I am trying to compute $i_v(dx\wedge dy\wedge dz):= (dx \wedge dy \wedge dz)(v,w_1,w_2)$, where $v$ denotes $\mathrm{grad} \ F$.
I tried to rewrite this as \begin{align*} i_v(dx\wedge dy\wedge dz)(w_1,w_2) &= (dx \wedge dy \wedge dz)(v,w_1,w_2) \\ & = dx(v) dy(w_1) dz(w_2) + dy(v) dz(w_1) dx(w_2) \\ & + dz(v) dx(w_1) dy(w_2) - dx(v) dz(w_1) dy(w_2) \\ & - dy(v) dx(w_1) dz(w_2) - dz(v) dy(w_1) dx(w_2) \\ \end{align*}
But I am stuck on evaluating $dx(v), dy(v), dz(v)$. The definition of exterior derivative that I am working with is $\mathrm{d} f = \sum_i \frac{\partial f}{\partial x^i} \mathrm{d} x^i$. If I assume $dx(v) = a, \ dy(v) = b, \ dz(v) = c$, then I can further write this as \begin{align*} & a dy(w_1) dz(w_2) + b dz(w_1) dx(w_2) \\ & + c dx(w_1) dy(w_2) - a dz(w_1) dy(w_2) \\ & - b dx(w_1) dz(w_2) - c dy(w_1) dx(w_2) \\ & = a (dy(w_1) dz(w_2) - dz(w_1) dy(w_2)) \\ & + b (dz(w_1) dx(w_2) - dx(w_1) dz(w_2)) \\ & + c (dx(w_1) dy(w_2) - dy(w_1) dx(w_2)) \\ & = a (dy \wedge dz)(w_1, w_2) + b (dz \wedge dx)(w_1, w_2) + c (dx \wedge dy)(w_1, w_2) \\ & = (a \, dy \wedge dz + b \, dz \wedge dx + c \, dx \wedge dy)(w_1, w_2) \end{align*}
EDIT: Thanks to the generous hint by Ted Shifrin, it occurs to me that I can write the gradient of $F$ as $v = \frac{\partial F}{\partial x}\partial_x + \frac{\partial F}{\partial y}\partial_y + \frac{\partial F}{\partial z}\partial_z$, so $dx(v) = \frac{\partial F}{\partial x}, dy(v) = \frac{\partial F}{\partial y}, dz(v) = \frac{\partial F}{\partial z}$. Then \begin{align*} i_v(dx\wedge dy\wedge dz)(w_1,w_2) &= (dx \wedge dy \wedge dz)(v,w_1,w_2) \\ & = dx(v) dy(w_1) dz(w_2) + dy(v) dz(w_1) dx(w_2) \\ & + dz(v) dx(w_1) dy(w_2) - dx(v) dz(w_1) dy(w_2) \\ & - dy(v) dx(w_1) dz(w_2) - dz(v) dy(w_1) dx(w_2) \\ & = F_x dy(w_1) dz(w_2) + F_y dz(w_1) dx(w_2) \\ & + F_z dx(w_1) dy(w_2) - F_x dz(w_1) dy(w_2) \\ & - F_y dx(w_1) dz(w_2) - F_z dy(w_1) dx(w_2) \\ & = F_x (dy(w_1) dz(w_2) - dz(w_1) dy(w_2)) \\ & + F_y (dz(w_1) dx(w_2) - dx(w_1) dz(w_2)) \\ & + F_z (dx(w_1) dy(w_2) - dy(w_1) dx(w_2)) \\ & = F_x (dy \wedge dz)(w_1, w_2) + F_y (dz \wedge dx)(w_1, w_2) + F_z (dx \wedge dy)(w_1, w_2) \\ & = (F_x \, dy \wedge dz + F_y \, dz \wedge dx + F_z \, dx \wedge dy)(w_1, w_2) \end{align*} So \begin{align*} d(F_x \, dy \wedge dz + F_y \, dz \wedge dx + F_z \, dx \wedge dy) &= \left(\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2}\right) dx \wedge dy \wedge dz \\ = (\Delta F) dx \wedge dy \wedge dz \\ \end{align*}
Is what I'm thinking mathematically correct? Thanks a lot in advance!