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For example if I have z = f(x,y), then dz = df/dx * dx + df/dy * dy. Then it is clear that dz is a linear approximation of change of z. (I don't know what is a differential analytically) But what is the differential of a function in integral $$y(s) = \int_0^s f(x) \,dx$$,

For example, under the follow equations, what is d$\phi$?

3 equations

It may be a dumb questions, but I can't see the differential and don't actually know what it is.

KelvinSK
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  • I would, shortly, say that it is an infinitesimal change, i.e. $dx\approx x+h$ for small $h$, being thus $\int_a^b f(x) dx$ a sum of rectangles of height $f(x)$ and base $dx$. Discretely one would have $\sum$ instead of $\int$, i.e. $\sum_i f(x_i)(x_i+h)$. – Martingalo Nov 30 '22 at 06:54
  • You really ought to edit into the question the function $\phi$ that you are looking at: it is of the form $\phi(s)=\int_{0}^{s} f(x,s) dx$. The textbooks tell you what conditions you need to deifferntiate this wrt $s$ and get something like $f(s,s)+\int_{0}^{s} \frac{\partial f(x,s)}{\partial s} dx$ – ancient mathematician Nov 30 '22 at 07:46

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