That sounds like the beginnings of intuition for exterior algebra. While the gradient is a vector (i.e. it lives in a vector space), it's better thought of as b

That sounds like the beginnings of intuition for exterior algebra. While the gradient is a vector (i.e. it lives in a vector space), it's better thought of as being different to a standard 'displacement' vector. Look at a topographical map of a landscape and note the contour lines. As you zoom into the contour lines (if they're detailed enough), they'll start to look more and more like parallel lines, densely spaced for a steep slope, or sparsely spaced for a mild slope. These parallel lines are the gradient. A gradient and a vector 'fit together' to give a real number. The more parallel lines the vector pierces, the bigger the number [0]. So a gradient 'eats' a vector, spitting out a real number, and vice-versa. Just like a row vector 'eats' a column vector and spits out a real number. I'm saying 'gradient' here, but really what I mean is 'one-form'. Language deliberately imprecise for all y'all mathematicians out there. [0] https://neutronstars.utk.edu/class/p616_s19/lect3/index.html#slide6 https://neutronstars.utk.edu/class/p616_s19/lect3/index.html...