Step 1

Divergence theorem relates surface integrals and volume integrals. By using the Gauss divergence theorem we can evaluate this surface integral.

The Gauss divergence theorem formula can be stated as follows:

\(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}=\int\int\int_{{V}}\div{F}.{d}{V}\)

\(\displaystyle\div{F}={\left(\partial\frac{{{z}^{{3}}{i}}}{{\partial{x}}}\right)}+{\left(\partial\frac{{-{x}^{{3}}{j}}}{{\partial{y}}}\right)}+{\left(\partial\frac{{{y}^{{3}}{k}}}{{\partial{z}}}\right)}\)

Step 2

div F = 0, since the partial derivative of \(\displaystyle{z}^{{3}}\) with respect to x is zero and partial derivative of \(\displaystyle-{x}^{{3}}\) with respect to y is zero and the partial derivative of \(\displaystyle{y}^{{3}}\) with respect to z is zero.

Then \(\displaystyle\int\int\int_{{V}}{0}.{d}{V}={0}\)

Since \(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}=\int\int\int_{{V}}\div{F}.{d}{V}\)

Then by Gauss Divergence theorem, we can say \(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}={0}\).

Step 3

So the final answer is zero by the Gauss Divergence theorem.

Divergence theorem relates surface integrals and volume integrals. By using the Gauss divergence theorem we can evaluate this surface integral.

The Gauss divergence theorem formula can be stated as follows:

\(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}=\int\int\int_{{V}}\div{F}.{d}{V}\)

\(\displaystyle\div{F}={\left(\partial\frac{{{z}^{{3}}{i}}}{{\partial{x}}}\right)}+{\left(\partial\frac{{-{x}^{{3}}{j}}}{{\partial{y}}}\right)}+{\left(\partial\frac{{{y}^{{3}}{k}}}{{\partial{z}}}\right)}\)

Step 2

div F = 0, since the partial derivative of \(\displaystyle{z}^{{3}}\) with respect to x is zero and partial derivative of \(\displaystyle-{x}^{{3}}\) with respect to y is zero and the partial derivative of \(\displaystyle{y}^{{3}}\) with respect to z is zero.

Then \(\displaystyle\int\int\int_{{V}}{0}.{d}{V}={0}\)

Since \(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}=\int\int\int_{{V}}\div{F}.{d}{V}\)

Then by Gauss Divergence theorem, we can say \(\displaystyle\int\int_{{S}}{F}.{N}{d}{S}={0}\).

Step 3

So the final answer is zero by the Gauss Divergence theorem.