What's the geometrical interpretation of the magnitude of gradient generally? In the following picture, the author of the _Field and Wave Electromagnetics_ shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum. !alt text But what about the geometrical interpretation of the **magnitude** of gradient **generally** or maybe is there a **geometrical interpreation** of the magnitude of graident? thanks.

Here's a geometric way of thinking that might be helpful: Consider a family of level surfaces $f(x,y,z)=C$ for some evenly spaced values of $C$ (where the spacing should be fairly small). These level surfaces will lie closely stacked in space near points where $|\
abla f|$ is large, and farther apart near points where $|\
abla f|$ is small.

(The two-dimensional counterpart is curves of constant elevation on a map; they are densely packed where the slope of the terrain is steep.)