Why did no one tell me the wave vector is actually a one-form? This caused me so much confusion in undergrad
Please expand. Physicist here who learned to live with the wave factor, but never really got it.
>>16960691Math garbage, they need 4 years just to "prove" kepler laws from Newton. They are useless
>>16960691A 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
>>16960639Isn't the gradient a 1-form too?
>linear functionala linear function that maps a vectorspace to its fieldfeed it a vector, it poops out a scalar>1-forma 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 scalarconsider 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 scalarwhen 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.
>>16960639I 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?>>16960734When 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]
>>16960639I took college trig and it took me awhile to understand the entire pi chart.
