The geometry of fields is discussed on this page here we discuss how to represent fields using multidimensional algebra types.
A vector field has a vector value for every point in a space:
For instance, if we have a very large mass at the origin, then we can plot the gravitational field, as in the diagram above. Its difficult to find a suitable way to show this, especially in 3 dimensions, an alternative might be to plot lines of constant field force.
The above situation can be shown as a scalar field. That on the following diagram the height (y-dimension) shows the scalar value.
We can always get a scalar field from a vector field by using the divergence as shown on this page.