The difference between the termiologies : “domain of the xy plane” and “closed domain of the xy plane”

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I was studying about differential equation. There, I came accross $2$ variable functions. It happened so, that this is the first time, I am working with them explicitly and rigorously. Up until now, I worked only with one-variable functions. Also, all the terminologies related to one-variable function say domain,co-domain,range ,etc , I assumed their analogous definitions for multivariable functions when I encountered them. Say, for a two variable function, $f(x,y)$ I had a notion that the domain of a two variable function is a subset of $\space \Bbb R\times \Bbb R$
( from the idea of a one-variable function). But then, I came accross the termiologies
domain of the xy plane and closed domain of the xy plane. I am having a hard time, in finding out the difference between them. What do they mean when they say so ?

To be really explicit : I encountered this terminology, while studying Lipschitz's condition in the book Differential Equations, (3rd Edition) by S.L Ross , on Page-470, (DEFINITION H).

Best Answer

In the context of two-variable functions, the terms "domain of the $xy$ plane" and "closed domain of the $xy$ plane" refer to the subsets of the Cartesian plane $\mathbb{R}^2$ (or $\mathbb{R} \times \mathbb{R}$) on which the function is defined.

Let's clarify the meaning of these terms:

  1. Domain of the xy plane: This refers to the set of all possible input pairs $(x, y)$ for which the function $f(x, y)$ is defined. This set can be open, closed, or neither. An open set is a set where every point has a neighborhood within the set, and a closed set is a set that contains all its boundary points.
  2. Closed domain of the xy plane: This specifically refers to a domain that is a closed set. In this case, the function $f(x, y)$ is defined for all points $(x, y)$ within the set, including its boundary points.

For example, consider the function $f(x, y) = \frac{1}{x^2 + y^2 - 1}$. The domain of this function is the set of all $(x, y)$ pairs for which $x^2 + y^2 \neq 1$. The domain of the xy plane for this function is $\mathbb{R}^2$ minus the unit circle. This domain is an open set because it does not include the boundary points (the unit circle).

On the other hand, if we have a function defined on a square region, say, $g(x, y) = x^2 + y^2$ for $(x, y) \in [-1, 1] \times [-1, 1]$, the domain is a closed set because it includes the boundary points of the square.

So, the "domain of the $xy$ plane" refers to the set of all input pairs $(x, y)$ for which a two-variable function is defined. A "closed domain of the $xy$ plane" is a specific type of domain where the function is defined on a closed set, meaning that it includes all its boundary points.

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