Taylor expansion for f(x,y) about (x0,y0) ?

In summary, the Taylor expansion for f(x,y) about (x0,y0) is a method of approximating a function at a specific point (x0,y0) by using a series of derivatives and evaluating them at that point. This involves taking a Taylor expansion in x around x0, and then taking the Taylor expansion in y of each term. The h^2(x) term in the third term comes from the 1D Taylor expansion in the x-variable, with h_x representing the difference between x and x0.
  • #1
izzy93
35
0
Can someone please explain the Taylor expansion for f(x,y) about (x0,y0) ?

Would really appreciate some sort of step by step answer :)

thankyou
 
Physics news on Phys.org
  • #2
Simple, just take a Taylor expansion in [itex]x[/itex] around [itex]x_{0}[/itex]
[tex]
f(x_{0}+h_{x},y_{0}+h_{y})=f(x_{0},y_{0}+h_{y})+h_{x} \partial_{x}f(x_{0},y_{0}+h_{y})+h_{x}^{2}\partial_{x}^{2}f(x_{0},y_{0}+h_{y})+\cdots
[/tex]
Then take the Tavlor expansion in [itex]y[/itex] of each term. Simple but tedious.
 
Last edited:
  • #3
Thanks for the reply, it looks logical but I'm stuck on how it all comes together/
and where does the h^2(x) term in the 3rd term on the right come from
 
  • #4
I am applying the 1D Taylor expansion to the x-variable. I am assuming, you know about the 1D Taylor expansion right?
 
  • #5
yes I do using this equation, f(x) = f(x0) + f '(x0)/1! (x-x0) ...
 
  • #6
The notation I am using:
[tex]
h_{x}=x-x_{0}
[/tex]
 
  • #7
ok, I think i get it, bit slow atm! thankyou!
 

Related to Taylor expansion for f(x,y) about (x0,y0) ?

1. What is a Taylor expansion?

A Taylor expansion is a mathematical method used to approximate a function using a series of terms, each of which is a polynomial of increasingly higher degree. It is named after mathematician Brook Taylor and is commonly used in calculus and other areas of mathematics.

2. How do you calculate a Taylor expansion for a function f(x,y) about a point (x0,y0)?

To calculate a Taylor expansion for a function f(x,y) about a point (x0,y0), you need to first find the partial derivatives of the function with respect to x and y. Then, plug in the values of x0 and y0 into the derivatives and evaluate the resulting expression. Finally, substitute the values into the Taylor series formula, which is f(x,y) = f(x0,y0) + fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) + ...

3. What is the purpose of a Taylor expansion?

The purpose of a Taylor expansion is to approximate a function using a polynomial series. This allows for the estimation of values of the function at points near the original point, as well as the calculation of derivatives and integrals of the function at that point.

4. Can a Taylor expansion be used for any function?

No, a Taylor expansion can only be used for functions that are infinitely differentiable, meaning that they have derivatives of all orders at every point. If a function is not infinitely differentiable, its Taylor series may not converge or may not accurately approximate the function.

5. How can a Taylor expansion be used in practical applications?

A Taylor expansion can be used in various practical applications, such as in physics, engineering, and economics, to approximate complicated functions and make calculations easier. It is also used in numerical analysis to approximate solutions to differential equations and in optimization problems to find the maximum or minimum value of a function.

Similar threads

I Taylor expansion of f(x)=arctan(x) at infinity
    • Calculus
Replies
2
Views
2K
I Taylor Expansion Question about this Series
    • Calculus
Replies
2
Views
1K
A Series expansion of ##(1-cx)^{1/x}##
    • Calculus
Replies
3
Views
2K
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
    • Calculus
Replies
19
Views
3K
B Mean-Value Theorem, Taylor's formula, and error estimation
    • Calculus
Replies
3
Views
1K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
1K
I Understanding the Taylor Expansion of a Translated Function
    • Calculus
Replies
3
Views
4K
I Taylor expansion of an unknown function
    • Linear and Abstract Algebra
Replies
5
Views
2K
B Why do I not understand what seems to be basic Calculus?
    • Calculus
Replies
11
Views
2K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
    • General Math
Replies
3
Views
909

Hot Threads

Recent Insights

Back
Top