I can not understand the practical way to calculate the following:
Let $X=a\frac{{\partial }}{{\partial x}}+b\frac{{\partial }}{{\partial y}}+c\frac{{\partial }}{{\partial z}}$ show that $dx\wedge dy\wedge dz(X,v_1,v_2)=ady\wedge dz-bdx\wedge dz+cdx\wedge dy.$
I have this: $i_X(dx\wedge dy\wedge dz)(v_1,v_2)=dx\wedge dy\wedge dz(X,v_1,v_2)=dx(X,v_1,v_2)\wedge (dy\wedge dz)(X,v_1,v_2)\\ =dx(X,v_1,v_2)\wedge (dy(X)dz(v_1)-dy(v_1)dz(X)+dy(X)dz(v_2)-dy(v_2)dz(X)+dy(v_1)dz(v_2)-dy(v_2)dz(v_1))\\ =[dx(v_2)dy(X)dz(v_1)-dx(v_2)dy(v_1)dz(X)]+[dx(v_1)dy(X)dz(v_2)-dx(v_1)dy(v_2)dz(X)+dx(v_2)dy(v_1)dz(v_2)-dx(X)dy(v_2)dz(v_1)]\\ =[bdx(v_2)dz(v_1)-cdx(v_2)dy(v_1)]+[bdx(v_1)dz(v_2)-cdx(v_1)dy(v_2)+ady(v_1)dz(v_2)-ady(v_2)dz(v_1)]$
(The only term that gives me is $ady \wedge dz$ the others do not fit me with the sign to form what I want. How would it be? I do not handle this multiplication game well.
Remember that $$\begin{align} dx \wedge dy \wedge dz & = dx \otimes dy \otimes dz + dy \otimes dz \otimes dx + dz \otimes dx \otimes dy \\ & - dx \otimes dz \otimes dy - dy \otimes dx \otimes dz - dz \otimes dy \otimes dx \end{align}$$
Therefore, $$\begin{align} (dx \wedge dy \wedge dz)(X,Y,Z) & = dx(X) \otimes dy(Y) \otimes dz(Z) + dy(X) \otimes dz(Y) \otimes dx(Z) \\ & + dz(X) \otimes dx(Y) \otimes dy(Z) - dx(X) \otimes dz(Y) \otimes dy(Z) \\ & - dy(X) \otimes dx(Y) \otimes dz(Z) - dz(X) \otimes dy(Y) \otimes dx(Z) \end{align}$$
In this case, $X = a \partial_x + b \partial_y + c \partial_z,$ so $dx(X) = a, \ dy(X) = b, \ dz(X) = c,$ which gives $$\begin{align} (dx \wedge dy \wedge dz)(X,Y,Z) & = a dy(Y) \otimes dz(Z) + b dz(Y) \otimes dx(Z) \\ & + c dx(Y) \otimes dy(Z) - a dz(Y) \otimes dy(Z) \\ & - b dx(Y) \otimes dz(Z) - c dy(Y) \otimes dx(Z) \\ & = a (dy(Y) \otimes dz(Z) - dz(Y) \otimes dy(Z)) \\ & + b (dz(Y) \otimes dx(Z) - dx(Y) \otimes dz(Z)) \\ & + c (dx(Y) \otimes dy(Z) - dy(Y) \otimes dx(Z)) \\ & = a (dy \wedge dz)(Y, Z) + b (dz \wedge dx)(Y, Z) + c (dx \wedge dy)(Y, Z) \\ & = (a \, dy \wedge dz + b \, dz \wedge dx + c \, dx \wedge dy)(Y, Z) \end{align}$$