The local heat flux

At the bottom of page 553 author defines the local heat flow $L(z) = \frac{\partial G}{\partial n}$ and this makes sense to me, it is another form of $- \nabla G$ and tells us how much fluid flows out of this boundary, as we move in a perpendicular direction. However on the next page, in (25) author says "At a boundary point $z$, the local heat flux is equal to the amplification of $f: L(z) = |f'(z)|$". Two things doesn't make sense to me, first if we take the derivative of $f(z)$, which is a mapping from $R$ to $D$, (figure [31]) then we can talk about the boundary point at $z$ but rather $w$ right? The mapping $f(z)$ is taking us from $R$ to $D$ so when we differentiate it we are already in $D$. Or we have a collision of naming here and $f(z)$ is a function of the vector field in $R$?
Second, how are we sure that $|f'(z)|$ is the vector $H$ shown of the RHS of [31] which signifies the local heat flux?

PS: Looks like author abandoned the use of Greek alphabet here, what are these $L$, $G$ like symbols?

Thank you.