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
This is a very short section and is here simply to acknowledge that just like we haddifferentials for functions of one variable we also have them for functions of more than one variable. Also, as we’ve already seen in previous sections, when we move up to more than one variable things work pretty much the same, but there are some small differences.
Given the function \(z = f\left( {x,y} \right)\) the differential \(dz\) or \(df\) is given by,
\[dz = {f_x}\,dx + {f_y}\,dy\hspace{0.5in}{\mbox{or}}\hspace{0.5in}df = {f_x}\,dx + {f_y}\,dy\]There is a natural extension to functions of three or more variables. For instance, given the function \(w = g\left( {x,y,z} \right)\) the differential is given by,
\[dw = {g_x}\,dx + {g_y}\,dy + {g_z}\,dz\]Let’s do a couple of quick examples.
There really isn’t a whole lot to these outside of some quick differentiation. Here is the differential for the function.
\[dz = \left( {2x{{\bf{e}}^{{x^2} + {y^2}}}\tan \left( {2x} \right) + 2{{\bf{e}}^{{x^2} + {y^2}}}{{\sec }^2}\left( {2x} \right)} \right)dx + 2y{{\bf{e}}^{{x^2} + {y^2}}}\tan \left( {2x} \right)dy\]Here is the differential for this function.
\[du = \frac{{3{t^2}{r^6}}}{{{s^2}}}dt + \frac{{6{t^3}{r^5}}}{{{s^2}}}dr - \frac{{2{t^3}{r^6}}}{{{s^3}}}ds\]Note that sometimes these differentials are called thetotal differentials.