Please expand. Physicist here who learned to live with the wave factor, but never really got it.

>>16960691
Math garbage, they need 4 years just to "prove" kepler laws from Newton. They are useless

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>>16960691
A one-form is basically a function in differential geometry that maps a vector to a scalar. You can visualize it as series of "surfaces", and the number of surfaces a vector crosses is the scalar value of the one-form and the vector.

This is basically how the wave vector is visualized as well (as least how I was taught it). The distance between surfaces (wavefronts) is the wavelength and the wavefronts themselves specify a path. The wave "one-form" and the position vector together specify the phase (scalar) of a wave. The wave vector being a "vector" is basically a mismoner

>>16960639
Isn't the gradient a 1-form too?

>linear functional
a linear function that maps a vectorspace to its field
feed it a vector, it poops out a scalar

>1-form
a differential form over a manifold that, when combined with a vector-valued function over the manifold, results in a 0-form (scalar function) that can be integrated to get a scalar
consider a space with a vector field, feed the 1-form the vector field, and it poops out a scalar field that can be integrated to get a scalar

when thinking about plane waves, k-vectors can be thought of as linear functionals measuring phase given a position vector.
maybe when thinking about k-space, one can think about k-vectors (with an inner product) as measuring how much of a 3d function that k-vector contributes to? i don't usually think of it like that, but it just sounds like a more formal way to talk about fourier transforms. i usually think of those in terms of vectorspace basis expansions, where basis vectors are replaced with basis functions and sums are replaced with integrals.

>>16960639
I mean, k dot x or p dot x means k or p is the dual object to x. Does it reveal anything significantly interesting that would've aided your undergrad understanding?

>>16960734
When it takes in a direction and outputs the directional derivative, yeah.

>>16960864
>When it takes in a direction and outputs the directional derivative, yeah.
The components of the gradient of a function transform like a covector

vector, covector, differential form,
once a collection of isomorphic notions is used often enough, I don't think it's worth really making the disction.

I mean almost every vector with components [math]w_i[/math] you'd work with is used as map [math]x\mapsto \sum_ w_i \cdot x_i [/math].
Even the projections [math] \pi_j(w) = (w, e_i) [/math] are of this form.
Hence, hard to find counter-examples. In those cases, you probably don't the vector addition and work with lists.

[math] x \mapsto \sum_i w_i \cdot x_i [/math]

>>16960639
I took college trig and it took me awhile to understand the entire pi chart.

>>16961625
People often miss the physics of a situation like.

When the frequency of (lambda) is different in two regions, the phases will seperate, creating a -phase transition boundary, creating something like a refractive boundary, meaning that the two frequency sections will still seperate again by cos O, sin 0 (roughly, by another 90degrees) as two frequency areas will seperate by non matching lambda values.

So it will always require a further 2 x ( X,Y. + 2c( kcos0 +ksin0) to cross a described -phase boundary.

In a very loose format

leave it to mathfags to obfuscate a very simple physics concept lol

wavelength is a scalar. just take the inverse of it to get the wave number in meters. it's that fucking easy.

>>16963113
this

>>16960639
>>16960691
retards it's LITERALLY the cyclic frequency of wavelength. WHAT THE FUCK IS THERE TO GET???

PERIOD => FREQUENCY ~ CYCLIC FREQUENCY

WAVELENGTH => 1/WAVELENGTH ~ WAVE NUMBER

so are -forms just a generalization/rigorization of differentials or is it A interpretation of what they could also be?
and why are these things studied in differential geometry and not just regular analysis where you initially encounter AND DEFINE differentials?

>>16963411
please understand. they have severe autism. to them a row-vector and a column-vector require different names. one of them is called an n-form, while the other has some retarded name i cannot for the life of me recall. row-vector and column-vector are too self-explanatory and accessible to the public, so they instead introduce idiosyncratic, idiotic verbiage to alienate everyone and feel smart doing it.
>insert mathfag screeching about some minutiae of details like how it's ACKSHUALLY a row-vector of differentials!
case in point.

>>16963469
>t. finite dimensional nigga