I started reading this ODE book and it first starts by introducing the concept of the differential of a function of one independent variable. Here is the definition:(adsbygoogle = window.adsbygoogle || []).push({});

Let y = f(x) define y as a function of x on an interval I. The differential of y, written as dy (or df) is defined by

[tex](dy)(x,\Delta x) = f'(x)\Delta x[/tex]

Then it goes on to say:

To distinguish between the function defined by y = x and the variable x, we place the symbol ^ over the x so that [itex]y = \hat{x}[/itex]. If [itex]y = \hat{x}[/itex] then

[tex](dy)(x,\Delta x) = (d\hat{x})(x,\Delta x) = \Delta x[/tex]

since f'(x) = 1. The text generalizes further by restating the first equation as

[tex](dy)(x,\Delta x) = f'(x) (d\hat{x})(x,\Delta x)[/tex]

I'm still not clear why this substitution is made. It then goes on and states: The relation [the equation above] is the correct one, but in the course of time, it became customary to write [the equation above] in the more familiar form dy = f'(x) dx. So, as I understand it, [itex]\Delta x = dx[/itex]!? I'm I missing something.

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# Meaning of the Differential

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