Vector Field Plotter
Visualize 2D and 3D vector fields F(x,y,z) with dynamic arrows.
Define Vector Components
Note: Use x, y (and z in 3D). Common patterns:
Rotation: P=-y, Q=x
Explosion: P=x, Q=y
Legend
Field Visualization
Drag to Rotate (3D only)What is a Vector Field?
A Vector Field assigns a vector (an arrow with magnitude and direction) to every point in space. It’s the mathematical way we describe things that are “flowing” or exerting force everywhere at once.
Real World Examples
- Wind Map: At every city, wind has a speed (length) and a direction (North/South).
- Gravity: Near Earth, gravity pulls everything toward the center. The force is a vector field.
- Fluid Flow: Water in a river flows at different speeds and directions around rocks.
The Notation
We write a vector field F as a function of position:
F(x,y) = P(x,y)i + Q(x,y)j
P determines the Left/Right strength.
Q determines the Up/Down strength.
Visualizing the Math
When you look at the plot, notice two things:
1. Direction: Where are the arrows pointing? Do they swirl (curl) or flow out from a center (divergence)?
2. Length/Color: Longer or redder arrows mean the force or speed is stronger at that point.
Frequently Asked Questions (FAQ)
Q: What is a “Source” or “Sink”?
A: If all arrows point away from a point, it’s a Source (positive divergence). If they point inward, it’s a Sink (negative divergence). Think of a faucet vs. a drain.
Q: What is a Conservative Field?
A: A field is conservative if it is the gradient of a potential function (like Gravity). In these fields, moving in a closed loop requires zero net work.
Q: Why do 3D fields look complicated?
A: 3D fields add a ‘z’ component (Up/Down). Visualizing them on a 2D screen requires projecting 3D space, which can make arrows overlap. Rotating the view helps you see the true structure.