A scalar field is described by a function that takes the coordinates of the points of a space and returns a single numerivcal value. That vale could be the gravitational potential, the height above some level, the electrostatic potential or something more exotic.
The value of the function will change from point to point, and the change in the funvtion at some neighbouring point will depend on both the original and the neighbouring point. To take account of this every scalar field has an associated vector field which expresses the rate of change of the scalar field in each direction.
A sclar field defined by a function
has an associated vector field
In the above expression
is an operatore:
Example
At
