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Let the equation of an ellipsoid be:

$$2x^2+y^2+2z^2=5$$

And the point on the surface of ellipsoid be:

$$(1,1,1)$$

How to compute the normal to the ellipsoid at the point on the surface of ellipsoid?

I read some article says that I can compute it by gradient, but I am not sure how to do it...

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If $c$ is a regular value of a smooth function $f$, then $S=f^{-1}(c)$ is a surface for which $\nabla f(p)$ is normal to $T_pS$. Here $f(x,y,z) = 2x^2+y^2+2z^2$ and $p = (1,1,1)$, so the normal to the ellipsoid at $p$ is $$\nabla f(1,1,1) = (4x,2y,4z)\big|_{x=y=z=1} = (4,2,4).$$

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    $\begingroup$ just want to make sure. if the equation is ax^2 + by^2 + cz^2 and p = (d,e,f), the normal to the ellipsoid at p will be (2ad, 2be, 2cf). Is this correct? $\endgroup$ – Mars Lee Sep 15 '16 at 1:41
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    $\begingroup$ Yup, you got it. $\endgroup$ – Ivo Terek Sep 15 '16 at 1:47

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