I started getting into the topic of multilinear forms and differential forms. I find it quite hard to get into.
if I solve $$ \Phi ( \begin{pmatrix} 1 \\ 2\\3 \end{pmatrix} , \begin{pmatrix} 4 \\5\\6 \end{pmatrix}) $$ $n=3 , \Phi= \triangle_{(1,2)}-2 \triangle_{(1,3)}+ 3 \triangle_{(2,3)} $
Is this way right?:
$$ \begin{vmatrix} 1 & 4\\ 2& 5 \end{vmatrix} - 2 \begin{vmatrix} 1 & 4\\ 3 & 6 \end{vmatrix} +3 \begin{vmatrix} 2 & 5\\ 3 & 6 \end{vmatrix}= -3+12-9=0$$
I am not sure how to deal with :
$$ ( df )( \pi , 0, \frac12 ) (\begin{pmatrix} -1 \\ \frac{1}{ \pi} \\ \sqrt{3} \end{pmatrix} )$$
where $n=3 $ and
$ f(x,y,z)= \sin x+e^y \arccos z $
any ideas?