Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

Question

I came up with this in my mind, Just wondering if it is true

I am thinking about it too, will post my observations, if any

Answer 1

The cross-product $U\times V$ of vectors $U$ and $V$ can be written in coordinates as the sum $$dy\wedge dz(U,V)\textbf{ i }- dx \wedge dz(U,V)\textbf{ j }+ dx \wedge dy(U,V)\textbf{ k }$$ (note the sign). Since the length of the cross product $U\times V$ is the area of the parallelogram spanned by them, the formula you wrote is quite suggestive. Note that it follows that the square of the area of the parallelogram spanned by $U,V$ is the sum of the squares of the areas of the parallelograms spanned by the orthogonal projections to the three coordinate planes.