Using the divergence theorem, evaluate `int_S F * hatn dS `  where `F = 9x*i + y*cosh^(2)(x)*j - z*sinh^(2)(x)*k` and `S` is the ellipsoid `4*x^(2) + y^(2) + 9*z^(2) = 36.` Sketch the...

Using the divergence theorem, evaluate `int_S F * hatn dS `  where

`F = 9x*i + y*cosh^(2)(x)*j - z*sinh^(2)(x)*k` and

`S` is the ellipsoid `4*x^(2) + y^(2) + 9*z^(2) = 36.`

Sketch the ellipsoid using matlab


(The volume of an ellipsoid is  `4/(3)*pi*abc` )

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

The divergence theorem states that `int_S (F*hatn) dS = int_V (grad*F) dV,` where `S` is a closed surface, `V` is the volume inside it and `F` is a good enough vector field defined inside `S` and on it. The symbol `grad` denotes the divergence operator.

In our case `S` is an ellipsoid, a smooth closed surface, the vector field `F` is defined everywhere and is infinitely differentiable. The divergence of `F` is

`(del/(del x), del/(del y), del/(del z))*(F_x, F_y, F_z) = del/(del x) F_x + del/(del y) F_y + del/(del z) F_z`

where `F_x, F_y, F_z` are the components of `F.`

In our case  `F_x=9x, F_y=y*cosh^2(x), F_z = z*sinh^2(x),`  thus

`grad F = 9 + cosh^2(x) - sinh^2(x) = 9` and the integral becomes a very simple one:

`int_S (F*hatn) dS = int_V (grad*F) dV = int_V (9) dV = 9*vol(V) = 12 pi abc.`

The equation of our ellipsoid is equivalent to  `x^2/3^2 + y^2/6^2 + z^2/2^2 = 1,`  therefore the semiaxes are `3,` `6` and `2` and the final answer is  `12 pi abc = 12*3*6*2 pi = 432 pi.`

 

I used WolframAlpha to sketch the ellipsoid.

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