# Differential forms

## Table of Contents

- math my thoughts on that
- DONE diffforms.pdf mathstudy
- a 1-diff form is an
**expression**F(x, y) dx + G(x, y) dy mathstudy - total differential is an example of diff. form: mathstudy
- a diff form is similar to vector field mathstudy
- a form is exact if it's a total differential of a scalar function mathstudy
- closed – if pF/py = pG/px mathstudy
- exact means that integral is path independent! mathstudy
- wedge space: given vector space V: mathstudy
- 2-form: an expression, built using wedges on pairs of 1-forms mathstudy
- rules for derivative: mathstudy
- orientation: for a curve, direction; for a surface – normal direction mathstudy
- so basically, by definition: integral of 2-form would be mathstudy
- what would it mean to eval surface integral of dx ^ dy + dx ^ dz + dy ^ dz over a sphere? mathstudy
- ah! so it's actually a flux: int int
_{S}F \dot dS = ∫ int_{S}F_{1}dy^{dz}+ F_{2}dz ^ dx + F_{3}dx ^ dy mathstudy - NOTE arclength is NOT a differential form (always > 0) mathstudy
- the manifold has to be parameterised for defining diff forms (or at least, for defining integrals?) mathstudy

- page 50: Maxwell's equations. EM field is a 2-form mathstudy

- a 1-diff form is an
- https://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions math
- math differential forms ^ : wedge/exterior product
- mathtensor ok, so diff forms are special types of tensors. Not all tensors are diff forms
- DONE calculus - What does \(dx\) mean in differential form? - Mathematics Stack Exchange math
- TODO what are the other examples of 1-forms? math
- gradient – ok, definitely pretty intuitive math
- TODO hmm. gradient is a vector and total derivative is covector?? https://math.stackexchange.com/questions/47618/definition-of-the-gradient-for-non-cartesian-coordinates math
- The 1-form ̃dfsuch that for any infinites-imally small vectordxfromTp ̃df(dx) =df(3.5)is called the gradient at point P . ok – that makes way more sense! math
- TODO dual 1-form. dual covector field for vector field; is that a thing??? math
- TODO ugh. I guess alla of them are basically derivatives since well.. there isn't much else apart of derivatives there and they form complete basis math

- gradient – ok, definitely pretty intuitive math
- TODO some intuitive stuff from NotesOnVectors.pdf math
- math Дифференциал (дифференциальная геометрия) — Википедия https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB_(%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)
- math differentiation in nLab https://ncatlab.org/nlab/show/differentiation
- math differential geometry - What is a covector and what is it used for? - Mathematics Stack Exchange
- math Closed and exact differential forms - Wikipedia
- START [B] A Visual Introduction to Differential Forms and Calculus on Manifolds | Jon Pierre Fortney | Springer mathviz
- mathviz ok, so illustrations are ok and there are many of them. explanations seemed ok as well I guess

## ¶ my thoughts on that math

so basically, treat dx, dy as formal symbols? you can multiply them by scalar, add them up NOTE in agda it would be just n-form with ordered requirement? or just a subset?

## ¶DONE diffforms.pdf mathstudy

### ¶a 1-diff form is an **expression** F(x, y) dx + G(x, y) dy mathstudy

### ¶total differential is an example of diff. form: mathstudy

df = pf/px dx + pf/py dy

### ¶a diff form is similar to vector field mathstudy

### ¶a form is exact if it's a total differential of a scalar function mathstudy

### ¶closed – if pF/py = pG/px mathstudy

exact impllies closed, but not in reverse

### ¶exact means that integral is path independent! mathstudy

#### ¶TODO this is sort of similar to complex analysis? mathstudy

### ¶wedge space: given vector space V: mathstudy

wedge rules: v^{u} = -u^{v}; u ^ u = 0; and linearity

build the space: \Wedge^{2} V = {Sum_{i} u_{i} ^ v_{i} | u_{i} ^ v_{i} | u_{i}, v_{i} in V} are imposed

### ¶2-form: an expression, built using wedges on pairs of 1-forms mathstudy

#### ¶TODO hmm. are all formal experessions looking like F dx^{dy} + G dx^{dz} + H dy^{dz} looking like that?? mathstudy

### ¶rules for derivative: mathstudy

linearity

d(f alpha) = df ^ alpha + f d alpha

d (dx) = d (dy) = d(dz) = 0 # TODO hmm what does that one mean??

### ¶orientation: for a curve, direction; for a surface – normal direction mathstudy

basically, **continuous** normal field. If it exists, there are two of them: n and minus n

parameterise the surface by two coordinates u, v so that du ^ dv is the direction of normal

#### ¶so basically, by definition: integral of 2-form would be mathstudy

int F dx^{dy} + G dy^{dz} + H dz ^dx = int int [ F p(x, y)/p(u, v) + G p(y, z)/p(u, v) + H p(z, x)/p(u, v)] du dv

#### ¶what would it mean to eval surface integral of dx ^ dy + dx ^ dz + dy ^ dz over a sphere? mathstudy

#### ¶ah! so it's actually a flux: int int_{S} F \dot dS = ∫ int_{S} F_{1} dy^{dz} + F_{2} dz ^ dx + F_{3} dx ^ dy mathstudy

#### ¶NOTE arclength is NOT a differential form (always > 0) mathstudy

but what we can do is to define a metric tensor

#### ¶the manifold has to be parameterised for defining diff forms (or at least, for defining integrals?) mathstudy

### ¶page 50: Maxwell's equations. EM field is a 2-form mathstudy

## ¶https://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions math

### ¶TODO By the universal property of exterior powers, this is equivalently an alternating multilinear map math

mm, I guess this bit is crucial to my understanding of alternating property

## ¶ differential forms ^ : wedge/exterior product math

f(x, y, z) dx ^ dy ^ dx – integrated over volume, gives volume

d operation – exterior derivative, results in k+1 form

ok so treat dx, dy, dz formally in form expression

### ¶TODO do in agda?… mathagda

## ¶ ok, so diff forms are special types of tensors. Not all tensors are diff forms mathtensor

https://en.wikipedia.org/wiki/Antisymmetric_tensor

a completely antisymmetric covariant tensor – p-form; contravarian – p-vector

https://en.wikipedia.org/wiki/Volume_form#Riemannian_volume_form

## ¶DONE calculus - What does \(dx\) mean in differential form? - Mathematics Stack Exchange math

https://math.stackexchange.com/questions/664648/what-does-dx-mean-in-differential-form/664674#664674

dx_{1} is a differential 1-form, linear map acting on tangent space. Action is: (1, 0, …. 0)

dx_{i} is a covector field. {dx_{i}}_{i} span all covector fields. For f ∈ R^{n} -> R, you can write df = [f_{1}(x) f_{2}(x) .. f_{n}(x)] as f_{1}(x) dx_{1} + … + f_{n}(x) dx_{n}

in that sense, dx_{1} is the derivative of coordinate function f(x_{1…x}_{n}) = x_{1}

### ¶right, here it's a 1-form! so no concerns about antisymmetry etc.. math

so, to compute the 1-form along the path, we:

split the path gamma(t); t from 0 to 1 into k parts

each part can be considered a tangent vector at gamma_{i}, and plugged into the form. then sum up the numbers

### ¶TODO ok, but what if it was curvilinear? What would dx mean then?? math

## ¶TODO what are the other examples of 1-forms? math

### ¶gradient – ok, definitely pretty intuitive math

#### ¶TODO hmm. gradient is a vector and total derivative is covector?? https://math.stackexchange.com/questions/47618/definition-of-the-gradient-for-non-cartesian-coordinates math

hence gradient depends on metric

ok, so I guess definition might involve coordinates; but otherwise they are coordinate free once we prove they transform properly

#### ¶The 1-form ̃dfsuch that for any infinites-imally small vectordxfromTp ̃df(dx) =df(3.5)is called the gradient at point P . ok – that makes way more sense! math

#### ¶TODO dual 1-form. dual covector field for vector field; is that a thing??? math

#### ¶TODO ugh. I guess alla of them are basically derivatives since well.. there isn't much else apart of derivatives there and they form complete basis math

## ¶TODO some intuitive stuff from NotesOnVectors.pdf math

Examples of how you can picture contravariant and covariant vectors. A contravari-ant vector is a “stick” with a direction to it. Its “worth” (or “magnitude”) is proportional tothe length of the stick. A covariant vector is like “lasagna.” Its worth is proportional to thedensity of noodles; that is, the closer together are the sheets, the larger is the magnitude ofthe covector. These and other pictorial examples of visualizing contravariant and covariantvectors are discussed in Am.J.Phys.65(1997)1037.Figure 3: Pictorial representation of the innerproduct between a contravariant vector and a co-variant vector. The “stick” is imbedded in the“lasagna” and the inner product is equal to thenumber of noodles pierced by the stick. The in-ner product shown has the value 5.

## ¶ Дифференциал (дифференциальная геометрия) — Википедия https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB_(%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F) math

Пусть {\displaystyle M} M — гладкое многообразие и {\displaystyle f\colon M\to \mathbb {R} } {\displaystyle f\colon M\to \mathbb {R} } гладкая функция. Дифференциал {\displaystyle f} f представляет собой 1-форму на {\displaystyle M} M, обычно обозначается {\displaystyle df} df и определяется соотношением {\displaystyle df(X)=d_{p}f(X)=Xf,} {\displaystyle df(X)=d_{p}f(X)=Xf,} где {\displaystyle Xf} Xf обозначает производную {\displaystyle f} f по направлению касательного вектора {\displaystyle X} X в точке {\displaystyle p\in M} p\in M.

hmm, that actually makes sense too!

## ¶ differentiation in nLab https://ncatlab.org/nlab/show/differentiation math

1. Idea Differentiation is the process that assigns to a function f:X→Y f : X \to Y its derivative, sometimes denoted df d f. The derivative is a function that, roughly speaking, assigns to each point x∈X x \in X the linear transformation dfx d f_x that maps infinitesimal differences y−x y - x (for points y y infinitesimally close to x x) to infinitesimal differences f(y)−f(x) f(y) - f(x).

## ¶ differential geometry - What is a covector and what is it used for? - Mathematics Stack Exchange math

## ¶ Closed and exact differential forms - Wikipedia math

https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d. For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α. Since the exterior derivative of a closed form is zero, β is not unique, but can be modified by the addition of any closed form of degree one less than that of α. Because d2 = 0, any exact form is necessarily closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.