Could anyone help me understand the second equality of the expression below? I have already tried to interpret as the derivative in relation to the time of the composition of two functions, ie the wedge product composed with the application that takes functional into functional, but could not.
This is found in Jesen's book, "Surfaces in Classical Geometries," page 227. Link http://library1.ga/_ads/802B98464DDA8737736FDF48F69E3F49
You obviously didn't search the whole book. The relevant equations are at the bottom of p. 225. When you use the product rule, you'll have $$\frac d{dt}\Big|_{t=0}\omega^1_t\wedge\omega^2_t = \left(dg^1+\sum g^j\omega_j^1\right)\wedge\omega^2 + \omega^1\wedge\left(dg^2+\sum g^j\omega_j^2\right).\tag{$\star$}$$ Now when we proceed, we seem to discover that they have a sign wrong. \begin{align*} d(g^1\omega^2-g^2\omega^1) &= dg^1\wedge\omega^2 + g^1d\omega^2 -dg^2\wedge\omega^1-g^2d\omega^1 \\ &=dg^1\wedge\omega^2 + g^1\sum\omega^j\wedge\omega_j^2 - dg^2\wedge\omega^1 - g^2\sum\omega^j\wedge\omega_j^1 \\ &=dg^1\wedge\omega^2+\omega^1\wedge dg^2 + g^1\omega^1\wedge\omega_1^2-g^2\omega^2\wedge\omega_2^1. \end{align*}
Expand the right-hand side of ($\star$) and you'll get exactly this plus the $g^3$ terms in their equation (taking into account their comments).