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

• I
• JorgeM
In summary: I'm capable of typos.In summary, the conversation discusses the general expression for the 2nd order Taylor series in 3 or more variables. The conversation also mentions the use of multi-index notation for higher order expansions and provides an example of a 6th order expansion of a function. The conversation concludes with a concise notation for a third order expansion in powers of a function.
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

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

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

Dr Transport
JorgeM said:
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}.

----------

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.

JorgeM and Delta2

## 1. What is a 2nd order Taylor Series for a function in 3 or more variables?

A 2nd order Taylor Series is a mathematical representation of a function in 3 or more variables. It is used to approximate the behavior of a multivariable function near a specific point.

## 2. How is a 2nd order Taylor Series calculated?

The 2nd order Taylor Series is calculated using partial derivatives of the function at a specific point. The formula for a 2nd order Taylor Series is f(x,y) = f(a,b) + (x-a)fx(a,b) + (y-b)fy(a,b) + (1/2)((x-a)2fxx(a,b) + 2(x-a)(y-b)fxy(a,b) + (y-b)2fyy(a,b)), where fx, fy, fxx, fxy, and fyy are partial derivatives of the function.

## 3. What is the purpose of using a 2nd order Taylor Series?

The 2nd order Taylor Series is used to approximate the behavior of a multivariable function near a specific point. It can be used to find the value of the function at a point, estimate the behavior of the function in a certain region, and find critical points of the function.

## 4. Can a 2nd order Taylor Series be used for any function in 3 or more variables?

Yes, a 2nd order Taylor Series can be used for any function in 3 or more variables. However, the accuracy of the approximation depends on the smoothness of the function and the distance from the point of approximation.

## 5. How can a 2nd order Taylor Series be extended to higher orders?

A 2nd order Taylor Series can be extended to higher orders by including more terms in the formula. For example, a 3rd order Taylor Series would include third-order partial derivatives of the function at the point of approximation. Higher order Taylor Series provide a more accurate representation of the function but require more computational effort to calculate.

Replies
3
Views
2K
Replies
7
Views
1K
Replies
7
Views
2K
Replies
4
Views
2K
Replies
5
Views
601
Replies
4
Views
1K
Replies
11
Views
1K
Replies
3
Views
466
Replies
4
Views
2K
Replies
10
Views
4K