# Odd/even for a multivariable function

• wumple
In summary, the definition of an odd/even function in multiple variables is when the function f(-x,-y)=-f(x,y). In some cases, there may be a family of odd or even functions involving two types of variables, and they are only odd or even in the first set of variables. For example, a family of even functions of x,y, parametrized by z, can be represented as a single function F(x,y;z) such that for all x,y,z we have F(-x,-y;z) = F(x,y;z). However, if the function is simply described as "even", it is expected to be even in all variables present.
wumple
Is the definition of an odd/even function in multiple variables what I would expect it to be, ie

$$f(-x,-y)=-f(x,y)$$

Thanks!

yes that is the standard definition. Sometimes I have a family of odd or even functions however, i.e. the functions involve two kinds of variables, the variables of the functions, and the variables that parametrize the family. Then they are only odd or even in the first set of variables.e.g. a family of even functions of x,y, parametrized by z, might be represented as a single function F(x,y;z) such that for all x,y,z we have F(-x,-y;z) = F(x,y;z).i.e. for each value of z, say z=c, the function F(x,y;c) is even in (x,y).

But if you just say "even", rather than "even in (x,y)", then yes I would expect it to be even in all variables present.

awesome thank you!

## 1. What is a multivariable function?

A multivariable function is a mathematical function that takes multiple input variables and produces a single output. It is also known as a multivariate function or a function of several variables.

## 2. How is odd/even determined for a multivariable function?

Odd/even for a multivariable function is determined by evaluating the function when all the input variables are replaced by their negative values. If the resulting function is equal to the negative of the original function, then the function is odd. If the resulting function is equal to the original function, then the function is even. If neither of these conditions are met, then the function is neither odd nor even.

## 3. What is the significance of determining if a multivariable function is odd or even?

Determining if a multivariable function is odd or even can provide insight into the symmetry of the function. Odd functions have symmetry about the origin, while even functions have symmetry about the y-axis. This information can also be useful in simplifying calculations and solving equations involving the function.

## 4. Can a multivariable function be both odd and even?

No, a multivariable function cannot be both odd and even. By definition, an odd function must have an origin as a point of symmetry, while an even function must have the y-axis as a point of symmetry. These two conditions cannot be met simultaneously.

## 5. How is odd/even used in practical applications?

Odd/even for a multivariable function is used in various fields of science and engineering, such as physics, economics, and statistics. It can help in analyzing and predicting behavior of systems, optimizing processes, and solving complex equations. It is also used in computer programming and data analysis to simplify and speed up calculations.

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