Basic Definitions

Two main types of functions #

$$f(x,y,z)=k$$

A value is associated for every point in space.

$$f(x,y,z)=<.,.,.>$$

A vector is associated with every point in space.

Let f(x,y,z) be a

Gradient operator $$ \vec{\nabla}:<\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}> $$

Gradient of f:

$$ \vec{\nabla}:<\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}> $$

Let $r=\phi(x,y,z)$ be a scalar point function.

By definition, $$\nabla \phi \cdot \vec{dr}=\delta r$$

If we nudge along the surface, δr=0

So, $\nabla \phi \cdot \vec{dr}=0$

∴ $\nabla \phi$ is perpendicular and hence, it’s normal to the surface