Taylor series with two variables

• JasonHathaway
In summary, the conversation discusses the correct equations for second-order and third-order total derivatives, including the notation for partial derivatives and the inclusion of the terms dxdy. It is noted that the partial derivative \frac{\partial^2 f}{\partial x\partial y} may not always be the same as \frac{\partial^2 f}{\partial y\partial x}.
JasonHathaway
Hi everyone,

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The Attempt at a Solution

It is correct so far.

ehild

So the equations of d^2f and d^2f are correct? What are they called? (Total derivatives?)

They are second-order and third-order total differentials. And you should write partial derivatives $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ instead df/dx and df/dy, and so on... And the partial derivative $\frac{\partial^2 f}{\partial x\partial y}$ are not always the same as $\frac{\partial^2 f}{\partial y\partial x}$.

And you omitted dxdy from the second term of d^2 f

ehild

Last edited:

1. What is a Taylor series with two variables?

A Taylor series with two variables is a mathematical representation of a function using an infinite sum of terms. It is used to approximate the value of a function at a specific point by considering the function's derivatives at that point. Unlike a Taylor series with one variable, which only considers the derivatives with respect to one variable, a Taylor series with two variables takes into account the derivatives with respect to both variables.

2. How is a Taylor series with two variables calculated?

A Taylor series with two variables is calculated using the following formula:
f(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) + (1/2!)[fxx(a,b)(x-a)2 + 2fxy(a,b)(x-a)(y-b) + fyy(a,b)(y-b)2] + ...
where f(x,y) is the function, (a,b) is the point of approximation, and fx, fy, fxx, fxy, fyy, etc. are the partial derivatives of the function at (a,b). The number of terms in the series depends on the desired level of accuracy.

3. What is the purpose of using a Taylor series with two variables?

A Taylor series with two variables is used to approximate the value of a function at a specific point. It can be useful in situations where the function is difficult to evaluate directly, or when only the derivatives of the function are known. Additionally, by including more terms in the series, a more accurate approximation of the function can be obtained.

4. Are there any limitations to using a Taylor series with two variables?

One limitation of using a Taylor series with two variables is that it only provides an approximation of the function at a specific point. It may not accurately represent the behavior of the function in other areas. Additionally, the accuracy of the approximation depends on the choice of the point of approximation and the number of terms included in the series.

5. How is a Taylor series with two variables used in real-world applications?

A Taylor series with two variables is used in various fields of science and engineering, such as physics, economics, and computer science. It can be used to approximate the behavior of complex systems and to make predictions based on known data. For example, in physics, a Taylor series with two variables can be used to approximate the motion of a particle in a magnetic field, while in economics, it can be used to model the relationship between two variables, such as supply and demand.

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