2nd order Taylor Series for a function in 3 or more variables?

[JorgeM]

TL;DR Summary

Hello there, I need to get the Taylor Series for f(r) and r is a function f(x,y,z))=r

I have taken a look but most books and Online stuff just menctions the First order Taylor for 3 variables or the 2nd order Taylor series for just 2 variables.
Could you please tell me which is the general expression for 2nd order Taylor series in 3 or more variables? Because I have not found nothing at all.
Thanks

 

[Svein]

Hint:
$df(x, y, z)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$

 

[hutchphd]

I usually don't like videos for teaching but this is not bad:

 

[WWGD]

This is just the tangent plane approximation to
surface. https://en.wikipedia.org/wiki/Taylor_series
following up on @Svein 's method

 

[Stephen Tashi]

Summary: Hello there, I need to get the Taylor Series for f(r) and r is a function f(x,y,z))=r

Some people make a distinction between a Taylor Series and a McLaurin Series. Let's assume you mean you want an expression for $f(x,y,z)$ in powers of $x,y,z$ rather than in powers of $(x-a),(y-b),(z-c)$

It is a strenuous exercise in LaTex to write it out higher order multi-variable expansions in ordinary notation. ( No wonder you only got hints - and I don't volunteer to write it out that way myself !)

You can find it written out in the "multi-index notation" https://en.wikipedia.org/wiki/Multi-index_notation
I'm not skilled at reading the mult-index notation. I'll try some concrete examples.

Example: consider the 6th order expansion of $f(x,y,z,w)$ What is the coefficient of $x^3 y^1 z^2 w^0$ in that expansion?

It is:
$\frac{ 1}{3!\ 1!\ 2!\ 0!}$ $\frac{\partial^3 f}{\partial x^3} \frac{\partial^1 f}{\partial y^1} \frac{\partial^2 f}{\partial z^2} \frac{\partial^0 f}{\partial w^0}$

where the partial derivatives are evaluated at x = y = z = w = 0 and keeping in mind the definition 0! = 1! = 1 and using the convention that $\frac{\partial^0 f}{\partial w^0} = 1$

The above expressions implicitly use the multi-index {3,1,2,0}.

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For a third order expansion in powers of $f(x,y,z)$, we could employ some concise (and atrocious?) notation such as saying $T(a,b,c)$ will abbreviate $\frac{1}{a! \ b! \ c!} \frac{\partial^a f}{\partial x^a} \frac{\partial^b f}{\partial y^b} \frac{\partial^c z}{\partial z^c} x^a y^b z^c$ and define $T(0,0,0) = f(0,0,0)$

$f(x,y,z) =$
$T(0,0,0)$
$+ T(1,0,0) + T(0,1,0) + T(0,0,1)$
$+ T(2,0,0) + T(1,1,0) + T(1,0,1) + T(0,2,0) + T(0,1,1) + T(0,0,2)$
$+ T(3,0,0) + T(2,1,0) + T(2,0,1) + T(1,2,0) + T(1,1,1) + T(1,0,2) + T(0,3,0) + T(0,2,1) + T(0,1,2) + T(0,0,3)$

Someone should check my work.