Question

The equation giving a family of ellipsoids is u = (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) . Find the unit vector normal to each point of the surface of this ellipsoids.

Answer 1

`\hat{n}\ =\ \ \frac{(x)/(a^2)\hat{i}+\ (y)/(b^2)\hat{j}+\ (z)/(c^2)\hat{k}}{\sqrt{((x)/(a^2))^2+((y)/(b^2))^2+((z)/(c^2))^2}}`

Explanation:

Given equation of ellipsoids,

`u\ =\ (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)`

The vector normal to the given equation of ellipsoid will be given by

`\vec{n}\ =\textrm{gradient of u}`

`=\bigtriangledown u`

`=\ (\frac{\partial{}}{\partial{x}}\hat{i}+ \frac{\partial{}}{\partial{y}}\hat{j}+ \frac{\partial{}}{\partial{z}}\hat{k})((x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2))`

`=\ \frac{\partial{((x^2)/(a^2))}}{\partial{x}}\hat{i}+\frac{\partial{((y^2)/(b^2))}}{\partial{y}}\hat{j}+\frac{\partial{((z^2)/(c^2))}}{\partial{z}}\hat{k}`

`=\ (2x)/(a^2)\hat{i}+\ (2y)/(b^2)\hat{j}+\ (2z)/(c^2)\hat{k}`

Hence, the unit normal vector can be given by,

`\hat{n}\ =\ \frac{\vec{n}}{\left|\vec{n}\right|}`

`=\ \frac{(2x)/(a^2)\hat{i}+\ (2y)/(b^2)\hat{j}+\ (2z)/(c^2)\hat{k}}{\sqrt{((2x)/(a^2))^2+((2y)/(b^2))^2+((2z)/(c^2))^2}}`

`=\ \frac{(x)/(a^2)\hat{i}+\ (y)/(b^2)\hat{j}+\ (z)/(c^2)\hat{k}}{\sqrt{((x)/(a^2))^2+((y)/(b^2))^2+((z)/(c^2))^2}}`

Hence, the unit vector normal to each point of the given ellipsoid surface is

`\hat{n}\ =\ \ \frac{(x)/(a^2)\hat{i}+\ (y)/(b^2)\hat{j}+\ (z)/(c^2)\hat{k}}{\sqrt{((x)/(a^2))^2+((y)/(b^2))^2+((z)/(c^2))^2}}`