- #1
abcdefg10645
- 43
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As mentioned in the title~
Could anyone give me a hint or an idea ?
Thanks~
Could anyone give me a hint or an idea ?
Thanks~
Taylor's expansion for multivariable function involves expanding a function with multiple variables around a point, whereas the one-variable case only involves expanding a function around a single point. This means that in the multivariable case, there are more terms in the expansion that involve different combinations of the variables.
The formula for Taylor's expansion for multivariable function is:
f(x,y) = f(a,b) + (x-a)∂f/∂x + (y-b)∂f/∂y + (1/2!)((x-a)^2∂^2f/∂x^2 + 2(x-a)(y-b)∂^2f/∂x∂y + (y-b)^2∂^2f/∂y^2) + ...
The purpose of Taylor's expansion for multivariable function is to approximate a multivariable function with a polynomial function. This allows for easier calculation and understanding of the behavior of a function around a specific point.
To prove Taylor's expansion for multivariable function, we can use the multivariable Taylor's theorem, which states that if a function has continuous partial derivatives up to order n+1 in a neighborhood of a point (a,b), then the function can be expanded around that point using the formula mentioned in question 2. This can be proved using the same techniques as proving one-variable Taylor's theorem, such as using the mean value theorem and induction.
No, Taylor's expansion for multivariable function can only be used for functions that have continuous partial derivatives up to the desired order at a specific point. If a function does not meet this criteria, then the expansion may not provide an accurate approximation of the function.