91Ó°ÊÓ

A k-form is an antisymmetric, multilinear function that takes k vectors as input and outputs a scalar. If \(V\) is a vector space, a k-form is an element of the exterior power \(\Lambda^k (V^*)\), where \(V^*\) is the dual vector space. In our case, the 1-forms \(\omega_1, \dots, \omega_k\) are elements of \(V^*\).

The exterior product (also called wedge product) is an algebraic operation that combines two differential forms to create a new differential form. Given two forms \(\omega\) and \(\eta\), their exterior product is denoted as \(\omega \wedge \eta\). The exterior product has the following properties: 1. Antisymmetry: \(\omega \wedge \eta = -\eta \wedge \omega\) 2. Linearity: \((a \omega + b \eta) \wedge \xi = a(\omega \wedge \xi) + b(\eta \wedge \xi)\) 3. Distributive law: \(\omega \wedge (\eta + \xi) = \omega \wedge \eta + \omega \wedge \xi\)

We want to show that \(\omega_1 \wedge \cdots \wedge \omega_k\) is a k-form. We will use the properties of the exterior product and the definition of a k-form. Consider the product \(\omega_1 \wedge \cdots \wedge \omega_k\). We want to show that this product is a k-form, i.e., it is antisymmetric and multilinear. 1. Antisymmetry: We just need to show that if we change any two of the 1-forms, the sign of the k-form changes. Let's swap two 1-forms, say \(\omega_i\) and \(\omega_j\) where \(i \neq j\). Using the antisymmetry of the exterior product, we have: \(\omega_1 \wedge \cdots \wedge \omega_i \wedge \cdots \wedge \omega_j \wedge \cdots \wedge \omega_k = -\omega_1 \wedge \cdots \wedge \omega_j \wedge \cdots \wedge \omega_i \wedge \cdots \wedge \omega_k\) This shows that the wedge product of k 1-forms is antisymmetric. 2.Multi-linearity: This product is clearly multilinear since each of the 1-forms \(\omega_1, \dots, \omega_k\) are linear, and we can use the linearity of the exterior product to show that the full product is multilinear. Consider the vectors \(u_1,\dots,u_k\) as arguments for the k-form: \((a\omega_1+b\omega'_1) \wedge \omega_2 \wedge \cdots \wedge \omega_k (u_1,\dots,u_k) = (a\omega_1 \wedge \omega_2 \wedge \cdots \wedge \omega_k + b\omega'_1 \wedge \omega_2 \cdots \wedge \omega_k )(u_1,\dots,u_k)\) Therefore, the wedge product of k 1-forms is indeed multilinear. By showing that the wedge product of k 1-forms is both antisymmetric and multilinear, we have demonstrated that \(\omega_1 \wedge \cdots \wedge \omega_k\) is a k-form.