[x3d-public] Need PhD for Vector Calculus problem

[Vincent Marchetti]

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.
> _______________________________________________
> x3d-public mailing list
> x3d-public at web3d.org
> http://web3d.org/mailman/listinfo/x3d-public_web3d.org
> _______________________________________________
> x3d-public mailing list
> x3d-public at web3d.org
> http://web3d.org/mailman/listinfo/x3d-public_web3d.org