I'm trying to solve the following question : Which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on $\mathbb R^4$:
\begin{align*} g(x, y) &= x_1y_3 - x_3y_2 \\\ h(x, y) &= x_1^3y_2^3 - x_2^3y_1^3 \end{align*}
a similar question has been asked before, but I didn't really understand the answer, here is the link to the other question : Which of these maps are alternating tensors?
Attempt at a solution :
The function h is not a tensor since its not linear in $ x,y $, $ c*h(x,y) \not= h(cx,y)$ and similarly for y For g i'm not really sure, if create a 3x3 matrix to calculate the derivative for g, one of the row will be completely zero, and hence, $g(y,x) \not= -g(x,y)$ This is what I think I'm not exactly sure, is there a standard way to approach questions like these, in particular if h was linear in each of its variables how would you approach that and similarly say how would you attempt this question for lets say $g(x,y) = x_1y_3 - x_3y_2 + x_2y_1 $
You need at least $g(x,y)=-g(y,x)$ which when you test it is not satisfied. This also holds for your alternate expression. You can anti-symmetrize any tensor as $$ \frac12(g(x,y)-g(y,x)). $$