This will not break into the calculus I think. The reason that sin and cosine are so big in calculus has little to do at all with their relation to angles and lengths, it has to do with these relations:
d^2/(dx)^2 f(x) = -f(x) e(jx) = cos(x) + j*sin(x)
(Yes, I know, those equations are related)
These quadrances and spreads won't scale that way. For an acute angle, the spread is equal to the square of the sin. Obviously, the solution to the simple harmonic equation is going to be rather difficult to express in terms of spread rather than phase! I think the cos and sin's place in calculus is pretty safe.
Maybe you can resolve a question I have always wondered about, but never asked. How do astronomers account for the fact that Doppler shift is directional? I understand that the universe is expanding, in general, but for ultra-fast things like these shouldn't certain of the matter be coming towards us and some away from us, and not only that, but at different angles relative to the line connecting our position and its? I tried to derive the doppler shift once and came up with the following:
where f_r is the frequency at the receiver, f_s is the frequency at the source (the "true" frequency), c is the speed of light, v_s is the velocity of the source at the time the signal was emitted, v_r is the velocity of the receiver at the time the signal was received and u_R is a unit vector in the direction of the propagation path of the wave. The part that I always wondered about was the dot product in the denominator. All the physics books I had when I looked up the doppler shift assumed the source was moving directly away from the receiver or directly toward it. Any thought on how that's resolved? Seems like the bands should be a big smear from the redshift to the blueshift with that directional dependence, assuming equal amounts of matter moving towards and away...
Sounds like a question that's answered by the father of modern philosophy, Descartes. "I think, therefore I am." The sentence is a recursive structure, because the very "I think" implies that there is a consciousness there to do the thinking. So it's a consciousness knowing that it is thinking that it is knowing, etc. A computer doesn't think, it processes. It may have data on itself, the program that it is running may have access to its serial number, its processor type, and the contents of all its registers. But nobody knows what self-awareness really means, not the most advanced psychological, neurological, or philosophical theory. Until we get a handle on what it is, we really can't even think about granting it to other objects. All that we know is that "I am", and logic implies that if one human is, than others probably are as well. On the other hand, you might all be figments of my imagination and I am the only self-aware entity in existence. Who knows?
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This will not break into the calculus I think. The reason that sin and cosine are so big in calculus has little to do at all with their relation to angles and lengths, it has to do with these relations:
d^2/(dx)^2 f(x) = -f(x)
e(jx) = cos(x) + j*sin(x)
(Yes, I know, those equations are related)
These quadrances and spreads won't scale that way. For an acute angle, the spread is equal to the square of the sin. Obviously, the solution to the simple harmonic equation is going to be rather difficult to express in terms of spread rather than phase! I think the cos and sin's place in calculus is pretty safe.
Maybe you can resolve a question I have always wondered about, but never asked. How do astronomers account for the fact that Doppler shift is directional? I understand that the universe is expanding, in general, but for ultra-fast things like these shouldn't certain of the matter be coming towards us and some away from us, and not only that, but at different angles relative to the line connecting our position and its? I tried to derive the doppler shift once and came up with the following:
\frac{f_r}{f_s} = \frac{\sqrt{c^2 - |\vec{v_s} - \vec{v_r}|^2}}{c - (\vec{v_s} - \vec{v_r}) dot \vec(u_R)}
where f_r is the frequency at the receiver, f_s is the frequency at the source (the "true" frequency), c is the speed of light, v_s is the velocity of the source at the time the signal was emitted, v_r is the velocity of the receiver at the time the signal was received and u_R is a unit vector in the direction of the propagation path of the wave. The part that I always wondered about was the dot product in the denominator. All the physics books I had when I looked up the doppler shift assumed the source was moving directly away from the receiver or directly toward it. Any thought on how that's resolved? Seems like the bands should be a big smear from the redshift to the blueshift with that directional dependence, assuming equal amounts of matter moving towards and away...
Sounds like a question that's answered by the father of modern philosophy, Descartes. "I think, therefore I am." The sentence is a recursive structure, because the very "I think" implies that there is a consciousness there to do the thinking. So it's a consciousness knowing that it is thinking that it is knowing, etc. A computer doesn't think, it processes. It may have data on itself, the program that it is running may have access to its serial number, its processor type, and the contents of all its registers. But nobody knows what self-awareness really means, not the most advanced psychological, neurological, or philosophical theory. Until we get a handle on what it is, we really can't even think about granting it to other objects. All that we know is that "I am", and logic implies that if one human is, than others probably are as well. On the other hand, you might all be figments of my imagination and I am the only self-aware entity in existence. Who knows?