Thread: Generic Calculus Questions

No sorry, I was just posting the general projectile equation... it's safe enough to put in numbers there and you get the quad pretty much straight away, but of course your form is what you're ultimately aiming for.

Saw no reason to open another thread for something that fits squarely into this one, so without further ado...

Spoiler for It's nothing really..:

Today was the last day of instruction in my differentials class, and the professor was going over practical applications of the Laplace Transform to model complex currents and voltages over elements in a circuit that can have multiple inputs with varying frequencies. I was so blown away by the ease of it all, especially now that I've had a long-time question answered.

More to the point, I wanted to ask about the origins of the Laplace Transform.. How exactly was it worked out by Mr. Laplace? I've already given a few attempts at finding out on my own but it appears that every description I run into demands I have a more in-depth knowledge of mathematics concepts I've never heard of. Is there an explanation in Layman's terms? I truly appreciate how useful this is. I just can't imagine how the end result (the transform of a function in time domain) was achieved.

Any halps are of course greatly appreciated.

Just so you know, electon positions can't be calculated that way.

Does that mean we don't have an expression that dictates an electron's motion with respect to time, or that the expression can't be transformed back and forth, or that it's useless to do so? :O
Enlighten me, señor.

Broken image methinks.

I haven't actually studied Laplacian Transforms yet, but from what I've seen, you basically take a Taylor series and generalise it to a continuous case. Obviously you can't explicitly give the coefficients any more if there are infinite number of them, so they're written as a function - this is the function that you 'take the transform' of, and it gives you the new function.

It's pretty straightforward working out what the transform should be with this in mind, and then it's also easy to see that this is equivalent to the way that it's normally written (it's in a slightly weird form because it turns out that calculations are just easier when it's written the way that it is).

From my understanding, Euler had been working on using integrals as solutions to differential equations. Laplace was extending the work. The key property of the transform is that it turns L[f'(t)] into sL[f(t)] - f(0). This can be seen by integration by parts. So once you see that, it's only a matter of time before it occurs to you (if you're as smart as Laplace) to use that property to transform differential equations into algebraic equations.

With regards to the electron, I think Ninja was saying that you can't calculate a position for an electron using an equation like that (from one of the earlier questions asked on the thread).