Let D be a set in 
. A vector field on is a function
that assigns to each point (x,y) in D a two-dimensional vector
.
Let E be a set in 
. A vector field on is a function
that assigns to each point (x,y,z) in E a three-dimensional
vector 
.
A gradient field is a field derived as the gradient of a scalar
function f. Such a vector field is called conservative, and
f is called a potential function for .
Vector fields are all around us: any phenomenon which associates with a point a vector can be represented using the concept of a vector field. The wind at evey point; the temperature gradient in a room (an example of a gradient field, derived from the scalar temperature function at each point); the velocity of a spatially-defined epidemic (like my raccoon rabies problem).
The gradient fields will have special properties, as we will see, and are extremely important in physics. Gravition and Electric charge are examples of these, and have the special property that they share of being ``inverse distance square'' laws.