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 16.1 : Vector Fields
4. Compute the gradient vector field for \(\displaystyle f\left( {x,y,z} \right) = {z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \ln \left( {\frac{{xy}}{z}} \right)\).
Show SolutionThere really isn’t a lot to do for this problem. Here is the gradient vector field for this function.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\nabla f = \left\langle {2x{z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \frac{1}{x},4{z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \frac{1}{y},2z{{\bf{e}}^{{x^{\,2}} + 4y}} - \frac{1}{z}} \right\rangle }}\]Don’t forget to compute partial derivatives for each of these! The first term is the derivative of the function with respect to \(x\), the second term is the derivative of the function with respect to \(y\) and the third term is the derivative of the function with respect to \(z\).