This java applet demonstrates various properties of vector fields. You can select from a number of vector fields and see how particles move in the field if it is treated as either a velocity or a force field. This helps you visualize the field. Also you can see the values of the divergence and curl of the field.

When you start the applet, you will see 500 particles moving in the "1/r single line" field, which is a field that attracts particles to the center. By default the particles are treating the field as a velocity field, which means that the field vectors determine how fast the particles are moving and in what direction. In this case, the particles just move toward the center and fall in. The velocity of all the particles at a certain point on the grid is always the same. If the field is treated as a force field, then the field vectors determine the acceleration of the particles, but their velocity may vary depending on where they started.

If the vector field is a gradient field then you will see the particles moving on a surface determined by the potential function associated with the field. They will roll downhill on the surface.

If the vector field is not a gradient field, then you will just see a 2-dimensional view of the particles moving around in a grid.

The Setup popup will allow you to select a vector field. Here is a list of possible vector fields.

The Color popup will allow you to select what the color of the floor means. Generally the floor color will indicate the value of some quantity at that point, where green means positive, gray means zero, and red means negative. Here are the possibilities:

• Field Magnitude means the floor indicates the strength of the field. It will be green if the field is strong and gray if it is weak or not present.
• Potential means the floor indicates the value of the potential function, if any.
• None means the floor is gray.
• Divergence means the floor indicates the divergence of the field.
• Curl Z means the floor indicates the Z component of the field's curl. (The curl can't have any other components because the field vectors are always in the x-y plane, and the curl must be perpendicular to the field vectors.) Green means positive curl (counter-clockwise rotation) and red means negative curl (clockwise).

The Floor popup allows you to select whether grid lines, equipotentials, or streamlines are drawn on the floor, or none of the above.

The Flat View checkbox allows you to select a two-dimensional view of the vector field rather than showing the particles on a potential surface. This is done automatically when viewing vector fields which are not gradient fields, when viewing field vectors or curl detectors, or when taking a line or surface integral.

The Display popup will allow you to select how the particles will move:

• Display: Particles (Vel.) means particles will move through the field, with the field vectors determining their velocity.
• Display: Particles (Force) means particles will move through the field, and the field vectors determine their acceleration.
• Display: Field Vectors shows you the field vectors at an array of locations.
• Display: None doesn't show any particles or field vectors.
• Display: Curl Detectors means the field will be treated as a velocity field, but the particles will be little plus signs instead of dots, so you can see if the field is making them rotate. This can be used to determine if the vector field has curl.

The Mouse popup controls what happens when you click on the view. If you set it to Adjust Angle or Adjust Zoom, you can adjust the orientation or size of the 3-d view by clicking and dragging.

If you select Line Integral from the mouse popup, you can perform a line integral by clicking and dragging out a rectangular area. This will show the circulation around that area. It should be proportional to the total curl inside, by Green's theorem.

If you select Surface Integral from the mouse popup, you can perform a surface integral by clicking and dragging out a rectangular area. This will show the flux through the bounds of that area. It should be proportional to the total divergence inside, by the divergence theorem.

The Stopped checkbox will stop the particles.

The Reverse checkbox will reverse the direction of all the field vectors.

The Reset button can be used to reset the positions of all the particles to random values.

The Kick button can be used to give all the particles a random acceleration in some direction. This is only allowed if the particle movement is set to "Force". It can be useful if none of the particles are moving, or if they are all moving in the same direction.

The Field Strength slider makes the field stronger or weaker, and also adjusts the brightness of the field vectors if you have Display: Field Vectors selected.

The Vector Density slider controls the number of vectors present if you have Display: Field Vectors selected.

The Number of Particles slider allows you to reduce the number of particles, which can be useful if you want to watch the behavior of just a few of them. Also it might speed things up if you have fewer particles.

A few additional field-specific sliders may be present, depending on the field you have selected.