VC4.34

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i).\nabla \times F={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\2x^{2}-3z&-2xy&-4x\end{vmatrix}}=j-2yk\, on simplification

Take a column parallel to z-axis as the elementary volume. This cuts the boundary at z=0 and z=4-2x-2y.Also the projection of 2x+2y+z=4 on xy-plane is bounded by x=0,y=0 and x+y=0. So,limits of y are from 0 to 2-x and the limits of x are from 0 to 2.

Here dV=dx dy dz adn hence we get

\iiint _{V}\nabla \times FdV\,

=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{2-x}}\int _{{z=0}}^{{4-2x-2y}}(j-2yk)dxdydz\,

=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{2-x}}(j-2yk)[z]_{{0}}^{{4-2x-2y}}dxdy\,

=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{2-x}}(j-2yk)(4-2x-2y)dxdy\,

=\int _{{x=0}}^{{2}}[(4y-2xy-y^{2})j-2{2y^{2}-xy^{2}-{\frac  {2}{3}}y^{3}}k]_{{0}}^{{2-x}}dx\,

=\int _{{x=0}}^{{2}}[(2-x^{2})j-{\frac  {2}{3}}(2-x)^{3}k]dx={\frac  {8}{3}}(j-k)\,

ii). \nabla \cdot F=[{\frac  {\partial }{\partial x}}i+{\frac  {\partial }{\partial y}}j+{\frac  {\partial }{\partial z}}]\cdot ((2x^{2}-3z)i-2xyj-4xk)\,

={\frac  {\partial }{\partial x}}(2x^{2}-3z)+{\frac  {\partial }{\partial y}}(-2xy)+{\frac  {\partial }{\partial z}}(-4x)=2x\,

Therefore,\iiint _{V}\nabla \cdot F=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{2-x}}\int _{{z=0}}^{{4-2x-2y}}2xdxdydz\,

=2\int _{0}^{2}\int _{{0}}^{{2-x}}x[z]_{{0}}^{{4-2x-2y}}dxdy\,

=2\int _{0}^{2}\int _{{0}}^{{2-x}}(4x-2x^{2}-2xy)dxdy\,

=2\int _{0}^{2}(2x^{3}-8x^{2}+8x)dx={\frac  {8}{3}}\,

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