Question

Find the domain 1.f(x,y,z)= z/(x^2-y^2)
2. f(x,y)= x sin(1/x)+ y sin(1/y)
3. f(x,y,z)= 1/(√ (4-x^2-y^2-z^2

Answer 1

1.$f(x,y,z)= \frac{z}{x^2-y^2}$

$dom(f)=\{(x,y,z)\in \R^3\ | \ x^2-y^2\ne 0\}=\{(x,y,z)\in \R^3\ | \ (x-y)(x+y)\ne 0\}= \R^3\setminus\{(x,x,z),(x,-x,z))\ |\ x,z\in\R\}.$

2. $f(x,y)= x \sin(\frac{1}{x})+ y \sin(\frac{1}{y})$

$dom(f)=\{(x,y)\in\R^2\ |\ x\ne 0, y\ne 0\}=\R^2\setminus\{(x,0),(0,y)\ |\ x,y\in\R\}.$

3. $f(x,y,z)= \frac{1}{\sqrt{4-x^2-y^2-z^2}}$

$dom(f)=\{(x,y,z)\in\R^3\ |\ 4-x^2-y^2-z^2>0\}=\{(x,y,z)\in\R^3\ |\ x^2+y^2+z^2<4\}.$