|1st March 2008, 12:02 PM||#31|
Join Date: Jan 2006
Lets see triple 17, double 9's and a triple 15 from 233, multiplied by several loud mates, a couple of hot barmaids, a buzillion flies, way too many Emu Exports and a hot day of snorkling. Math was much more fun in those days.
|4th March 2008, 06:28 PM||#32|
Join Date: Nov 2006
Blog Entries: 1
If you know the basics of Calculus (even if you've forgotten how to actually crank out many actual answers with it), the best thing I can think of would be to learn the basics of Differential Equations.
Differential Equations, to me, are mostly what Calculus is good for. Calculus was great to learn, and useful, and had some appreciable 'wow-factor' and beauty to it. But, compared to that, differential equations are stunningly, exquisitely, gorgeous!! Once you see what they are, and what they can do, it's definitely a 'slap-your forehead' "epiphany" type of moment. Almost everything starts there. You can model virtually anything with a differential equation, or at least with a set of them (which just becomes one familiar-looking differential equation again, if you subsitute a vector symbol for the variables and a matrix symbol for the coefficients).
The formal solution of a differential equation is... an algebraic equation! BUT, that's exactly what you wanted, having previously known only the differential relationships, which are what are usually most-intuitive, or what are available from the known 'rules' of 'Mother Nature' (physics, et al).
Once you see how differential equations work, all of the arcane 'formulas' might suddenly make much more sense, because it's easy to see where they came from. For example, I can't imagine taking a Physics course that doesn't start with differential equations for everything (to which they apply), because then you can't really see what's actually going on, and just have to memorize 'formulas', instead of having them be 'obvious', or quickly re-derivable from the underlying differential equations, which usually make much more sense, or are much more intuitive (not to mention HUGELY more-general!). Gravity, dynamics/mechanics, thermal dynamics, electricity and magnetism, and almost everything else that 'changes' in any way, or causes change... ALL are most-fundamentally described by differential equations (at least in the 'classical'-physics sense of things).
Even coooler, maybe, is the fact that the differential equations often have exactly the same form, for completely-different phenomena, which really helps to eventually 'unify' the mathematics, AND the ways that things actually work, in your mind. For example, RLC electronic circuits and spring-mass-damper mechanical systems both have EXACTLY the same differential equation, and, therefore, *exactly* the same behavior. And there are many other similar analogues.
For audio, and electronics in general, you'd also want to learn about Fourier Series (i.e. ANY periodic waveform can be expressed as a sum of sinusoidal waveforms [possibly an infinite number of smaller and smaller ones, but that's no big deal]). After that, it would be very helpful to at least have a descriptive understanding of Fourier Transforms, with which you can mathematically transform any steady-state periodic signal from the time-domain to the frequency-domain (think 'from oscilloscope display to spectrum analyzer display').
Even MORE useful is the LaPlace Transform, which works for both the transient and steady-state components of things (e.g. of signals and system responses), simultaneously.
THEN comes another 'slap-your-forehead' moment! With LaPlace Transforms comes the idea that you can transform any differential equation from the time-domain to the frequency-domain. Sounds boring? Well get this(!): The nasty-looking differential equation gets transformed into (gasp!) a simple ALGEBRAIC equation! And most of us can still solve those, if we HAVE to (or, just re-arrange and inspect them, if all we want is poles and zeros, for example). Many signals can also be transformed to simple LaPlace-domain expressions. So, now, finding the output of an RLC electronic network, for example, for a given input signal (sine, step, impulse, etc etc), is not so difficult. After the "s-domain" algebraic equation is solved, you can convert the solution back to the time domain and see the resulting signal, just as it would appear on an oscilloscope, and with transient-responses included, too. Filter Schmilter. (Reminds me of the airliner from Poland that crashed, after the pilot told the passengers about something that could be seen from the starboard windows. It turned out that there were "too many Poles in the right half plane". Groan!)
Differential equations ALSO lend themselves very well to discrete-time domains (e.g. "digital stuff"). It's *literally* trivial to convert a Differential Equation to a discrete-time "Difference Equation", where each derivative (i.e. 'rate of change' of one variable versus another e.g. time) is simply expressed as the variable's value at any time t minus its value at time t-deltaT, divided by deltaT. Higher-order derivatives are similarly trivial. Put it all in a computer-program loop that increments t by deltaT (with deltaT small-enough), set some initial conditions if you want to, press Start, and... you have just simulated the actual behavior of the system that was described by your original 'analog' differential equation!
(Is this boring, yet? "Heck no!", I hope.) And JUST like the LaPlace Transforms for continuous-time systems, there are Z-Transforms for discrete-time systems, which convert time-domain Difference Equations into z-domain algebraic equations. Cool, huh?
There is much, much more, if you get enthusiastic, later... like non-linear differential equations, probabilistic systems and signals theory, automatic control theory (i.e. "classical feedback theory": highly recommended; not really difficult; very relevant to audio and electronics), time-varying differential equations, and on and on, that almost all build on the foundation of your basic 'Linear Ordinary Differential Equations'.
(Aside: Regarding 'time-varying differential equations': Those of you who do/did know diff eqs might be interested to know that you cannot use ordinary differential equations for the seemingly-simple system consisting of a child on a swing! It's similar to the simple pendulum equation. BUT(!), the length of the pendulum is now TIME-VARYING, as the child's legs move up and down to keep the swing going, or to make it go higher. And that pendulum-length was just a constant coefficient, in the differential equation for the simple pendulum. That type of time-varying system is also called a 'parametric amplifier', by the way. It cannot be solved with the 'normal' differential equations that most engineers study in their third semester, for example.)
Sorry to have blathered-on about all of that. But even if you can't derive some or all of that stuff, and can't even work most (or any) of the textbook problems, it sure seems like just knowing that it's there, and knowing the basics of what it can do and how it does it, would make a LOT of other things much easier to understand, or at least easier to be more comfortable with.
(Aside: I was first exposed to calculus and differential equations when I asked my home-town librarian to find a book for me that explained how to make a "trajectory", so I could figure out how to implement a calibrated, settable range for a small 'catapult', so I could more-accurately LOB things at other neighborhood kids(!), when we played 'war'. I was ten or eleven years old and never did figure it out, back then. But I definitely learned how to do it later, with (yup) differential equations, even for the case when both the firing platform and the target are moving and accelerating in three dimensions, and even if the firing platform is also rotating and accelerating around all three of its rotational axes [and even if it was also flexing and twisting and vibrating in all six of those dimensions]. We even learned how to come up with automatic control systems that would actually control the firing platform's motion, based on the target's (and firing platform's) motion et al, to be able to hit the target with 'non-intelligent' ballistic projectiles, 'in an optimal fashion'.)
If you can get to a larger university bookstore, at a university where they have a school of electrical engineering or something similar (or, go online, I guess), some of the "Schaum's Outline Series" types of books might be quite helpful. They usually just have the more-practical parts, fairly condensed, and logically presented, and are available for everything mentioned above. They're good for review and can also be good just to 'survey' something new.
That's about all I have, for now. Sorry if I was being overly enthusiastic!
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