I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 12.5 : Functions of Several Variables
3. Find the domain of the following function.
\[f\left( {x,y,z} \right) = \frac{1}{{{x^2} + {y^2} + 4z}}\] Show SolutionThere really isn’t all that much to this problem. We know that we can’t have division by zero so we’ll need to require that whatever \(\left( {x,y,z} \right)\) is it will need to satisfy,
\[{x^2} + {y^2} + 4z \ne 0\]Since this is the only condition we need to meet this is also the domain of the function.
Let’s do a little rewriting on this so we can attempt to identify the domain a little better.
\[4z \ne - {x^2} - {y^2}\hspace{0.5in} \Rightarrow \hspace{0.5in}z \ne - \frac{{{x^2}}}{4} - \frac{{{y^2}}}{4}\]So, it looks like we need to avoid points, \(\left( {x,y,z} \right)\), that are on the elliptic paraboloid given by the equation above.