[x3d-public] Need PhD for Vector Calculus problem
Vincent Marchetti
vmarchetti at kshell.com
Mon May 15 13:39:22 PDT 2023
The Taylor series defined for real numbers, say near 0, is
F(x) = a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 ....
with the series being absolutely convergent for some region around 0; that is |x| < eps for some positive eps > 0
Then, taking a variety of routes, we evenuallt get to a relation between the coefficients ant the derivatives of F(x)
There is no bid deal about all of ths a_i themselves being vectors, or matrices, perhaps even functions on some domain.
BUT
If the x values are intended to be elements of a vector space, or other hgher dimensional space, there's no useful way to define the powers x, x^2, x^3 ....
Hence, no Taylor series on vector space domains.
Vince
The challenge here is that for a domain that's more than one-dimensional, there's no obvuj
> On May 15, 2023, at 4:29 PM, John Carlson <yottzumm at gmail.com> wrote:
>
> Think of the sum of a taylor series being a vector, matrix, or field. This is like
>
> f(V) = ...something like a Taylor Series where V is a vector
>
> f(M) = ... ditto where M is a matrix
>
> f(F) = ... ditto where F is a field (as in physics)
>
> I'm looking more for a name of the thing than anything else. Field series?
>
> On Mon, May 15, 2023 at 2:47 PM GPU Group <gpugroup at gmail.com> wrote:
> Taylor Series use: linearizing a non-linear equation for iterative linear solutions. Take the first 2 terms of the taylor series expansion, and that's a straight line equivalent to the tangent line to the non-linear curve. Solve for a new guess on each iteration until the error term is small.
> -Doug
>
> On Mon, May 15, 2023 at 12:52 PM John Carlson <yottzumm at gmail.com> wrote:
> Is there a kind of Taylor Series that produces a vector or vector field instead of a single number?
>
> Or maybe just an advanced undergraduate?
>
> I guess it's kind of like a matrix sum, where the terms are matrices.
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