Understanding Taylor's Theorem w/ Two Variables

In summary, the conversation is about understanding Taylor series proof with one variable and how it works with two variables. The person asking the question has a basic understanding of the proof and wants to know if it is similar for two variables. The person answering confirms that it is essentially the same and explains the use of the symbol $D^\alpha f(\mathbb a)$ for different values of $\alpha$.
  • #1
Petrus
702
0
Hello MHB,
I understand taylor series proof with one variable but how does it work with Two variabels? is it pretty much the same? The one I understand is

Taylor's theorem - Wikipedia, the free encyclopedia
Go to proofs Then it's the one under "Derivation for the integral form of the remainder"

Regards,
\(\displaystyle |\pi\rangle\)
 
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  • #2
Petrus said:
Hello MHB,
I understand taylor series proof with one variable but how does it work with Two variabels? is it pretty much the same? The one I understand is

Taylor's theorem - Wikipedia, the free encyclopedia
Go to proofs Then it's the one under "Derivation for the integral form of the remainder"

Regards,
\(\displaystyle |\pi\rangle\)

Yep. It's pretty much the same.
Note that the symbol $D^\alpha f(\mathbb a)$ is a vector for $\alpha = 1$, a matrix for $\alpha = 2$, a 3-dimensional matrix for $\alpha = 3$, and so on.
 

FAQ: Understanding Taylor's Theorem w/ Two Variables

1. What is Taylor's theorem with two variables?

Taylor's theorem with two variables is a mathematical tool used to approximate a multivariable function with a polynomial. It is an extension of Taylor's theorem for one variable and is useful in analyzing functions of multiple variables.

2. How is Taylor's theorem with two variables different from the one-variable version?

Taylor's theorem with two variables is different from the one-variable version in that it takes into account the partial derivatives of the function with respect to each variable. This means that the polynomial approximation will have terms for each variable instead of just one.

3. What is the purpose of using Taylor's theorem with two variables?

The purpose of using Taylor's theorem with two variables is to approximate a multivariable function with a simpler polynomial form. This can make it easier to analyze and understand the behavior of the function, and can also be used to make predictions or estimate values for the function at certain points.

4. How is Taylor's theorem with two variables used in real-world applications?

Taylor's theorem with two variables has many applications in fields such as physics, engineering, and economics. It is used to approximate the behavior of complex systems and to make predictions about their future behavior. For example, it can be used to approximate the motion of a projectile or the value of a stock over time.

5. Are there any limitations to using Taylor's theorem with two variables?

Yes, there are limitations to using Taylor's theorem with two variables. The polynomial approximation will only be accurate within a certain range of values, and the accuracy will decrease as the distance from the point of approximation increases. Additionally, the accuracy of the approximation depends on the smoothness of the function and the number of terms in the polynomial used for the approximation.

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