Step 1

\(\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}-\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}\)

Add \(\displaystyle\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}\) to both sides,

\(\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}+\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

\(\displaystyle\int\int{\left({f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

Step 2

Using the product rule for derivatives,

we can write \(\displaystyle{f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}={f}\nabla.{\left(\nabla{g}\right)}+\nabla{f}.\nabla{g}=\nabla.{\left({f}\nabla{g}\right)}\)

\(\displaystyle\int\int_{{D}}\nabla.{f{{\left(\nabla{g}\right)}}}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

\(\displaystyle\int\int_{{D}}\div{\left({f}\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

\(\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}-\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}\)

Add \(\displaystyle\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}\) to both sides,

\(\displaystyle\int\int_{{D}}{f}\nabla^{{2}}{g}{d}{A}+\int\int_{{D}}\nabla{f}.\nabla{g}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

\(\displaystyle\int\int{\left({f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

Step 2

Using the product rule for derivatives,

we can write \(\displaystyle{f}\nabla^{{2}}{g}+\nabla{f}.\nabla{g}={f}\nabla.{\left(\nabla{g}\right)}+\nabla{f}.\nabla{g}=\nabla.{\left({f}\nabla{g}\right)}\)

\(\displaystyle\int\int_{{D}}\nabla.{f{{\left(\nabla{g}\right)}}}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)

\(\displaystyle\int\int_{{D}}\div{\left({f}\nabla{g}\right)}{d}{A}=\oint_{{C}}{f{{\left(\nabla{g}\right)}}}.{n}{d}{s}\)