Definition
A scalar field is a function in -dimensional space, , which maps a scalar value onto every point in that space. Unlike a vector field, a scalar field assigns no direction to each point.
A trivially simple scalar field in 2D space might be , where is some constant real number, thus giving a scalar field that is uniform everywhere. A slightly more complex one might be , giving a field that changes linearly along the x-axis.
Wikipedia gives a good visualization of a scalar field in 2D space:
