Finding the interval of which a multivariable function is defined Find the interval in which f(x,y,z)=z+ln(1−x^2−y^2) is defined So all that is need to to check for which values ln(1-x^2-y^2)>=0 That mean 1-x^2-y^2>= 1 x^2+y^2>= 0 But how do I find the specific x terms and y terms?

Goundoubuf

Goundoubuf

Answered question

2022-11-24

Finding the interval of which a multivariable function is defined
Find the interval in which f ( x , y , z ) = z + l n ( 1 x 2 y 2 ) is defined
So all that is need to to check for which values l n ( 1 x 2 y 2 ) 0
That mean 1 x 2 y 2 1 x 2 + y 2 0
But how do I find the specific x terms and y terms?
Or is x 2 + y 2 0 is sufficient?

Answer & Explanation

Lilyana Simon

Lilyana Simon

Beginner2022-11-25Added 7 answers

You are only partly right: if the only thing that you want is for f to be defined, then ln must be defined, which means that its argument must be > 0. In formulae: 1 x 2 y 2 > 0, which means x 2 + y 2 < 1, which is the interior of the circle of radius 1 and center ( 0 , 0 ). There is absolutely no reason why the logarithm should be positive, as you say.
Urijah Zhang

Urijah Zhang

Beginner2022-11-26Added 2 answers

it is sufficient, you don't need separate limits on x and y

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