1) If $f^1∧f^2...f^k$ are real valued smooth functions, then show that $$d(df^1∧df^2...df^k)=0$$
2) If $f^1∧...∧f^k : U⊂R^n→R$ are smooth functions, then $$df^1∧df^2...df^k=\displaystyle \sum_{i_1<...<i_k}\left|\frac{\partial(f^1,...,f^k)}{\partial(x^{i_1},...,x^{i_k})}\right|dx^{i_1}∧...∧dx^{i_k}$$ where $$\left|\frac{\partial(f^1,...,f^k)}{\partial(x^{i_1},...,x^{i_k})}\right|$$is the Jacobian determinant.
In 1), the wedge product of functions gives me a square matrix which it's left diagonal is zero. The part I'm missing that I differentiate it,why it becomes a zero matrix.
Sadly I have no opinion in 2).