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2

$dl'$ is equivalent to "$d|\mathbf{r}|$", it is essentially a "scalar length measure". The electrodynamics integral you wrote here is a vector-valued integral, so no dotting happens. If you use a linear coordinate system, it may be evaluated as three scalar line integrals, one for each coordinate. Vector valued integrals cannot really be evaluated using a ...

1

"Comfortable ride" is a tricky thing to quantify. Jerk is not the right metric to use. The reason it works for roller coaster design is the fact that in a roller coaster, you brace yourself against the rather large low-frequency acceleration. If you make a sharp turn to the left, you will want to lean left. If you then suddenly make a large turn to the right,...

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To formalize the comments (now in chat here): Associated jerk is probably what you want to calculate, as it is the measure of how violently something is shaken.$^1$ Jerk is the derivative of the acceleration with respect to time. To properly calculate this, you would use the formula $\left| a \right| = \sqrt{a_x^2+a_y^2 +a_z^2}$ (from the Pythagorean ...

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Physical laws are usually described in terms of hyperbolic differential equations. In such equations, the Cauchy problem is well-posed, i. e., if you know the value of your variable and some of its derivatives at the initial time, you can integrate them locally in order to find them at other times. That is the case of classical mechanics and quantum ...

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How are we sure that we don't need real values (for example initial values at zero time)? In all experiments I have been involved in, we have initial values at zero time, with the +/- experimental errors. Can we determine initial values at zero time (for example u(0) )? ditto Is it possible to have a physics by other mathematics ...

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Stephen Wolfram has been a great advocate for re-building physics on discrete models, and Gerard 't Hooft has done a cellular automa interpretation of quantum mechanics. But frankly those kinds of thing aren't going to push calculus out of the way because we don't use it out of some kind of abstract reverence, but because it works. That really is the final ...

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