Caption: A diagram of a generic 2-dimensional vector field.

One can mentally use field lines to connect the arrows → in the direction of the vector field (see vector_field_field_lines.html).

Features:

1. In physics, a field is a quantity that has value at every point in space and time or at least some region of space and time.

   Thus, there are a continuum of points with values as opposed to a discrete set of points with values.

2. A field with only a real-number value or some other kind of number value) at every point in space and time is a scalar field. Examples are density, pressure, and temperature.

3. A vector field value has magnitude (the absolute value of a real-number value or some other kind of a number) and a direction at every point in space and time or at least some region of space and time.

   Formally, to be a vector some other properties beyond magnitude and direction are needed---but let's NOT go into all that arcana.

4. How does one represent a vector field in a diagram?

   One standard way is to label a representative set of points with arrows. The tail end of the arrow is set at the point.

   The arrows point in the vector direction and and have length proportional to the vector magnitude.

   Except for the displacement vector, the vector extends in its own abstract space, NOT in space space: e.g., the velocity vector extends in velocity space.

   The vector direction is in space space.

   In a diagram, one sort of superimposes the abstract spaces on the space space.

5. One chooses a scale factor for the arrows for diagramatic convenience: e.g., chose a scale factor (0.1 in/(1 m/s)) and a 10 m/s velocity vector gets an arrow 1 in long.

6. Note one can only use a representative set of arrows. One CANNOT put an arrow at every point. There is an infinite continuum of points.

   One mentally interpolates between the arrows to visualize the whole vector field.

7. The vector field in the diagram could be, for example, the velocity vector field of a fluid flow.

   There is a converging flow along the upper left to lower right diagonal and a diverging flow along the upper right to the lower left diagonal

   The point where the velocity zero is called a stagnation point in fluid dynamics.

8. An altnerative to using representative arrows to represent a vector field is to use field lines.

   A field line is drawn along a path such that at every point the field line points in the direction of the vector at that point.

   The field line is given the direction of the vector.

   One draws a representative set of field lines to represent the vector field and mentally interpolates between them to visualize the rest of the vector field.

   Note that field lines CANNOT cross, except where the vector field goes to zero since a vector can't point two ways, except, in a sense, when it is zero and has no defined direction.

9. Field lines are often the preferred visualization tool since they seem to give a good idea of the overall structure of the vector field.

   Field lines are often used for electric fields and magnetic fields.

   Field lines were, in fact, invented by Michael Faraday (1791-1867) to aid in understanding electric fields and magnetic fields.

10. In fluid dynamics, the field lines are called streamlines.