While this is true, note that this special relationship between gradient, divergence, and curl only exists in 3D, due to the fact that 2 + 1 = 3, and the presen

While this is true, note that this special relationship between gradient, divergence, and curl only exists in 3D, due to the fact that 2 + 1 = 3, and the presence of the [hodge star](https://en.wikipedia.org/wiki/Hodge_star_operator https://en.wikipedia.org/wiki/Hodge_star_operator) operator that allows us to convert "areas" into "normal vectors to areas". A curl is really a "infinitesimal area", which we then choose to interpret as "a normal vector". This is captured by the [De Rham cohomology](https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/De_Rham_cohomology), whose calculations are sometimes simplified by using the [Penrose notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation https://en.wikipedia.org/wiki/Penrose_graphical_notation) as outlined above. Having used the notation, I don't think it helps for vector calculus. And I 100% agree with you that we should teach the intuition for div, grad, and curl first, but I wanted to bring to the attention of anyone who's never seen the "fully general" form of these objects as to what it looks like.