Taylor expansion of a vector function

In summary, Taylor expansion of a vector function is a mathematical technique used to approximate a vector function using a series of polynomial terms. It is useful for simplifying complex vector functions and can be applied in various fields such as physics and engineering. The steps involved include finding derivatives, evaluating them at a specific point, and constructing the polynomial series. However, this method has limitations as it is only an approximation and may not accurately represent the function in all cases. The accuracy also depends on the order of the polynomial used and may not be applicable to all types of vector functions.
  • #1
xBorisova
4
0
Could someone please explain how does this taylor expansion work:
1/|r-r'| ≈ 1/r+(r.r')/r3

possibly you have to taylor expand twice to get this result, an attempt at which led me nowhere,
surely it cannot be this complicated.

any useful comment about this would be greatly appreciated

thank you :rolleyes:
 
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  • #2
Hello, xBorisova. Welcome to PF!

As a hint, note that for any vector ##\textbf{A}##, ##|\textbf{A}| = \sqrt{\textbf{A}\cdot \textbf{A}}##.
 

What is Taylor expansion of a vector function?

Taylor expansion of a vector function is a mathematical technique used to approximate a vector function using a series of polynomial terms. It is similar to the Taylor series used to approximate a single-variable function, but in this case, the function takes vector inputs and outputs vector values.

Why is Taylor expansion of a vector function useful?

This technique is useful for simplifying complex vector functions into a series of simpler polynomial terms. It can also be used to approximate the behavior of a vector function at a point where the function is not well-defined.

What are the steps involved in Taylor expansion of a vector function?

The steps involved in Taylor expansion of a vector function include finding the derivatives of the function, evaluating those derivatives at a specific point, and then using these values to construct the series of polynomial terms that will approximate the function.

What are some applications of Taylor expansion of a vector function?

Taylor expansion of a vector function is commonly used in physics, engineering, and other fields where vector quantities are involved. It can be used to approximate the motion of objects in space, the behavior of electric and magnetic fields, and many other physical phenomena.

What are the limitations of Taylor expansion of a vector function?

Taylor expansion of a vector function is only an approximation and may not accurately represent the behavior of the function in all cases. The accuracy of the approximation also depends on the order of the polynomial used in the expansion. Additionally, this technique may not be applicable to all types of vector functions, particularly those with discontinuities or singularities.

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