VC5.59

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Given surface is \phi (x,y,z)={\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}+{\frac  {z^{2}}{c^{2}}}-1=0\,

From (1),we have

\nabla \phi =[i{\frac  {\partial }{\partial x}}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}]({\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}+{\frac  {z^{2}}{c^{2}}}-1)\,

=2[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]\,

Therefore,n=unit normal vector={\frac  {2[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]}{{\sqrt  {({\frac  {4x^{2}}{a^{4}}}+{\frac  {4y^{2}}{b^{4}}}+{\frac  {4z^{2}}{c^{4}}})}}}}\,

={\frac  {{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k}{{\sqrt  {{\frac  {x^{2}}{a^{4}}}+{\frac  {y^{2}}{b^{4}}}+{\frac  {z^{2}}{c^{4}}}}}}}\, --(2)

Now,equation of the tangent plane to (1) at(x,y,z) is

{\frac  {xX}{a^{2}}}+{\frac  {yY}{b^{2}}}+{\frac  {zZ}{c^{2}}}=1\, --(3) where (X,Y,Z) are the coordinates of current point on the tangent plane (3)

Therefore,p=length of the perpendicular drawn from (0,0) to plane (3) or

p={\frac  {1}{{\sqrt  {{\frac  {x^{2}}{a^{4}}}+{\frac  {y^{2}}{b^{4}}}+{\frac  {z^{2}}{c^{4}}}}}}}\, --(4)

From (2) and (3)n=p[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]\, --(5)

Therefore,we have {\frac  {1}{p}}=[{\frac  {1}{p}}n]\cdot n=[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]\cdot n\, --(6)

Now,\iint _{S}{\frac  {1}{p}}dS=\iint _{S}[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]\cdot ndS\, using (6).

=\iiint _{V}{\mathrm  {div}}[{\frac  {x}{a^{2}}}i+{\frac  {y}{b^{2}}}j+{\frac  {z}{c^{2}}}k]dV\,

=\iiint _{V}[{\frac  {\partial }{\partial x}}({\frac  {x}{a^{2}}})+{\frac  {\partial }{\partial y}}({\frac  {y}{b^{2}}})+{\frac  {\partial }{\partial z}}({\frac  {z}{c^{2}}})]dV\,

=[{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}+{\frac  {1}{c^{2}}}]\iiint _{V}dV\,

=[{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}+{\frac  {1}{c^{2}}}]\times (volumeoftheellipsoid)\,

=[{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}+{\frac  {1}{c^{2}}}][{\frac  {4}{3}}\pi abc]\,

={\frac  {4\pi }{3}}{\frac  {a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{abc}}\,

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