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I'd love to seen an ADEPT method explanation of Laplace Transforms. I got through most of math fairly easily by having a mental model of what was going on and

by theobon 10y ago

I'd love to seen an ADEPT method explanation of Laplace Transforms. I got through most of math fairly easily by having a mental model of what was going on and could always check that I was on the right track as it made sense in my mental model. However, when I got to Laplace transforms I never figured out how to visual what that meant. Everything collapsed into transform into the magical space where you can do some things easier and then you can transform out into a new place. I could never be sure of how I got from a to b without a tedious examination of every step to ensure I applied the rules correctly. I'd love to have a mental model for Laplace Transforms. As a generalization, how does one explain things where no good mental model exists?

copperx 10y ago
> As a generalization, how does one explain things where no good mental model exists? Papert et al. are convinced that computers ought to help with this problem. The idea is that visualizations, even if they are interactive, aren't very useful. Instead, the learner should iteratively build and play with a simple version of a model (a microworld) until the learner gets into the full-fledged model. Preferably by programming the model itself. It's sad that this idea never caught much traction beyond educating children.
Noseshine 10y ago
I have not check out these videos in particular, but when I went through Khan Academy I thought they were doing a remarkable job at explaining. Try https://www.khanacademy.org/math/differential-equations/laplace-transform https://www.khanacademy.org/math/differential-equations/lapl...
kalid 10y ago
Kalid from BetterExplained here, my quick intuition: The Fourier Transform breaks a signal into its "cycle recipe" (what circular paths are present?). The LaPlace Transform breaks a signal into its "spiral recipe". Circles are made from a type of exponential (given by e^ix), and spirals are the more general version, where the radius changes (if s=a+bi, then e^is = e^a * e^bi, aka a circular path where the radius changes exponentially). The LaPlace transform actually deals with decaying spirals (negative s) -- why is this useful? Well, perfect circles that never decay (the Fourier Transform) are nice for analyzing audio samples, as in music. (Repeated drumbeat throughout the song.) Decaying spirals model things in the real world, where friction, etc. dampen the signal over time. The Laplace transform can cleanly represent this scenario, whereas you need an infinite number of cancelling terms in the Fourier Transform to represent the "decays over time" setup. Engineering applications prefer Laplace, compression mechanisms may prefer Fourier. (Separately, Laplace/Fourier make differential equations easier to solve by writing functions in terms of exponentials, which are easy to derive/integrate. The Laplace transform is more general and powerful in this regard, since it can handle any rate of decay, including 0. The Fourier Transform is embedded within the Laplace.) Just some quick thoughts from an amateur on this :).

by theobon 10y ago

copperx 10y ago

> As a generalization, how does one explain things where no good mental model exists? Papert et al. are convinced that computers ought to help with this problem. The idea is that visualizations, even if they are interactive, aren't very useful. Instead, the learner should iteratively build and play with a simple version of a model (a microworld) until the learner gets into the full-fledged model. Preferably by programming the model itself. It's sad that this idea never caught much traction beyond educating children.

Noseshine 10y ago

I have not check out these videos in particular, but when I went through Khan Academy I thought they were doing a remarkable job at explaining. Try https://www.khanacademy.org/math/differential-equations/laplace-transform https://www.khanacademy.org/math/differential-equations/lapl...

kalid 10y ago

Kalid from BetterExplained here, my quick intuition: The Fourier Transform breaks a signal into its "cycle recipe" (what circular paths are present?). The LaPlace Transform breaks a signal into its "spiral recipe". Circles are made from a type of exponential (given by e^ix), and spirals are the more general version, where the radius changes (if s=a+bi, then e^is = e^a * e^bi, aka a circular path where the radius changes exponentially). The LaPlace transform actually deals with decaying spirals (negative s) -- why is this useful? Well, perfect circles that never decay (the Fourier Transform) are nice for analyzing audio samples, as in music. (Repeated drumbeat throughout the song.) Decaying spirals model things in the real world, where friction, etc. dampen the signal over time. The Laplace transform can cleanly represent this scenario, whereas you need an infinite number of cancelling terms in the Fourier Transform to represent the "decays over time" setup. Engineering applications prefer Laplace, compression mechanisms may prefer Fourier. (Separately, Laplace/Fourier make differential equations easier to solve by writing functions in terms of exponentials, which are easy to derive/integrate. The Laplace transform is more general and powerful in this regard, since it can handle any rate of decay, including 0. The Fourier Transform is embedded within the Laplace.) Just some quick thoughts from an amateur on this :).