Step 1

Given vector field is \(\displaystyle{F}={\left\langle{z}-{x},{x}-{y},{2}{y}-{z}\right\rangle}\).

Compute the derivative of the above vector field.

\(\displaystyle\nabla\cdot{F}=\frac{\partial}{{\partial{x}}}{\left({z}-{x}\right)}+\frac{\partial}{{\partial{y}}}{\left({x}-{y}\right)}+\frac{\partial}{{\partial{z}}}{\left({2}{y}-{z}\right)}\)

=-1-1-1=-3

Step 2

Now calculate the net outward flux.

\(\displaystyle\int\int_{{D}}\int\nabla\cdot{F}{d}{V}=\int\int_{{D}}\int{\left(-{3}\right)}{d}{V}\)

=-3(Volume of a sphere)

\(\displaystyle=-{3}{\left(\frac{{4}}{{3}}\pi{\left({{r}_{{2}}^{{3}}}-{{r}_{{1}}^{{3}}}\right)}\right)}\)

\(\displaystyle=-{4}\pi{\left({4}^{{3}}-{2}^{{3}}\right)}{\left[\because{r}_{{1}}={2},{r}_{{2}}={4}\right]}\)

\(\displaystyle=-{224}\pi\)

Given vector field is \(\displaystyle{F}={\left\langle{z}-{x},{x}-{y},{2}{y}-{z}\right\rangle}\).

Compute the derivative of the above vector field.

\(\displaystyle\nabla\cdot{F}=\frac{\partial}{{\partial{x}}}{\left({z}-{x}\right)}+\frac{\partial}{{\partial{y}}}{\left({x}-{y}\right)}+\frac{\partial}{{\partial{z}}}{\left({2}{y}-{z}\right)}\)

=-1-1-1=-3

Step 2

Now calculate the net outward flux.

\(\displaystyle\int\int_{{D}}\int\nabla\cdot{F}{d}{V}=\int\int_{{D}}\int{\left(-{3}\right)}{d}{V}\)

=-3(Volume of a sphere)

\(\displaystyle=-{3}{\left(\frac{{4}}{{3}}\pi{\left({{r}_{{2}}^{{3}}}-{{r}_{{1}}^{{3}}}\right)}\right)}\)

\(\displaystyle=-{4}\pi{\left({4}^{{3}}-{2}^{{3}}\right)}{\left[\because{r}_{{1}}={2},{r}_{{2}}={4}\right]}\)

\(\displaystyle=-{224}\pi\)