1/?) Time for an experimental series of threads about a tricky and subtle concept: differential forms!

Forms are at first glance somewhat mysterious objects! Hopefully I will make you understand how natural they are and what they are there for!

Let's get started!

2/?) Forms were introduced to generalize the concept of gradient of a function to arbitrary smooth manifolds. Let's start with a definition: the gradient is the vector valued function such that its dot product with a vector v is the directional derivative along v

3/?) Vector valued functions can be generalized to smooth manifolds, we call them vector fields. They assign a tangent direction (i.e. an "arrow") to each point in the space. it seems natural to generalize the gradient as the vector field that obey our previous definition right?

4/?) Let's try! The gradient of the function f is the only vector field that, at each point p of the space, gives the directional derivative of D_v f(p) as the dot product with the tangent vector v.

But wait! the dot product is not defined in an arbitrary vector space!

5/?) The dot product is an additional structure on a vector space that can be used to define lengths, angles and orthogonality. It is difficult to imagine but in a general vector space those concepts are not defined! vector spaces are rubbery objects!

6/?) The tangent space of a smooth manifold doesn't have a dot product (i.e. a metric)! Smooth manifolds are like rubber sheets that can be stretched at will. Does it mean that we can't define the gradient there? Nope, we can! Enter the dual space!

7/?) Don't worry it's not scary as it sounds. Every vector space has a twin: the dual space. An element of the dual w is a linear function that eats vectors and gives numbers! Note the similarity with dot product, which is an operation that takes two vectors and give you a number

8/?) This suggest a method to define a gradient! The gradient is a function that assigns to each point in the a smooth space an element of the dual of the tangent space that gives you the directional derivative when you give it v as input! A bit of a mouthful, read it twice!

9/?) These functions (more properly fields) from the smooth manifolds to dual element of their tangent spaces are called differential forms! They eat tangent vectors and spit out numbers. the gradient of a function is an interesting special case! The gradient of f is the form df!

10/?) Now consider arbitrary coordinates x,y,z,.. that chart the smooth manifolds. These are just functions that you can use for "knowing where you are" in the manifold/ Each one assign a number for each point.

11/?) The coordinates are functions and we can therefore take their gradients! We get the differential form dx, dy, dz and so on! These is just gradients like df. It is defined as follows: df(v) = D_v f

12/?) Now remember that the dual is a vector space with the same dimensionality of the tangent space (it's a twin!). This allows to write the gradient df as linear combination of the gradient of the coordinates: df = (df/dx) dx + (df/dy) dy + (df/dz) dz + ...

13/?) This looks a lot like the gradient of vector calculus! It is a linear combination of unit vectors where the coefficient are the partial derivatives! The only difference is that it is noz a linear combination of linear functions (duals)!

14/?) Let's now try to give a directional vector to the gradient and use some algebra: df(v) = (df/dx) dx(v) + (df/dy) dy(v) + (df/dz) dz(v) + ...

15/?) What is df(v)? By definition it is the directional derivative of the coordinate x along v. This derivative is equal to zero in a N - 1 dimensional subspace (where N is the dim of the manifold) since it is the null space of a linear function from V to R.

16/?) The null space of df is therefore well defined and it suggests a geometric interpretation. It is the space of constancy of the function where df(v) = 0. The form df can be interpreted a stack of these spaces of constancy along the direction of the steepest change of f.

17/?) If the form is a gradient, these stacks of constancy spaces can be joined into the constancy lines of the function. This is a generalization of a regular height map that assign a value to a line of constant height!

18/?) Nota that those stacks are signed, meaning that they know the direction where the function increases/decreases (we do gradient descent right?). Also the magnitude can be visualized as the density of those stacks like in a regular height map!

19/?) However, not all forms are gradients! A general form

Xi = a dx + b dy + c dz + ...

can still be interpreted locally as a stack of "costancy" null spaces such as Xi(v) = 0. However, those spaces cannot always be joined coherently to create heightmap of a function!

20/?) For example, you can have an Escherian form that gives you a perpetually increasing stack of spaces along a closed loop so that it it were a gradient it would define a function that does not return in itself if you move along a closed loop from a point p to itself!

21/21) There is so much more to say but we are out of time. If you want more content like this please remember to like and retweet! These threads takes some time to make and you can pay me back with a dopamine burst! ^^

Thank you and see you in the next threads about forms!