Answer to Question #289802 in Physics for Zara

Question

show that unit vector normal to the surface n= ∇φ ÷ | ∇φ | . find n for the ellipsoid.

φ = a(x^2) + b(y^2) +c(z^2)

Answer 1

Given:

$φ = ax^2 + by^2+cz^2$

The gradient of scalar field is given by

$$\nabla \varphi=\frac{\partial \varphi}{\partial x}\hat i+\frac{\partial \varphi}{\partial y}\hat j+\frac{\partial \varphi}{\partial z}\hat k$$

Hence

$$\nabla \varphi=2ax\hat i+2by\hat j+2cz\hat k$$

$$|\nabla \varphi|=2\sqrt{(ax)^2 +(by)^2+(cz)^2}$$

$${\vec n}=\frac{\nabla \varphi}{|\nabla \varphi|}=\frac{ax\hat i+by\hat j+cz\hat k}{\sqrt{(ax)^2 +(by)^2+(cz)^2}}$$