An improved version of the fundamental theorem of engineering?

Ah, ****, here we go again.

The fundamental theorem of engineering.

Please don't use it, guys. I mean, no, but...

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sin x = x, really?

Well, before I start ranting again...

Let's take a look at what the second result of the fundamental theorem of engineering actually points at.

sin x = x

Let f: R -> R be a function. Let's suppose that f is smooth enough, so it's continuous and differentiable as many times as you want on R.

So, if we take a point t, and h small enough so that t+h is really close to x, it's possible to approximate f(t+h) in function of f(t).

As you can see, there's still some error in that approximation. But in order to lessen the error, you need to understand this..

We can try to approximate a function around a point by using some polynomial expression., which coefficients include f and its successive derivatives. It works the same way as this.

Of course, the more you haven, the less error you get.

This is called Taylor expanding a function at a point t0.

Now, how does that apply to sin t = t?

Well, by expanding with 1 extra term near t0 = 0, as h -->0:

There, a rather minimalist approach to the fundamental theorem of engineering.

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Application: Euler's method

Taylor expanding can come very handy if you want to approximate solutions within the bounds in which the Taylor expansion remains accurate enough.

Let's see an application of that: falling down with air friction.

After having been summoned to Disboard, Sora and Shiro (modeled by a point M at height z(t)) fall down from the sky without initial speed.

System: M

Frame of referential: Terrestrial, supposed to be galilean

Forces: The weight P and the friction force F, as shown above.

Note: f is a constant here. F is proportional to the square of the speed for high speed falls. For slower falls, it's better to model F as proportional to the speed.

Furthermore, we should expect the speed to converge because the fall gets slown down by the air friction.

Newton's second law, projected on ez, gives, for any t:

We don't know how to solve non linear differential equations.

But we can at least approximate the solutions more or less accurately.

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Now, assuming that during the fall v can be modeled as a smooth function, Taylor expanding to the first order for a small turation h would give:

So, what does this mean here?

Suppose that you know what at time t, what v(t) and dv/dt(t) are.

Then, you can approximate v(t+h) using the formula above, then approximate dv/dt(t+h) by using the differential equation:

Now, doing this step by step, you can actually more or less find the numeric values of v and v' at the times t = nh, for n an integer. You just have to guess the value of a point from the value of the previous one.

Now, let's punch that onto Python. To ease the problem, we'll name K = f/m.

Now. Punching that onto the interpreter and plotting a graph for you:

As expected, the falling speed actually stabilizes around (converges to) a certain value:... Roughly, 35m/s, which is around 126km/h.

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Conclusion

Euler's method is a very powerful algorithm that can numerically solve most differential equations, linear or not. Taylor expand any derivative of your function until the last derivative, then find the latter by using the differential equation.

As you can see, there is a certain domain of validity to the fundamental theorem of engineering. You don't have to discard it right away, but PLEASE do try to stay as close as possible to the domain of validity if you want to have consistent results.

Thank you for listening to my long ass rant, and see you next time.

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Post scriptum

The non differential equation here is actually solvable. I'll discuss the proof in another post.