Interior product of differential 3-form

Question

I know how to compute interior products of differential 1 form and 2 forms, but I am not sure how to compute an interior product of a differential 3 form. Is there a general expression for calculating interior products of 3-forms?

Answer 1

If $\alpha$ is a $p$-form and $\beta$ is a $q$-form, then $i_X(\alpha\wedge\beta) = (i_X\alpha)\wedge\beta + (-1)^p\alpha\wedge(i_X\beta)$. It follows that if $X = \sum_{i=1}^nX^i\frac{\partial}{\partial x^i}$, then

$$i_X(dx^{i_1}\wedge\dots\wedge dx^{i_k}) = \sum_{j=1}^k (-1)^{j-1}X^{i_j}dx^{i_1}\wedge\dots dx^{i_{j-1}}\wedge dx^{i_{j+1}}\wedge\dots\wedge dx^{i_k}.$$