Tom Mattson

That's great. Here we have this rare, magnficent creature of which we know next to nothing, and the best thing these people can think of to do is catch it and gut it. Didn't anyone think of tagging and monitoring it? Tom

Do the partial derivatives one at a time. You wrote this as a quotient, but I like to rearrange complicated functions so that I never have to use the quotient rule, like so: f(x,y,z)=(sin((x2+z2)1/2))(x2+z2-y)-1 fx=(cos((x2+z2)1/2))(1/2)(x2+z2)-1/2(2x)(x2+z2-y)-1+(sin((x2+z2)1/2))(-1)(x2+z2-y)-2(2x) fx=xcos(((x2+z2)1/2))*(x2+z2)-1/2*(x2+z2-y)-1-2xsin(((x2+z2)1/2))*(x2+z2-y)-2 Happily, x and z occur symmetrically in this function, so you can find fz by simply swapping x<-->z in the above derivative. Finally, fy=sin((x2+z2)1/2)(x2+z2-y)-2 Tom

No. Your assumption of "no universal rest frame" is an assumption of relativity (buried in the relativity postulate), and the conclusion you state does not follow from that. Specifically, if we assume relativity the upper limit on velocity is c, period. Tom

The underlined text in my last post are links (wasn't sure if that was obvious). You can get more detailed info on Goedel, his theorem, and fuzzy logic by following them. Tom

It was the one asking "If a family has one girl, what is the probability that she has a sister?" I said 2/3, and it was 1/3. :slaphead: Tom

1 wrong-->93.75%

This is not about fuzzy logic, but about formally undecidable propositions. You may have heard of Kurt Goedel. He had many theorems, but the one known as "Goedel's Theorem" was published in a paper called On Formally Undecidable Propositions, and it states that every formal system at least as complicated as arithmetic is either incomplete or inconsistent. A consequence of this is that all formal systems break down under self-reference (that is, when a statement refers to its own truth value). Fuzzy logic is a different animal altogether, as it relaxes the restriction of a two-valued logic. Tom

*ahem* You may want to edit the title. It's a little misleading. Tom

I'm not a bio person, but I'm going to take a shot. I think you're talking about the epiglottis, that flap of tissue that keeps food from going down your windpipe. Tom

The scalar potential should be familiar from Physics II as "energy per unit charge". The vector potential is not so different from that, as it is a kind of "energy per unit current density" (don't take that too literally, as A is still a vector, and energy is not). The Lagrangian and Hamiltonian formulations of dynamics are in terms of energy, and so when dealing with the dynamics of charged particles, terms such as "eV" and "j.A" are thrust upon us. The bit about gauge invariance comes in because the potentials V and A are not uniquely defined for a given E and B. Tom

Not really, Raider. In Newtonian mechanics, the space is a Euclidean 3-space from which time is just something separate. It is not logically entailed by the space, but it is also not inconsistent with it. In SR, however, the Minkowski spacetime includes the time component as a necessity. Tom

The first thing to realize is that they are not physical fields, but mathematical functions from which physical fields are derived. As I said, they are not physical fields, but they are the elementary fields that appear in the Lagrangian or Hamiltonian. This is because the energy associated with a field is of the form "eV" (charge times scalar potential) and "j.A" (scalar product of current density and vector potential). Since Hamiltonians and Lagrangians deal with energies, you get terms like that. Yes: For a 4-potential Au, any transformation of the form: Au-->Au'=Au+:pdif:uX with :pdif:u:pdif:uX=0 will leave the physical fields unchanged. Tom

That is the unequivocal testimony of the physics community. No; only 3D figures in time. There is no motion without time. This effort is a non-starter, because the Minkowski space of Special Relativity is a legitimate 4D vector space, and time is its 4th dimension. What exactly about that do you have a problem with? Tom

My guess: Your calculator was in "degree" mode, and you entered the limits of integration in radians. Tom

An operator is any mathematical object that acts on any other mathematical object to return a third mathematical object. Operators arose in QM because of the discrete spectra of radiators. Heisenberg first noted that the experimental results could be reproduced if one were to regard the state of an atom as a vector and represent the physical observable as a matrix. Schrodinger developed his version of QM (the one we still use today) not in terms of matrix operators, but in terms of differential operators. It turns out that the two approaches of QM are equivalent. Indeed, it is known from mathematics that matrix operators and differential operators are equivalent. All observables in QM (momentum, energy, position, angular momentum,...) are represented by an operator. Also, many non-observables are represented by operators, for convenience. Examples are the ladder operators the Radical Edward mentioned. Tom

In classical mechanics, the Hamiltonian is the total energy. In quantum mechanics, it becomes an operator. When that operator acts on the wavefunction, you get the Schrodinger equation. p=momentum Tom

What I mean is, the MIT course would be a prerequisite for the last set of notes you posted. In more detail: MIT Course Prerequisites: Physics I/II (mechanics + EM), Calculus (differential + integral), Basic Linear Algebra, Basic Differential Equations Zettili Course Prerequisites: MIT QM Course, Advanced Calculus (including vector calculus, calculus of variations), Linear Algegra (including vector spaces), Differential Equations (including Fourier analysis, PDE), Abstract Algebra (especially groups) Tom

Those notes are good. The only thing I would advise on is that they are clearly not meant for a first course in QM. On the other hand, the MIT Open Course Ware notes at the top of the thread are a first course. You might want to consider that when deciding which one you want to cover first. Tom

I had a post explaining why your situation is impossible, but then I saw the picture. The point is, if the spaceship sees him die, then he dies in every frame. Try playing with the Lorentz transformation for the following: Event 1: Laser fired Event 2: Message sent Event 3: Laser hits Event 4: Message arrives You will certainly be able to construct scenarios in which the message arrives before the laser hits (by making the path length for the laser sufficiently long), but you will never be able to construct an case in which the man dies in one frame and not in another. Tom

I thought these lonely forums could use some traffic. In Chicago's Field Museum of Natural History resides the largest, best-preserved T-Rex skeleton ever found., named Sue. The restoration was done just off the main lobby so that visitors could see, which was kind of cool (I was there). When I went back in Nov. 2001, she was standing up at full height on exhibit. Check it out: http://www.fmnh.org/sue/default.htm Tom

Is anyone going through these? Would anyone like to, here in this thread? I will flesh out the details, if there is sufficient interest. Tom

The induced EMF is: EL=-L(di/dt) Note the negative sign. That means that the induced EMF has opposite polarity to that of the battery. This reflects the physical fact that an inductor always works against changes in magnetic flux. So, in a sense the inductor is pushing the current, causing (di/dt) to shrink. I don't really like doing people's homework for them. Can you try to start the discussion, and I will help you through the rough spots? Tom

fafalone: "Isn't the spin quantum number for a photon 1? Yes. However, by "additive quantum number" I mean something that adds algebraically (like charge or strangeness). Spin adds like a vector. fafalone: "Energy doesn't create gravity, mass creates gravity." You are stuck on Newton. Look at the GR equations again, and you will see that it is energy-momentum that is the source term for spacetime curvature (and thus gravity). Tom

If anyone wants to get started on this, let me make a quick note: The energy-momentum tensor for EM: Tuv=(1/4:pi: )(FurFvr-(1/4)guvFrsFrs) guv is, of course, the unknown metric for which you have to solve. But Fuv is known ahead of time. Its definition is: Fuv=duAv-dvAu, the EM field strength tensor. where du=(d/dt,-grad) is the 4-gradient Au=(V,A), the 4-vector potential of EM For EM waves, V=0 and A should work out such that you get sinusoidal E and B waves (A is not unique, because of the gauge freedom). Tom

Radical Edward: "so, discounting diffraction, two infinitely long parallel laserbeams would bend towards one another?" Now, you are pushing the limits of what I know (GR is not my strongest suit). On my "To Do" list is read this document: http://arxiv.org/PS_cache/gr-qc/pdf/9811/9811052.pdf in which they analyze the problem. I will let them speak for it. fafalone: "Is there such a thing as an anti-photon? :/" The photon is its own antiparticle. Antiparticle states are understood in QFT as charge conjugated states. The charge conjugation operator changes the sign of all the additive quantum numbers, but for a photon these are all zero. Tom