DrRocket

Beyond the quadratic it gets ugly fast. http://en.wikipedia.org/wiki/Discriminant

1. I said that a knowledge of real and complex analysis is needed to be adequately prepared to TEACH calculus, not to use it in the monkey-see monkey-do manner that you describe. It is needed for more sophisticated application of physical principles, most assuredly including fluid dynamics. 2. I have see quite a few applications in which the Navier-Stokes equation and the nature of solutions are very important. I have seen millions of dollars spent on such problems with billions at risk. 3. That MIT professor to whom I referred is a world-recognized expert in aerodynamics, and the Navier-Stokes equation is rather important to his work. His opinion is at odds with yours. I agree with him. 4. I have encountered quite a few PhD engineers who are woefully deficient in their understanding of mathematics. One more is not a big surprise. 5. You are quite right that the "average engineer" uses very little calculus. The "average engineer" also has a tendency to find himself in over his head with non-routine problems. I have been involved in several major failure and accident investigations as a result. These issues put lives at stake. 6. I am well-acquainted with refineries and chemical plants. Your argument does not impress me. 7. "Sizing a pump or flow rate through a packed bed" are useful topics, but of little use in the case of supersonic flow involving compressible gas. There are many facets to fluid dynamics and a head-in-the ground perspective that limits one's horizons is counter-productive to a broad understanding of the discipline. 8. I would certainly not hire anyone with the sort of blinders that you describe in what was my fluid dynamics group. There multi-phase analysis of reacting flow is not just an academic exercise, it is an everyday analysis task. 9. If you would like to see applications of real and complex analysis to air foil design you need look no farther than the work involving conformal maps or the work of Abraham Robinson. 10. Deep understanding of physics by virtue of understanding the associated mathematics may not reflect the work of an "average undergrad engineer", but "average undergrad engineers" do not handle much beyond routine problems. To do exceptional work, you need exceptional understanding. 11. The problem with your statement that you "don't see it as issues that even very experienced and good engineers with only an undergraduate degree need to be aware of" (referring to the basis of derived equations) is that unless one is at least "aware of" such issues one does not recognize when a situation arises in which they are crucial and the result can be catastrophic. This attitude is precisely why an understanding of the fundamentals is important. At the very least the average engineer needs to recognize when help is needed. You have done nothing but convince me that more attention to fundamentals, not less, is what is needed. Your attitude is a symptom of the problem, not the solution. Your last statement that "Obviously, your mind is made up," is an unwarranted statement, commonly adopted by someone who wishes to deflect attention that their own mind is closed.

See the post above. It is all based on the, reasonable, assumption that the universe is pretty much the same everywhere. That assumption is quite reasonable, but it is indeed an assumption. You can also assume that the universe, and the laws of physics themselves, vary. But that too is an assumption, and there is not the slightest bit of empirical evidence to support it. It is great for selling books, and advertising time on TV programs, but then so is magic.

You lack understanding of the burden of proof. You have made an unqualified assertion. The burden of proof is on you. The expansion of the (entire) universe is based on several things 1) the observed red shift 2) general relativity and 3) the cosmological principle which asserts the homogeneity and isotropy of the universe on the largest scale. The first two are veery well supported by evidence. The cosmological principle is an outright assumption that is consistent with what is observed. All of this is consistent with the big bang model based on general relativity. It may or may not ultimately be true, but unless the accelerated expansion of everything that we can detect stops, it doesn't matter as it applies to everything with which we are or ever will be causally connected. As soon as you violate the cosmological principle, there are a great many (really infinitely many) cosmological models available since you are then free to change physics willy-nilly as a function of space. That sells books, but it violates the basic requirement of good science -- falsifiability in principle. Michu Kaku is a poor source. Now, back up your assertion. Fair warning, I am quite aware of such speculations and of the very shaky basis for them. The fact that that you may have read such stuff in popularizations is not adequate basis to assert the truth or even likelihood of it. There are huge gaps in those models.

Prove it.

That is differentiating functions whose graph is a straight line. But that is an important class of functions. The whole point of differential calculus s that a great deal can be said about a very large class of functions by approximating them locally by straight lines.

An easy one would be [math] id: \mathbb R \rightarrow \mathbb R[/math] with the usual topology in the domain and the discrete topology in the range. OK so far, though the language is a bit stilted. You have the concept of a homeomorphism understood. [math]\mathbb{R}^1\setminus \{0\} \times \mathbb{R}^1 \setminus \{0\}[/math] is not homeomorphic to [math]\mathbb{R}^2 \setminus \{0,0\}[/math]. It is homeomorphic to [math] \mathbb R \times \mathbb R \setminus \{(\mathbb R \times \{0 \}) \cup (\{0 \} \times \mathbb R)\}[/math], which is not connected. It is true that [math]\mathbb{R}^1[/math] is not homeomorphic to [math] \mathbb{R}^2[/math] but that does not follow from your argument. It is also true that [math]\mathbb{R}^2 \setminus\{0,0\}[/math] is connected and therefore cannot be mapped continuously onto [math]\mathbb{R}^1\setminus \{0\}[/math] which is not connected. What might surprise you is that [math]\mathbb{R}^1\setminus \{0\}[/math] can be mapped continuously onto [math]\mathbb{R}^2 \setminus\{0,0\}[/math] (this is far from obvious and requires theorems on "space filling curves"). This does not seem to follow from even the fact that no two of [math]\mathbb{R}^1[/math], [math] \mathbb{R}^2[/math], [math]\mathbb{R}^1\setminus \{0\}[/math] and [math]\mathbb{R}^2 \setminus\{0,0\}[/math] are homeomorphic. Moreover it is quite easy to have an open mapping that is not a homeomorphism, say [math] (x,y) \rightarrow x [/math] from [math] \mathbb R^2[/math] to [math] \mathbb R [/math]. How this implies that open maps need not be continuous (a true statement) is not evident. As an aside you may be interested in theinvariance of domain theorem for maps between Euclidean spaces of like dimension that shows that injective continuous maps are open.

How hard can it be to buy sulfuric acid? It is rather common and readily available. http://www.dudadiesel.com/search.php?query=sulfuric&affiliate_pro_tracking_id=17:25:

Plenty. As I said this question is really a matter of opinion and experience. It is not a question of established fact. The opinion of people who have both taught and practiced mathematics and engineering at a high level is certainly germane. Everyone has a right to an opinion, but in a question like this all opinions are not of equal value. Apparently you don't understand logic, the nature of fallacy, any better than you understand this problem. I did not mention your credentials, and I frankly don't care about them. I have heard your arguments many times, from whining sophomores and from mediocre practicing engineers. That does not imply that you are one of them, but it does serve to identify a common source of that argument and a common end product of that approach.

This is in large part a matter of experience and opinion. So, one one side we have an MIT professor of engineering and a PhD mathematician with an MS in engineering and about 25 years of industrial experience. On the otherbside I hear the standard response of undergraduate engineering students who just want to "plug and chug" and who in industry have to be watched closely, arguibg in favor of learning a subject only superficially. I think I'll stick with the side I am on.

dunno for sure. Ask Alan Jackson. Or Jimmy Buffet. But it apparently failed to save the dinosaurs.

This question is easily addressed. The impact occurred at five o'clock (somewhere).

No, it's a "make sure the data is real and the experiment is reproducible" problem.

I am generally not in favor of high school calculus. I once had this discussion with an extremely well known professor of engineering at MIT, and he and I agree. The difference between an average high school calculus class and a university calculus class lies in the understanding of the basic concepts, and not in facility in pushing symbols and doing calculations. To teach calculus properly requires an in-depth knowledge of real and complex analysis. Not many high school teachers have that understanding. The local university does not give credit for AP calculus below a "5" level on the AP test, and that strikes me as about right. I would much rather see high school students who really understand basic algebra and trigonometry. The problem is that all too often they don't. In fact many can't add fractions or compute the area of a rectangle. To address the problem what is needed are high school teachers with understanding of mathematics and science. In my state a senior emeritus professor of mathematics has taken on the task of developing a program to educate ex-professional people to suit them to teach high school science. Believe it or not one of the participants is a rather intelligent fellow of my acquaintance whose previous employment was in the National Basketball Association (above average height for a science teacher). Bottom line: I would rather see high school students learn algebra and trigonometry well than calculus superficially. Algebra and trig are quite sufficient for high school chemistry and introductory physics. Quite advanced physics can even be presented with little else. See "Physics for Future Presidents".

No, it does not. "Closed" in the theory of manifolds means compact without boundary. "Without boundary" means without an edge. The surface of a sphere is a closed 2-manifold.

ajb's advice is good. However, it might help you if itn is pointed out that your statement above is wrong. Such maps are called "open maps" and not all continuous maps are open. Continuous -- the inverse image of open sets is open Open -- the direct image of open sets is open Exercise -- find an example of an open map that is not continuous. Recommendation: Get a good course in point set topology under your belt before you tackle functional analysis and topological vector spaces.

The time derivative of position is velocity. The time derivative of velocity is acceleration. The time integral of velocity is change in position. The time integral of acceleration is change in velocity.

This is why experiments are checked and rechecked. Good work. http://www.sciencenews.org/view/generic/id/331050/title/No_new_particle_from_second_detector

Patton might have been quoting Shaw. George C. Scott uttered the phrase in the movie "Patton".

Medicine is not science. The objective of science is understanding of natural phenomena. The objective of medicine is treatment and cure of disease and injury, using the best available knowledge and methods. Medicine makes use of science, principally chemistry and biology, but it is a sub-discipline of neither. In many regards medicine is similar to engineering, which also is a user of science, but is not science. A physician treats a patient using the knowledge and techniques available to him, despite acknowledged gaps in that knowledge and limitations in techniques. He is faced with a most complex and variable system, the human body, and offers treatment despite uncertainty in outcome and often without definitive knowledge of the root cause of an ailment. The art of medicine, like the art of engineering is practiced in the face of uncertainty, within time constraints, and is limited by cost considerations. Medicine even more so than engineering is constrained by licensing and other governmental regulation. Medical procedures and treatments are subject to extensive governmental review and the unavoidable influence of politics and lobbying. It is inappropriate to criticize medicine for failure to follow strict scientific discipline. It is equally inappropriate to criticize science on the basis of medical protocol.

At any given point the derivative of a function is the slope of a line tangent to the curve at that point. If a function is non-negative the integral of that function over the interval is the area between the graph of the function and the x-axis. Chemists have been known to integrate a curve by cutting it out of a piece of graph paper and weighing it on a sensitive laboratory balance. If the function takes on positive and negative values, areas above the x-axis are positive and areas below the x-axis are negative. Derivatives and integals are connected by the Fundamental Theorem of calculus, which relates integrals directly to "ant-derivatives" (if f is the derivative of g then g is an anti-derivative of f). There are much more sophisticated and more general ways to approach differential and integral calculus, but this is the basic idea.

Electrostatic force is the repulsive force between like charges or the attractive force between opposite charges. It is what makes your hair stand up when you comb it on a dry day. It is the electric component of the Lorentz force from electrodynamics, [math]F=q(E + v \times B)[/math] , [math]E[/math] being the electric field and [math]B[/math] being the magnetic field. The electron energy comes from the same place that the electron comes from. That is the energy associated with the weak force and its carrier, one of the W bosons, the [math]W^-[/math] boson in the case of electron emission, which itself comes from a down quark converting to an up quark. At this level neither conservation of mass nor conservation of energy hold, but rather it is mass/energy that is conserved, keeping in mind the equivalence of mass and energy ([math]E=mc^2[/math]). http://en.wikipedia....wiki/Beta_decay

If I take your model literally the you are proposing that the observable universe is just a 3-ball in some enormous spherical shell. That could be, but the universe could be a 3-dimensional patch in ANY 3-manifold, and that opens up infinitelt many possibilities. The usual assumption is the "cosmological principle" or "Copernican principle" that the universe is homogeneous and isotropic. That implies that space is a Riemannian manifold of constant curvature and is completely determined as follow; 1) in the case of positive curvature itis a 3-sphere 2) in the case of zero curvature it is Euclidean 3-space and 3) in the case of negative curvature it is hyperbolic 3-space. If the assumption of isotropy is relaxed then other possibilities, like a flat 3-torus open up. If I take your model not so literally, then maybe is is just case 1. Homogeneity and isotropy are not logical requirements, but they do seem to be consistent with observations on the largest scales. Relaxation of those assumptions generally is accompanied by rationale. Your "thick-shelled coconut" is spherically symmetric, but not homogeneous -- given two arbitrary points there is not necessarily an isometry that carries one point to the other. Neither is it isotropic -- the radially direction is quite different from the azimuthal direction (there are loops in one direction but not the other). So, the question is what observational or theoretical evidence suggests such a model ? Ans: none. There is a very good article by Geroch and Horowitz, "Global structure of spacetimes" in General Relativity An Einstein Centenary Survey, edited by Hawking and Israel. Your library should be able to get a copy through interlibrary loan if it is not in their stacks. What you are describing is simple connectivity -- a vanishing first homotopy group. Unfortunately the "coconut shell" has the homotopy type of a 2-sphere and is simply connected. To distinguish an n-sphere from the Euclidean space of the same dimension you need to go to the n-th homotopy group. A torus is not simply connected. The first homotopy group of an n-torus is the direct sum of n copies of [math]\mathbb Z[/math] But the usual argument against isotropy of the torus is that some geodesics are closed and others are dense (see "irrational flow on the torus"). I am not personally convinced that strict global isotropy is a good assumption. The model of av flat 3-torus is seriously considered in some circles (see for instance the "Pac Man" space in Brian Greene's The Hidden Reality (this is not an endorsement, and in general I dislike the book)).

There is a fairly long section on derivations in volume 1 of Zariski and Samuel (Commutative Algebra), but I didn't see anything pertinent to your question on a quick skim. Until your post I had not run across the term "outer derivation". I presume it is just a derivation that does not arise from a commutator, hence in the abelian case any non-trivial derivation would be "outer". I also assume that "centre simple" is British for "central simple". "Two people separated by a common language" -- George S, Patton BTW "A bee in your bonnet" has nothing to do with an insect in your automobile..

The only evidence of an end of space of which I am aware occurs when my puppy and I try to nap at the same time. Damn bed hog.