Differential forms

Table of Contents

[2018-11-24] 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

CREATED: [2018-11-10]

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: vu = -uv; u ^ u = 0; and linearity
build the space: \Wedge2 V = {Sumi ui ^ vi | ui ^ vi | ui, vi 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 dxdy + G dxdz + H dydz looking like that?? mathstudy

rules for derivative: mathstudy

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 dxdy + G dydz + 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 intS F \dot dS = ∫ intS F1 dydz + F2 dz ^ dx + F3 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 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

[2018-11-06] 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

[2018-11-25] ok, so diff forms are special types of tensors. Not all tensors are diff forms mathtensor
a completely antisymmetric covariant tensor – p-form; contravarian – p-vector

DONE [2018-11-14] calculus - What does \(dx\) mean in differential form? - Mathematics Stack Exchange math
dx1 is a differential 1-form, linear map acting on tangent space. Action is: (1, 0, …. 0)

dxi is a covector field. {dxi}i span all covector fields. For f ∈ Rn -> R, you can write df = [f1(x) f2(x) .. fn(x)] as f1(x) dx1 + … + fn(x) dxn
in that sense, dx1 is the derivative of coordinate function f(x1…xn) = x1

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 gammai, 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?? 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.

[2018-11-10] Дифференциал (дифференциальная геометрия) — Википедия 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!

[2018-11-10] differentiation in nLab 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).

[2018-11-18] differential geometry - What is a covector and what is it used for? - Mathematics Stack Exchange math

[2018-11-24] Closed and exact differential forms - Wikipedia math

 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.

START [B] [2019-01-24] A Visual Introduction to Differential Forms and Calculus on Manifolds | Jon Pierre Fortney | Springer mathviz

  • State "START" from "TODO" [2019-02-27]

[2019-06-12] ok, so illustrations are ok and there are many of them. explanations seemed ok as well I guess mathviz

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