Still don't understand tensors. How is a /sci/ tensor different from a /g/ tensor?
A /g/ tensor is always a matrix.
in /g/ everything is an array
>>17004755a tensor is a multilinear mapwhat does that mean?it means that it is a function that maps vectorspaces into vectorspaces, and that the function is linear in each of it's arguments as the other arguments are held constanthow do you represent a tensor and evaluate it?given a basis for each vectorspace, you just expand each input and output in terms of its associated basis and apply linearity and the uniqueness of basis expansions to represent it in terms of components multiplied by the remaining basis tensors.example:consider a (bilinear) map [math]B: U \times V \rightarrow W[/math], where [math]U, V, W[/math] are vectorspaces.let [math]\vec{u} \in U, \vec{v} \in V, \vec{w} \in W[/math] such that [eqn]\vec{w} = B(\vec{u}, \vec{v})[/eqn]assume you have bases for each vectorspace such that[eqn]\vec{u} = \sum_i^{N_u} u_i \vec{u}^i\quad \vec{v} = \sum_j^{N_v} v_j \vec{v}^j\quad \vec{w} = \sum_k^{N_w} w_k \vec{w}^k[/eqn]expand the vectors in the map using these bases[eqn]\sum_k w_k \vec{w}^k = B\left(\sum_i u_i \vec{u}^i, \sum_j v_j \vec{v}^j\right)[/eqn]and apply linearity in each argument to move the sums outside of the map[eqn]\sum_k w_k \vec{w}^k = \sum_i u_i B\left(\vec{u}^i, \sum_j v_j \vec{v}^j\right) = \sum_i u_i \sum_j v_j B\left(\vec{u}^i, \vec{v}^j\right) = \sum_i \sum_j u_i v_j B(\vec{u}^i, \vec{v}^j)[/eqn]notice how you only have to know how the basis vectors transform to know how to evaluate arbitrary inputalso notice that the map can be expanded in the [math]W[/math] basis[eqn]\sum_k w_k \vec{w}^k = \sum_i \sum_j u_i v_j \sum_k B_k(\vec{u}^i, \vec{v}^j) \vec{w}^k = \sum_i \sum_j \sum_k u_i v_j \sum_k B_k(\vec{u}^i, \vec{v}^j) \vec{w}^k[/eqn]by uniqueness of basis expansion, the components must be equal, so[eqn]w_k = \sum_i \sum_j u_i v_j B_k(\vec{u}^i, \vec{v}^j)[/eqn]or (with implicit sums)[eqn]w_k = u_i v_j B_k^{ij}[/eqn]where [math]B_k^{ij} = B_k(\vec{u}^i, \vec{v}^j)[/math]>>17004789a rectangle is always a square
>>17004755they're the same thing.scalar = dotvector = linematrix = flat 2d planetensor = 3d constructseems simple enough, if you follow the pattern, graph it and give it form.a 4d (object?) would be one that is effected by a ever changing external/internal source (user, observer, sensors, etc)a 5d (construct? idk) would be something that provides feed back/returns a value, and has direct effect on the formula itself, itd be like a living being....i'm a retard though, and i dont science or math, so who know (it made sense to some of you up until this point)
>>17005700i feel bad for the electrons used to compose, send, store, broadcast, and display your posti also feel bad for the electrons that compose your person
>>17005700Bot
I hardly know her
Use AI or youtubehttps://youtu.be/CliW7kSxxWU
>>17004755Matrices are for flat spaces, but if you want your math to work in a warpship or a gracity well youll want tensors, they have the mathematical machinery to handle the bends where a simple matrix would get lost
>>17005906Then why do we use the stress tensor in flat space?
>>17004848ayo that wabbit be tweakin out
