How to evaluate the operation of a differential form on a vector?

Question

While studying some Discrete Calculus following Grady and Polimeni (2010), I've found the following problem. I would like to know how the author found the result w(v) = -8. I've found that the vector v (given by the cross product definition) is given by [3 e1 ; -1 e2; 2 e3]. Is this right also?

Thanks in advance.

Problem Statement

where:

Definition

Answer 1

HINTS

• What is $$ \bar{e_i}\wedge \bar{e_j} \wedge \tilde{\sigma}^m\tilde{\sigma}^n $$

• What is $$2\cdot 7 + 3\cdot(-6) + (-1)\cdot 4 $$