Differential of square $d(w^2)$or square of differential$(dw)^2$!? in Curved space
it seems $d(w^2)=(dw)^2$ how is it possible!?

$$x^2+y^2+z^2+w^2=\kappa^{-1}R^2,$$
first:
$$dw=-w^{-1}(xdx+ydy+zdz),$$

$$\kappa^{-1}R^2-(x^2+y^2+z^2)=w^2,$$
$$dl^2 = dx^2 + dy^2 + dz^2+dw^2,$$

$$dl^2 = dx^2 + dy^2 + dz^2
+\frac{(xdx+ydy+zdz)^2}{\kappa^{-1}R^2-x^2-y^2-z^2}$$

Remark:
in general as i know $dx^2=2xdx$ so $dw^2$ should be:

$$d(w^2)=(2w)w^{-1}(xdx+ydy+zdz)=2(xdx+ydy+zdz),$$
but here

$$d(w^2)=(dw)^2=w^{-2}(xdx+ydy+zdz)^2,$$
how is it possible?
In general, $dx^2=2dx$, but you've been confused by notation enlightening, where the parenthesis are omitted because there should be no possible confusion. Imagine that page (or any work on general relativity) with every parenthesis explicitly written. My eyes'd start to bleed.
In general, $dx^2=2dx$, but you've been confused by notation enlightening, where the parenthesis are omitted because there should be no possible confusion. Imagine that page (or any work on general relativity) with every parenthesis explicitly written. My eyes'd start to bleed.