Sometimes I think about parallelepipeds then when I reassociate I'm smiling and my fiance is visibly wondering what I'm thinking about. "Don't worry hun, parallelepipeds again"

where × is the cross product and · is the dot product

pardon my ignorance but what the fuck is a "cross product" or a "dot product"? I assumed at first this was multiplication, but then I saw the dot and realized this isn't anything I've ever been taught

it’s used to calculate volumes of curved objects. basically you chop down the object into a lot of small parallelepipeds (mentally), and then calculate the volume of each of them small things and sum over them. Done.

For anyone not quite getting this (like me), to calculate the area under a curve in 2D we were taught to cut it into tiny thin rectangles and sum those up; intergration when those rectangles have a width tending to 0.

For 3D (e.g. a pond ripple) or higher curves, a simple rectangle wont cut it as the length of the rectangle might only get the top of a wave, but not capture the crest of it tangential to it. So you create slopey rectangles to approximate that space, and bring the limit to zero to get the area (I think).

My only confusion now is, if I'm deforming a rectangle from one side to approximate the curve just above it, am I not also deforming the bottom of that rectangle in the same way (for the parralel strcture to hold true), and creating a forgotten space just above the axis plane?