Taylor's theorem for multivariable functions

In summary, the conversation discusses Taylor's equation for a function z=f(x,y) with a constant y value. The total change of z is equal to the sum of two equations, and to get the total change, terms representing x and y changing together are needed, such as ∂²f/∂x∂y.
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To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y
 
  • #3
mathman said:
To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y

Thank you
 

What is Taylor's theorem for multivariable functions?

Taylor's theorem for multivariable functions is a mathematical theorem that allows us to approximate a function with a polynomial of a certain degree. It is an extension of the one-variable Taylor's theorem, which is used to approximate a function with a polynomial in one variable.

How is Taylor's theorem for multivariable functions different from the one-variable Taylor's theorem?

Taylor's theorem for multivariable functions takes into account multiple variables and their derivatives, whereas the one-variable Taylor's theorem only considers the derivatives of one variable. This allows for a more accurate approximation of a multivariable function.

What is the purpose of using Taylor's theorem for multivariable functions?

The main purpose of using Taylor's theorem for multivariable functions is to approximate a complicated function with a simpler polynomial function. This can be useful in many areas of science, such as physics, engineering, and economics, where complex functions are often encountered.

What are the assumptions for using Taylor's theorem for multivariable functions?

The main assumptions for using Taylor's theorem for multivariable functions are that the function must be differentiable up to the desired degree, and the derivatives must exist and be continuous in the region of interest. Additionally, the function must have a finite number of derivatives at each point.

Can Taylor's theorem for multivariable functions be used to find exact values of a function?

No, Taylor's theorem for multivariable functions is an approximation method and cannot give exact values of a function. However, as the degree of the polynomial increases, the approximation gets closer to the actual value of the function.

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