﻿What is the gradient?


In 3blue1brown, traditional calculus land, the gradient at a point is a vector at that point pointing in the most upwards direction. It’s useful in neural networks because you want to optimize a function (find the minimum), which you can do by following the gradient.


But saying that the gradient is a vector is dirty and wrong (although considering the existence of a metric, it’s acceptable). In reality the gradient is a covector.


So hang on a second. You have the directional derivative operator, which is quite cool and all, because the directional derivative is a lot less useful. But then something similar happens for the gradient. You have the gradient operator, which is basically a function d, which when given a specific f becomes a gradient (and in the world's biggest shock, it does this in the sneakiest way ever: it now waits for a vector X that it can use to apply the function f!!).


So the gradient operator d takes a function and returns a covector (gradient), which itself is a function that awaits a vector to measure it and become a real number (lol, ironic).


Next thing you do to find the “natural” covectors in a chart is that you use the chart coordinate functions are functions that you use with the gradient operator to receive your gradient vectors basis (quite sneaky if you ask me). You can confirm that this is indeed the correct dual basis to the natural tangent space basis vectors by showing that applying one to the other gives you the kronecker delta.


Looking at this gradient operator function more closely is potentially relevant, because d will end up being important in the grassmann algebra as a way of going from d-differential forms to d+1 differential forms.


Next is the notion of the push forward and pull back.


Consider you have a map from M to N. You can push forward curves, vectors (duh), and tangent vectors, and any other higher order tensors that are entirely covariant.


You can also pull back any functions on N, and gradients (covectors), and any other higher order tensors that are entirely contravariant.


The pushforward and pullback is the name of the actual map that takes these objects in one manifold to that of the other.


Push forwards for tangent vectors is also called the differential.


Eventually as I understand it, the pull back will be used to integrate on manifolds.


And I guess the push forward to differentiate on manifolds? I honestly don’t know.


Another interesting things is that if the map from M to N is a diffeomorphism, then you can also pull back vectors and push forward functions (opposite than what you expect).


What is a bundle, explained simply.