DrRocket

Any good university will have adequate or better programs in chemistry and chemical engineering. The various UC campuses are excellent. Job experience will depend on what you wind up doing. A chemist developing new molecules for the pharmacy industry will see different things from a chemical engineer working in an oil refinery, which is different from a propellant chemist compounding new rocket fuel, or a chemical engineer working on combustion processes for a power generation company. Eugene Wigner, the Nobel Laureate noted for introducing group theory methods to quantum mechanics, was educated as a chemical engineer. You are not confined narrowly by any degree.

Chemistry is science. It is basically the study of how atoms combine to form molecules and the resulting properties of those molecules. As with all sciences the primary objective is the discovery and understanding of the fundamental principles involved. Chemists very commonly deal with small scale reactions under very well controlled laboratory conditions. Chemical engineering is not science. It is engineering. The objective of engineering is the development of useful and beneficial products, coomonly involving complex and incompletely characterization of real-world systems, often resulting in the face of incomplete understanding of the underlying science, and nearly always withy schedule and economic constraints. Chemical engineers make extensive use of chemistry, but are not chemists. Chemical engineers commonly deal with large scale industrial chemical reactions, and control of those reactions so as to produce consistent product in large quantity. In industry there is much interplay between chemists and chemical engineers, and people educated in one discipline may find themselves doing work that might be seen as being associated with the other. It should be quite easy at the undergraduate college level to switch between the two disciplines, so you have a lot of time to make up your mind. There is much overlap in the first couple of years of study. University colleges often hold open houses for prospectivev freshmen. You might attend one or more and talk to both some chemists and chemical engineers. Most state universities have adequate departments in both disciplines. You can also seek information from engineering societies and the American Chemical Society.

Channge of scale.

Maxwell's equations mean exactly what they say, in any inertial reference frame, without need for reference to any "ether". It is the Lorentz invariance of Maxwell's equations and the fact that light is nothing more and nothing less than an electromagnetic wave that provide profound evidence for special relativity. Mathematics does not "explain everything". Mathematics provides the framework whereby physics explains those things that we do currently understand. That does not yet encompass everything. It does include quite a lot, and that lot is accessible only to those who understand the language.

Time does not speed up or slow down. Time always progresses, according to any accurate clock, at 1 second/second. What clocks measure is proper time. Proper time is associated with a world line, and two world lines joining joining two identical spacetime points can exhibit different proper times. This phenomena can account for "gravitational time dilation", the "twin paradox" and any other questions involving time. http://www.scienceforums.net/topic/54990-proper-time/

Vector addition is always commutative. Scalar multiplication distributes over vector addition. All of this comes from the definition of a vector space, and is more general than the "parallel law". The parallel law, and other geometric aspects of vector addition come from additional structure that can be imposed, such as a norm or an inner product. You can show commutativity of "parallelogram addition", by drawing pictures, which is basically how the operation is described. The definition of a vector space starts with an abelian group of vectors, hence commutativity of vector addition, coupled with an operation of scalar multiplication that is required to have the usual properties of associativity and distributivity. See any text on abstract algebra or linear algebra. Volume 3 of Nate Jacobson's three volume set Lectures in Abstract Algebra is one good source.

It appears that what you are describing is just the number of ways in which 8 objects can be sorted into (unordered) pairs -- each element in a pair being an end point of one of your lines. This has nothing to do with the geometric arrangement of those points -- as for instance points on the face of a square. For n elements, n even, the number of such pairs [math] \dfrac {n!}{ (\frac{n}{2})!2^{\frac{n}{2}}} [/math]. For the case of n=8, this number is 105. You can find this sort of comcinatorics problem discussed in elementary algebra texts (high school Algebra II) under the heading of "permutations and combinations" (this is a question of combinations).

Not only the shorthand, but the actual content. Popularizations and baby talk fail to convey anything other that superficialities -- the mere illusion of understanding. No, you don't. What has become abundantly clear since the advent of quantum mechanics and relativity, over a century ago, is that your everyday experience, essentially classical Newtonian mechanics, is simply wrong at the atomic level, when dealing with objects moving at high relative speeds, and when large gravitational fields are present.. The Newtonian model is a very good approximation when dealing with macroscopic bodies, at low speeds, in moderate gravitational fields. But it is only an approximation, and to understand "reality" requires a more sophisticated perspective, complete with the relevant mathematics. By denying yourself the necessary mathematical sophistication, you are limited to, at best, a nineteenth century understanding of nature. But what we understand of the world has changed a great deal, and you have opted out of that understanding by remaining ignorant of the mathematics which makes that understanding accessible. You are like the blind men attempting to understand the elephant. What is meant by time dilation, length contractiom speed and position is crystal clear to those who understand the relevant mathematics. Those phenomena are in fact a simple consequences of the invariance of the Minkowski metric.

What you have demonstrated is that you have no idea what mathematics is. Mathematics is not a scientific theory. In fact mathematics, because it has no reliance on experiment and no a priori tie to nature, is not really a science. Mathematics is not symbols. Neither is mathematics primarily equations or even solutions to equations. Mathematics is not only describable in words, it is in fact described in words -- see any good mathematics text. The symbols are merely shorthand for a great many words. Those words are important The use of words in mathematics is exquisitely precise, and you need to know what they mean. "Mathematics is the study of any kind of order that the human mind can recognize" -- Pasquale Porcelli, Professor of Mathematics Mathematics is also the language in which physics is formulated. You can no more understand physics without mathematics than you can understand French literature with no knowledge of the French language. Popularizations and baby talk just don't cut the mustard. If you do not understand mathematics then you are effectively illiterate in physics. Ignorance of mathematics is correctable. Willful ignorance is not.

"To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’. To those who do not know mathematics it is difficult to get across a real feeling as the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once." – Richard P. Feynman in The Character of Physical Law

They admit to not getting mathematics (some with pride). More troublesome, thought not readily admitted, are their problems with logic and ethics.

In that case you should have been able to answer the question yourself.

It would be very educational for you to read a book on special relativity so that you might comment on what the theory actually says rather than what you think it says. An Introduction to Special Relativity by Wolgang Rindler would be a good place to start.

Right. Relativity (special or general), just like classical mechanics, can be viewed in two ways: 1) As a mathematical theory, subject only to the foundational assumptions and rigorous mathematical logic. 2) As a predictive model of natural behavior. When viewed in the first way, both are airtight, mathematically consistent theories. No disproof by thought experiment is possible. When viewed in the second way, it has already been proved that classical mechanics is "wrong", and the evidence in favor of special relativity is that proof. If special relativity is wrong then the disproof will have to come in the form of experimental data. Since special relativity is just a local approximation to general relativity and neglects gravity, we know that it is not absolutely correct -- but it is still an excellent model when properly applied. We also know that general relativity (GR) and quantum mechanics are incompatible. Thus it is likely that general relativity is also not the last word. But it has the support of a huge body of experimental data and is a very good model when quantum effects are not important. Even in the absence of quantum effects there is a competing theory of gravitation. It is called Einstein-Cartan (EC) theory. EC theory differs from GR in not making the a priori assumption that spacetime is torsion free. This greatly complicates the mathematics of the differential geometry. At the current level of measurement technology the predictions of EC theory are indistinguishable from GR, and because the mathematics is so much more difficult EC theory is not so well known. It has the intriguing feature that at least some of the singularities associayed with exotic GR solutions do not occur in EC theory. The point here being that again we have two mathematically consistent theories, and only an advance in experimental capability will be able to determine which is the better model or "correct". Thought expereiments are not sufficient.

I don't need a curl definition for more than three dimensions. The integral of a gradient around a closed curve is still zero. You need to be more clear about what you mean by "dynamic" and how your function is defined. All that you have said thus far is that you have a gradient and have chosen to call one variable "t". The 3-dimensional vector analysis to which you refer extends to analysis in any dimension and to manifolds of arbitrary dimension. The subject is differential geometry. You might want to take a look at Mike Spivak's little book Calculus on Manifolds.

This makes no sense. It is not even pseudoscience. Dimensions don't work like that. You cannot "live in higher dimensions". You must live in ALL dimensions, even if some are compactified so thar you do not notice their effect.

The integral around every closed loop being zero is equivalent to the integral over an arbitrary path depending only on the end points of the path. You started by stating that you vectorvfield arose as a gradient. Potential fields are closed, therefore exact. Your field is necessarily conservative; i.e. line integrals are independent of path.

And therein lies the difference between you and owl.

The intelligence of a committee varies approximately as [math]\dfrac {1}{(no. \ of \ members)^4}[/math] The group intelligence of the U.S. Congres with 535 members is practically 0.

This top[ic is listed under "Other Sciences". I understand the applicability of "Other". Might someone explain the relevance of word # 2 ?

The issue of time travel has been beat to death in several threads. To repeat much of a post in one of them the situation is this: General relativity by itself does not preclude closed timelike curves. But other things may, at least at the macroscopic level. This is Hawking's "Chronology Protection Conjecture". Pretty much everyone expects that something like the chronology protection conjecture will be shown to be true. But it is not straightforward. Closed timelike curves do exist in some exotic solutions to the Einstein field equations. Gödel's lambda dust space, the Tipler cylinder and Kerr black holes have CTCs. On the other hand Hawking showed that if the weak energy condition holds then there are no CTCs. Quantum fields, as I understand it, generally do not conform to the weak energy condition (ajb chime in) so chronology protection may well not hold at the quantum level. There are Feynman diagrams that include non-causal branches. There is good reason to demand a mathematical proof based on general relativity. Odd things really do happen. When black hole solutions were first found, many physicists, Einstein included, did not believe that physical black holes would exist. One learns in physics by pushing the limits of what is known and investigating situations that might be termed "exotic". Exotic is not necessarily impossible. Any proof of the chronology protection conjecture will have to include some conditions that constrain the admissible solutions to the Einstein field equations. The known exotic solutions with CTCs serve to demonstrate that fact. General relativity alone is not enough. In a sense the deep problem is that causality appears to not strictly hold at the quantum level, but it seems to apply in "normal" macroscopic situations. So, where is line between the quantum world and the macroscopic world where causality kicks in ? This is well worth understanding.

I am not expert in Hodge theory, but it appears that you have things rather contorted, too contorted to attempt to work out here. A couple of suggestions: 1. Get a good book that treats Hodge Theory. Principles of Algebraic Geometry by Griffiths and Harris or Foundations of Differentiable Manifolds and Lie Groups by Frank Warner perhaps. 2. Learn to walk before running. This is rather sophisticated stuff and more than one quantum jump up from material in your earlier posts which you had not mastered.

You have a very much undefined and unconstrained problem statement. The line integral of the gradient of any differentiable function over a closed loop is 0. Other than that your problem is wide open.

A paradox occurs when valid reasoning produces contradictory conclusions. Whan you have produced is not a paradox. It is just a mistake. Any conclusion reached about a physical event, such as an explosion, in one reference frame is valid in any other reference frame. Events are invariant. What are not invariant are cooordinate dependent measurements such as time and length. A nuclear chain reaction involves more than just spatial geometry -- a ratio of volume to surface area -- as you suggest. The retention of energy within the mass is a dynamic process and also involves time. You neglected to include the effect of variation in time measurements between reference frames in your analysis. This is more difficult, and demonstrates why calculations can be greatly simplified by choosing a convenient reference frame. But in any case the mass will either explode in all reference frames, or no explosion will occur in any reference frame. No paradox. Just an incomplete analysis, a mistake.

Actually a sphere passing by at relativistic speeds LOOKS LIKE a sphere. The flattening is not observed, due to the finite speed of light. There is a big difference between the actual dimensions and what would be seen by an eye or a camera. This effect was first noticed by Roger Penrose, and you can find a discussion in his book The Road to Reality (highly recommended). On the other hand, an actual measurement, could it be performed, would show the sphere to be highly oblate. Appearances can be deceiving.