Calculus
Scalar Fields & Gradients
A scalar field assigns a number to every point in space, shown here as a 3D density cloud. Drag the small probe through it: an arrow points the way the field grows fastest, and its gradient magnitude is shown alongside.
Density here—
Gradient |∇f|—
Drag the cube to move the probe; drag the background to orbit; scroll to move it in depth. The arrow points toward the steepest increase, longer where the field changes faster. Shrink the cube to zoom into the local derivative.
What to observe
- Move the probe into a uniform region of the cloud: the arrow shrinks because the field is nearly flat, so |∇f| → 0.
- Sit on the edge of a bright core. The gradient pointsinto the density, always toward steepest increase, never along the surface.
- Shrink the probe. The arrow direction barely changes: you are looking at the local derivative, the limit as the sampling box shrinks.
Shortcuts: space run/pause · s step · r reset · f fullscreen