What Does the Norm of a Jacobian Matrix Represent?

[Buri]

In "Differential Equations, Dynamical Systems and Introduction to Chaos", the norm of the Jacobian matrix is defined to be:

|DF_x|
= sup |DF_x (U)|, where U is in R^n and F: R^n -> R^n and the |U| = 1 is under the sup.
...|U| = 1

DF_x (U) is the directional derivative of F in the direction of U. But I don't understand what this definition means?

Thanks

 

[Mark44]

The supremum is taken over all unit vectors U.

 

[Buri]

Ahh I see, thanks!