VC5.33

From Example Problems
Jump to: navigation, search

Let V be the volume enclosed by S. By definition,we get

{\mathrm  {div}}F={\frac  {\partial }{\partial x}}(4xz)+{\frac  {\partial }{\partial y}}(-y^{2})+{\frac  {\partial }{\partial z}}(yz)\,

=4z-2y+y=4z-y\, --(1)

Hence by the divergence theorm,we have

\iint _{S}F\cdot ndS=\iiint _{V}{\mathrm  {div}}FdV=\iiint _{V}(4z-y)dxdydz\,,by using (1)

=\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1}}\int _{{z=0}}^{{1}}(4z-y)dxdydz\,

=\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1}}[2z^{2}-yz]_{0}^{1}dxdy\,

=\int _{0}^{1}\int _{0}^{1}(2-y)dxdy=\int _{0}^{1}[2y-{\frac  {y^{2}}{2}}]_{0}^{1}dx\,

=\int _{0}^{1}[2-{\frac  {1}{2}}dx={\frac  {3}{2}}\int _{0}^{1}dx={\frac  {3}{2}}\,

Main Page