# How to switch from tensor products to wedge product

• A
victorvmotti
Suppose we are given this definition of the wedge product for two one-forms in the component notation:

$$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$

Now how can we show the switch from tensor products to wedge product below:

$$\epsilon=\epsilon_{\mu_{1}...\mu_{n}}dx^{\mu_{1}}\otimes...\otimes dx^{\mu_{n}}$$
$$=\frac{1}{n!}\epsilon_{\mu_{1}...\mu_{n}}dx^{\mu_{1}}\wedge...\wedge dx^{\mu_{n}}$$

Mentor
2022 Award
What happens to your equation, if ##A=B##?

victorvmotti
Is this computation below for the case of ##n=2## correct?

$$\epsilon= \frac{1}{2} ( \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\wedge dx^{\mu_{2}})$$
$$= \frac{1}{2} (\epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} - \epsilon_{\mu_{2}\mu_{1}}dx^{\mu_{2}}\otimes dx^{\mu_{1}})$$
$$= \frac{1}{2} ( \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} - (-1) \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} )$$
$$= \frac{1}{2}(2\epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}})$$
$$= \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}}$$