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"The second fundamental confusion of calculus"


Genady
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A cool little example of common confusion with partial derivatives, from Penrose's "The Road to Reality" (he attributes the words in the title to Nick Woodhouse.)

Let's consider a function of two coordinates, f(x,y), and a coordinate change

X = x, Y = y + x

Because the X coordinate didn't change and is the same as the x coordinate, one could expect that the corresponding partial derivatives are the same, fX=fx.

And, because the Y coordinate is different from the y, these partial derivatives, fY and fy, could be expected to differ.

In fact, this is just opposite:

fX=fx-fy

fY=fy

The confusion is caused by the notation: fX does not mean a derivative along X, but rather a derivative with a constant Y; and fY is not a derivative along Y, but a derivative with a constant X.

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