"The second fundamental confusion of calculus"

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, f_X=f_x.

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

In fact, this is just opposite:

f_X=f_x-f_y

f_Y=f_y

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