This partial derivative assumes that \(q\) and \(p\) are independent, i.e., it doesn't consider \(q\) as a function of \(p\) and only differentiates \(F\) w.r.t. its fourth argument.

\(t\) is the parameter which allows you to move along a \(\gamma_p\). We set this parameter so that \(\frac{dx}{dt} = F_p(x,y,z,p,q)\). which then means that,

$$ \frac{dx}{F_p} = dt = \frac{dy}{F_q}$$

$$ \implies \frac{dy}{dt} = F_q(x,y,z,p,q)$$

(2) is satisfied for some \(p\) for all points on the envelope(Monge cone).

Different \(p_0\) will be used for points on different \(\gamma_p\) ;

where \( \gamma_{p_0} \) is the set of all points

\((x,y,z) \in \mathbb{R}^3\) such that

$$z-z_0 = p_0(x-x_0) + q(p_0)(y-y_0)$$

and,

$$\partial_p z(x,y;p_0) = 0 $$

(5.) only holds when \((dx,dy)\) points in direction of \(\gamma_p\)

As we take small step \((dx, dy)\) along \(\gamma_a\) , the change in partial derivative of \(w\) w.r.t. \(a\) is 0 (Since partial derivative at every point on \(\gamma_a\) is 0, so the change in partial derivatve is also 0).

The change in partial derivative can also be written as

$$dw_a = \frac{\partial w_a }{\partial x}dx + \frac{\partial w_a}{\partial y}dy $$

where \(w_a\) is a function defined on \(\gamma_a\) (the partial derivative of \(w\) w.r.t \(a\) at \((x,y) \in \gamma_a\)) that is identically 0.