Can I visualize three variable functions through scalar and vector fields?

[LucasGB]

Graphics of one variable functions are two dimensional lines. Graphics of two variable functions are three dimensional surfaces. Three variable functions cannot be plotted.

But can I think of the usual 3D representations of vector and scalar fields as manners of visualizing a three variable function?

 

[Kyouran]

In fact, a scalar field IS a scalar function and likewise a vector field IS a vector function, so a representation of a 3D scalar field is in fact a representation of a 3 variable scalar function.

Any 3D representations of a 3D scalar or vector field on a 2D plane like your computer screen or a sheet of paper cannot be a continuous plot however, only some finite amount of points will be represented like the one in here:

http://upload.wikimedia.org/wikipedia/commons/d/d1/Vector_Field.gif

In the image above, the magnitude of each vector is represented by adding a colour dimension to the 3 spatial dimensions