ever thought possible.

I like Handbook of Stochastic Methods by Crispin W. Gardiner. The title makes it sound like it is very proof-orientated, but it is pretty applications-driven. As an engineer, I usually skip a lot of the larger proofs myself, and it wasn't necessary in this book. I also like Stochastic Processes in Physics and Chemistry by N. G. Van Kampen. Very applications driven, this book, which has just had a new edition published, really is a classic in the field.

Something struck me as odd about this thread, but it took me like 5 mins before I could stick my finger on it... it can't be "National" Sex Day if it's only one region or town, can it? Still a funny link, nonetheless.

But, I still fail to see anywhere that an extra time dimension (or more) is needed. The traveling of a body toward a gravity source can easily be handled in just 1 time dimension. It may not necessarily be solvable directly, but it's description using 1 time dimension is pretty easy. There are situations when using multiple time dimensions is appropriate. When describing a population of cells, one can use the cell's age as one of many descriptors. Then, there are two times -- the regular time that we all know, call it t, and the age of the cell, call it t'. The only difference between the two is that t' resets to zero every time a cell divides. But, the derivative between them is still 1 = dt'/dt They both pass at the same rate. But, I don't see how a derivative in time (which is a velocity, or an acceleration) "implies" another time dimension. If we take the a derivative in x, df(x)/dx, we don't say that that implies another dimension, y? So why would derivatives in time imply that? Especially since if we change the variables, instead of a stationary observer and observe with the moving particle, a derivative in space (d/dx) can be transformed into a derivative in time (d/dt). I don't see any need to invoke another dimension.

No, you are confusing the mathematical concept of a point with an actual point drawn on a piece of paper. Obviously, a real point on a piece of paper will have some height and width. Even the really small ones like the words written on a grain of rice, or on the head of a pin, or electronics microimprinting, the "points" will still have height and width, even is it is only microns. But, the mathematical concept of a point has no height and width. In 1-D, if I say x=5, then x is only exactly 5. If I needed x to have some "width" I'd have to write a statement like [math]x \in (4.99,5.01)[/math] so that x would have a "width" of 0.02. If there is a "height", then you'd have to invoke a second dimension, like y. But, if x=5, then that doesn't include anything else, not 4.9999...1000 more 9's... 99999 nor 5.000....1000 more 0's...0001. There is no width at all. Only 5, and because there is no width or anything else, a point is zero dimensional. Just because it is a physical impossibility to "create" such a point, that doesn't hamper its usefulness as a concept. Depending on the application, such as treating ideal gases, a molecule can very easily be considered a point. In real life, sure, it isn't a "point", the molecule has a width, height, and depth measured in angstroms, but working with the math treating it as a point predicts the behavior of the gas exceptionally well. Another example is gravity. If you have two objects that are spheres, and so long as the two objects don't get too close, you can predict the behavior of the interaction treating both objects as points. For example, using model of the solar system to predict how to launch a probe at Mars, the influence of Uranus and Pluto and other far away objects are important, but you don't need the level of detail of the density differences in the composition of the planets -- treating them as points is more than accurate enough. There are many examples of where something that isn't actually a "point" can be treated as a point in the mathematical sense and it is perfectly reasonably to do so.

The angle between vectors formula is valid for any number of dimensions: cos(theta) = a.b / (|a| |b|) because the dot product is valid in any number of dimensions. Therefore, you can find the angle between 10 D vectors, for example. This is about the only example I can think of, however, but there are probably many more, and this may not have been what you were looking for.

I knew it was a high number, but didn't realize it was up to 12. I remember when I took undergrad thermodynamics and the professor gave us the then-up-to-date phase diagram for water, with all the ice type phases known at the time -- what a mess!

To expand a little on what Sysyphus said, a few of the other ones that are less dense in their solid state than liquid state are silicon, antimony, gallium, and bismuth. Acetic acid is the only other compound that I know of. Considering the millions of compounds/elements that we know the physical properties of, it really is quite rare. And, then, to further confuse us all, not all ice is actually less dense than liquid water. However, that ice is only formed under rare circumstances, like high pressure and/or very cold temperatures. If I remember correctly, there are something like 6 different forms of ice -- maybe even more.

Just a few quick notes in reply here. 1) RE bolded part in your quote above: Energy and momentum are not necessarily linked. Potential energy doesn't have momentum in it, chemical energy stored in gasoline or ATP doesn't have momentum in it, the electrical energy stored in a battery doesn't have momentum in it, and yet all of these still use the same units. 2) Following your methodology there in your last paragraph, say I come up with a quantity that has units of mass*(length squared)/(time squared), however I got there, again how do I know if that is a unit of energy or a unit of torque? Just as was said above, two very, very different quantities, but the same unit. Lastly, I am not too sure what is really wrong with the current methodology. Take F=ma, force is defined as mass times acceleration, so force is defined to have units of mass*length/(time squared). Or Work dW=F*dl, Work=force times a distance, so work has to be mass*(length squared)/(time squared). Etc., etc. The units come from the definitions of the terms, and are what they are. Again, from your way, it looks to me like putting the horse before the cart, because just because a set of units come out to a certain form, doesn't mean that that combination is meaningful.

I heard about this on the radio about a month ago (NPR's Science Friday show), and the motivation was one that I think a lot of us can sympathize with. The original researcher was annoyed that his cell phone battery would die, and the phone would chirp in the middle of the night to alert the guy that its battery was dying. And he thought, why can't this be recharged without having to remember to actually plug it in. It's a good idea, but I actually don't see being too commonplace for household items. The trouble is the field that is generated, it will destroy a lot of electronic information that gets between the generators, like credit cards or hard drives. It's use will most likely be to power measuring devices that it would be impossible or very difficult to run a power line to it: something like a temperature probe in the middle of a reactor -- the probe may be resilient enough to survive the environment, but the wire to it may not be.

Snail, I am sorry, every day I log on and see this post and keep thinking "I really should recommend that book I read last year" but I never went to the shelf and got it out. Well, today, when a new post bumped you from the top, I finally got off my duff and did it: I read Games and Decisions. Introduction and Critical Survey by R. Duncan Luca and Howard Raiffa. I thought it was pretty good. It is a Dover book, so it lists for $15.95, but because it is a Dover book I think that it is a little dated. It was originally published in 1957, and Dover republished it in 1989. It doesn't have any modern work at all, but all the classical results are covered in very good detail. It has exactly what is sounds like you are looking for "starting with basic situations and use of matrices and onwards." For the cheap Dover price, it is well worth it. Game Theory was just one of many passing interests, so I never found a more modern book to read, though I am definitely interested. Someday...

Ann, it is one of this forum's rules that we won't do homework for you, and this at face value looks like a homework problem, so I won't do it for you. But, I will give you some hints -- the second sum goes from k=0 to infinity, while the first goes from 1 to infinity. I'd find a way to convert that second sum into k=1 to infinity or the first sum to k=0 to infinity so that you are dealing with the same sums. That should get you pretty close to finishing the problem.

This is admittedly going to be very non-specific and non-rigorous like completely formal mathematics can be. But, my understanding of dimension has always been that a separate dimension is needed for any quantity that cannot be described by any other dimensions. That's worded a little oddly, so, let me give you some examples: You cannot describe depth in any way whatsoever using only length and width, or, to put it another way, you cannot describe a point's position in z using only x and y. Unless there is a known relationship between x,y,& z, like tracing a curve or edge of a solid. Or another example, you cannot describe the velocity of a particle using only it's position. Or (taken from an example of my engineering work) a particle's size, or shape, or reactivity, or porosity, or concentration using only its position and/or velocity. Each of those requires using another variable, which can be thought of as another dimension, to describe the distribution of particles. One of the first examples of using something like this is the kinetic theory of gases, where they integrate over the 6 dimensional space of position and velocity to describe a population of gas molecules. So, again, the definition isn't very formal at all, but that's how I always thought of dimension -- it's a quantity that cannot be described using any of the other dimensions.

What about Knoxville, Iowa? They have the National Sprint Car Hall of Fame & Museum and the Knoxville Raceway dirt track. I bet the other 4 Knoxvilles (Georgia, Illinois, Maryland, and Pennsylvania) have something in them, too. But, seriously, CDarwin, did you just start working for the Tourism Board or something? There has to be some sort of back story why you'd start a post like this...

If pioneer would actually do the math, he could prove this for himself. Until he does the math, personally, I've given up on this thread. Like you said, Ben, we've told him 9 times now, is he waiting for 20? 50? Whatever it is, I'm not going to keep repeating myself.

Think if it this way, each unit square of the shell has a certain amount of gravity force, right? Let's get close enough to the wall so that only 1 unit square is in "front" of you. Sure, that unit square is going to exert a certain amount of force. But, with every single other unit squares behind you, they all exert a force the other direction. Some of them, the ones farthest away, will exert only a tiny amount, but because there are so many behind you, it adds up to be the same as the one in front of you. I said it above, but the combination of small amount of mass close to you completely equals the combination of a large amount of mass far away. It is exactly the same as being between 2 masses: a 1 kg mass 1 m away and a 4 kg mass 2 m away. The smaller but closer mass completely balances the farther away but larger mass. pioneer, this is a result that has been known a very long time. The math really isn't all that hard... have you done any of the math? Once you do it, you should be able to convince yourself that what the three of us are saying is true. If you have any problems, I personally volunteer to help you work through the math, post it to this thread, and I'll do my best to help. But, this is a problem that has been worked by literally tens of thousands of students, and the answer is very well known. So, please do the math to convince yourself of the zero gravity answer. Otherwise, this thread is just going to go on and on and on. But, even more meaningful, is that since your very first premise is flawed, nothing you derive or reason from it can be right, so you're doing nothing but building on a rotten foundation -- the house you build on a rotten foundation will not stand.

No, you can't agree and then say that the gravity is "highest at the surface of the shell." Inside a uniform shell, there is no gravity. Not in the center, not halfway between the center and the wall, not even on the inside of the wall. Zero gravity. Consider being on the surface of the shell. Despite being closer to the shell in one direction, there is more mass on the farther away side. The combination of farther away but with more mass completely balances the nearer but less mass. So everywhere inside a uniform shell has zero gravity.

Sisyphus is exactly right, it is a very classical result that a spherical shell of uniform density will have no gravity inside the shell. If your math says something different, you've made a mistake somewhere.

?????? [math]-1 \neq \sqrt{-1} [/math] why would you think that?

Sometime in Calculus II you will be introduced to a technique termed 'Trigonometric Integration' where you will substitute trigonometric functions for the variables in the integrand. And, the idea is that the trigonometric functions are easier to manipulate, i.e. via angle sum rules, angle product rules, etc, than the way the original integrand was written. You will need to know the trig functions forward and backward to pass this part of the class. So, there is a part of calculus where the trig functions are an absolute necessity. However, I do agree with the other posters that you can in all likelihood teach yourself all the necessary information. Another option, at least what I did at my undergrad institution, was a week-long trig review class one week before classes started. Something like that might be perfect.

I guess I just don't see the advantage to using 4-D when 3-D would do just as well. The real trouble with using 4-D when the situation is really 3-D is that one of the dimensions of the 4-D system will be dependent on the other 3. And, a really big one would be like John C said, resolution of forces or any other vector mathematics would become significantly more difficult, and it isn't all that easy in the first place. On a little bit broader note, what is wrong with [math]\sqrt{-1}[/math] anyway? It is well defined, it obeys all the logical rules, the mathematics of it are pretty well established and developed (as demonstrated by the size of the average Complex Analysis text). I personally just don't see a problem.

Well, paper burns at 451 degree Fahrenheit, and water boils at 212 degrees Fahrenheit.

Well, starting from a purely mathematical point of view, one way to define perpendicular is to define the angle between two vectors as [math]\cos \theta = \mathbf{n}_1 \cdot \mathbf{n}_2 [/math] and thusly when [math]\theta = \frac{\pi}{2}[/math] then [math]\cos \theta = 0[/math] and the vectors are considered perpendicular. In 2-D, to see if two lines are perpendicular, you just computed the dot product of vectors that go along the line. In 3-D, you usually used a vector that was normal to the plane and computed the dot product between those normals. (You could do exactly the same thing in 2-D.) For example, consider two planes in 3-D: [math]a_1 x + b_1 y + c_1 z + d_1 = 0[/math] [math]a_2 x + b_2 y + c_2 z + d_2 = 0[/math] Then the normals are [math]\mathbf{n}_1 = [a_1, b_1, c_1][/math] and [math]\mathbf{n}_2 = [a_2, b_2, c_2][/math] There is no reason that this same thing cannot be extended to 4-D or more. Certainly dot products can be extended to 4-vectors and higher, and normals to planes in higher dimensions can be calculated in just the same way. Finally, though, I want to come back to "In one dimension, only points can be perpendicular" This statement is essentially meaningless. Especially when you look at the definition above. The normal of every number in 1_D is "1" Then, you take the dot product between two "1"s? It pretty quickly becomes meaningless.