Paul's Online Notes
Home / Calculus III / Line Integrals / Vector Fields
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 5-1 : 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 Solution

There 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 $$z$$.