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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 \(z\).