The explanation of the domain of two variable function

[sayed5555]

can anyone explain that domain or give me any reference about it??

 

[HallsofIvy]

Do you know what the domain of a single variable function is? There really isn't any great difference!

The "domain" of a function, of any number of variables, is the set of all values of those variables for which the function is defined. Sometimes that is given as part of the definition of the function. For example, I can define "$f(x,y)= x^2- y^3$ for all positive x and y[/itex]". In that case, the domain is exactly as stated: all positive x and y: the first quadrant of R²; $\{(x, y)|x> 0, y> 0\}$.

Often, a function is "defined" simply by a formula, in which case the domain is the "natural domain", all values of the variables for which the formula can be calculated. If I just said, "$f(x,y)= x^2- y^3$" since we can square, cube, and subtract all numbers, there is no restriction- its natural domain is all of R²: all pairs (x, y).

But if I define $f(x,y)= 1/(x+ y)$, I cannot divide by 0 so x+ y cannot equal 0. That means that y cannot equal -x: The domain is all (x, y) such that $y\ne -x$, all of R² except the line y= -x.

Similarly, if I define $f(x,y)= \sqrt{x+ y}$, now I cannot take the square root of a negative number so x+ y cannot be negative. The domain is all (x, y) such that $x+y\ge 0$. That would be all of the points in R² on or above and to the right of the line y= -x.

 

[sayed5555]

thank you indeed
i would be grateful if you draw it or give me simples in papers