DrRocket

The fundamental relationship of the sine and cosine to the exponential function is Euler's formula : [math] e^{ix} = \displaystyle \sum_0^ \infty \dfrac {(ix)^n}{n!} = cos(x) + i \ sin(x)[/math] So [math]cos(x) = Re(e^{ix})[/math] and [math]sin(x) = Im(e^{ix})[/math] This is the way that sine and cosine are defined in a rigorous analytical treatment -- see for instance Rudin's Real and Complex Analysis. I don't see how defining sine and cosine in terms of your differential equation adds clarity. For instance, without going back to a power series I don't see a clear path via the DE to showing that the solutions are periodic.

The state university here, through the College of Science has a competitive scholarship program for freshmn women that has a significant stipend and arranges for hands-on participation with research faculty that starts in the summer before the fall freshman semester. It has been very successful. I would start by inquiring if your school has a similar program.

Most real numbers have infinite decimal representations. Without the real numbers you could not do the bulk of calculus, or physics. If you don't want to worry about infinite decinals, become an accoutant.

Nope. We mathematicians claim him. He was, after all, the Lucasian Professor of Mathematics. Mathematicians tend to be pretty versatile.

Feynman probably understood quantum yheory as well as anyone who has ever lived. There are different, valid, ways to view QM. The post above gives you his perspective. To say that Feynman is "wrong" would be pretty sporting.

Yes, mathematicians win. "To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’." . – Richard P. Feynman in The Character of Physical Law Before the physicists jump in to claim Newton, let me note that 1) I am a mathematician and 2) my academic geneology is directly traceable to Newton. Newton was a mathematician.

Here's Feynman on "wave-particle duality. From the book QED based on the Robb Lectures by Feynman: "Quantum electrodynamics "resolves" this wave-particle duality by saying that light is made up of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probabilities that a photon will hit a detector, without offering a good model of how it actually happens."

There is no paradox. .9999999............ = 1 There is a big difference between a finite decimal expansion and an infinite one. 1 = 0 + 1 + 0 + 0 + 0 + ... true 1 = 0 + 0 + 1 + 0 + 0 + ... true 1 = 0 + 0 + 0 + 1 + 0 + ... true 1 = 0 + 0 + 0 + 0 + 0 + ... false , you can't just make the "1" disappear 1=0 ridiculous

Actually, it is not. GR models spacetime as a Lorentzian manifold without boundary.

argh !

The only "arbitrarily small" non-zero number is 0. .999999............ = 1 As can be seen: x = .9999999............. 10x = 9.9999999.......... 9x = 9.99999999..... - .99999999.... = 9 x =1 This is a perfectly legitimate mathematical proof. If you desire gory detail here it is ; [math]\displaystyle \sum_{n=0}^N x^n = 1 + x \displaystyle \sum_{n=o}^N x^n - x^{N+1} [/math] [math](1-x)\displaystyle \sum_{n=0}^N x^n = 1-x^{N+1}[/math] [math]\displaystyle \sum_{n=0}^N x^n = \dfrac {1-x^{N+1}}{1-x}[/math] Similarly [math]\displaystyle \sum_{n=1}^N x^n = \dfrac {1-x^{N+1}}{1-x} -1 [/math] [math] = \dfrac {x-x^{N+1}}{1-x} [/math] So, if [math]|x|<1[/math] [math]\displaystyle \sum_{n=0}^\infty x^n[/math] [math]=\displaystyle \lim_{N \to \infty} \displaystyle \sum_{n=0}^N x^n = \displaystyle \lim_{N \to \infty} \dfrac {1-x^{N+1}}{1-x}[/math] [math] = \dfrac {1}{1-x}[/math] And [math]\displaystyle \sum_{n=1}^\infty x^n[/math] [math]= \displaystyle \lim_{N \to \infty} \displaystyle \sum_{n=1}^N x^n = \displaystyle \lim_{N \to \infty} \dfrac {x-x^{N+1}}{1-x}[/math] [math] = \dfrac {x}{1-x}[/math] [math]0.99999........ = \displaystyle \sum_{n=1}^\infty 9 (\dfrac{1}{10})^n[/math] [math] = 9 \displaystyle \sum_{n=1}^\infty (\dfrac{1}{10})^n[/math] [math] = 9 \dfrac {\frac {1}{10}}{1- \frac{1}{10}}[/math] [math] = 9 \dfrac {1}{9}[/math] [math] = 1[/math]

Proof applies to mathematics, not science. Relativity is supported by a mountain of experimental and observational evidence. But general relativity is also known to be incompatible with quantum mechanics, which is also supported by a large body of evidence. Either or both will probably eventually be supplanted by a theory that will refine and extend both.

maybe this http://arxiv.org/abs/1104.0699

There is no such proof, nor is there likely to ever be. Michio KJaku is a very poor source. He makes flamboyant statements with no basis, apparently in an effort to promote himself and sell books. If Kaku said that the sky was blue, I would immediately seek independent verification. I have several hundred science and mathematics books. I have one by Kaku -- a mediocre book on quantum field theory. I am very unlikely to ever own two.

No more so than the North Pole is the edge of the Eaarth.

I don't consider mathematics to be science. The major discriminator is that mathematics relies on logical proof, as opposed to evidence from experiment, in establishing "truth'. Mathematics is related to science, but is a discipline unto itself. Engineering is characterized by an end objective of the production of a useful product, commonly with budget and schedule constraints. The goal of science is the development of understanding. Engineering builds on that understanding to produce products, sometimes in the face of imperfect understanding of the details of the science involved, and often involving very complex systems for which first-principles modeling is impractical or impossible. While the academic subject matter in formal classes is sometimes similar, the ultimate objectives of science and engineering are quite different.

What is the book ? This material is usually treated in a text on complex analysis rather than one on functional analysis. Those hypotheses are a bit strange. Once you know that f is analytic, you know that it has continuous derivatives of all orders. See below But if the authopr is starting from scratch an intends to use Stokes Theorem then he needs the hypothesis of continuous derivatives. There are better ways to do this. Yes, see above. Most treatments simply call the theorem Cauchy's Theorem. The usual proof is done first for triangles in a convex region and then generalized to arbitrary curves in a convex region. An analytic function is one that is locally representable by a power series. This is considerably more restrictive that even having continous derivatives of all orders ( [math]C^\infty[/math]) What is striking about complex-valued functions of a complex variable is that any differentiable function is automatically analytc. This makes complex analysis very different from real analysis. Analytic functions are, in a sense, "nice". But you give up existence of partitions of unity, so complex manifolds are so difficult that very little theory exists, and Kahler manifolds (about which I know very little) are what are studied.

I don't follow your statement regarding sine as following from "elementary analysis" since extending the domain to complex numbers requires complex analysis, but the statement is true. You have outlined the proof that [math]\overline{f}(z) = f(\overline{z})[/math]. To wit: [math] \overline{z_1z_2} = \overline {z_1} \overline{z_2} [/math] clearly extends to show that [math]\overline{f}(z) = f(\overline{z})[/math] for any polynomial function [math]f[/math]. Since conjugation is continuous and since any analytic function is the uniform limit on compacta of polynomials the theorem now follows for entire functions. Since sine is entire, it follows for sine. Analyticity is not required for equality. Your counterexample is valid. The only real-valued analytic functions are constant. no

Just think of conjugation as a function then [math] \overline {f}(z) = \overline {f(z)} = (conjugation \circ f)(z)[/math] [math] f(\overline {z}) = (f \circ conjugation) (z) [/math]

What have you done to try to solve this problem ?

What have you tried ? These should be pretty simple if you were paying attention in class. D0 you know the definitions of: 1) linearly independent set, 2) spanning set, 3) basis, and 4) dimension ?

He was opining on difficultiy with relativity and stated that [math] F = m \frac {dv}{dt} [/math] which is not correct. In special relativity the correct relation is [math] F = \frac {d(\gamma m_0v)}{dt}[/math]. This makes me think he was simply reciting to the camera words that he did not understand. Yet he is being portrayed in some stories as the 12 year old genuis who is challenging Einstein's theory of relativity.

If you were to never utter again, that would be all right with me.

That is just fine. I have no idea why you might feel that you cheated. Imatfaal's stylistic comments are helpful and might make your explanation easier to follow, but that is just icing on the cake. You might want to take a look at the idea of "free body diagrams" as a device to make the relations among forces clear. For some reason they are not always taught in physics courses (I recall showing the technique to a nuclear physicist who was teaching a basic non-majors class and getting a "gee that's neat" response), but they are routine in engineering mechanics. See an engineering text on mechanics or statics, or just talk to an engineering student. Engineering calculations quite often have to be shown to a large, critical audience for approval and therefore clarity of exposition is as important as is getting the right answer. Free body diagrams are a big help.

As a result of conversations elsewhere what I have seen and learned is: 1. The kid was diagnosed with Asperger's syndrome which tends to result in some odd behavior. I have a nephew with Asperger's and he is also a bit odd, but not a prodigy. 2. He has made some statements about relativity that, though wrong, are getting some press. 3. He talked about integration by parts, coming from the product rule for differentiation, and got that right. That puts him ahead of most college freshmen, but does not make him a genius. 4. He has a pushy mother who seems to be behind the hoopla. 5. Time will tell if he ever produces anything original and startling. I am not holding my breath.