🧭 Gradient, Divergence & Curl

📖 The Physics

The operator ∇ ("del") = (∂/∂x, ∂/∂y, ∂/∂z) combines with a field in three ways. Which one you use depends on whether the field is a scalar (like temperature) or a vector (like flow velocity).

Gradient (scalar → vector):

Divergence (vector → scalar):

Curl (vector → vector):

Gradient of a scalar field points in the direction of steepest increase, and its length is how steep that slope is — here every arrow points toward the top of the hill. Divergence measures how much a vector field spreads out of a point: positive is a source, negative a sink. Curl measures how much it rotates about a point.

For the radial field F = a(x, y), ∇·F = 2a everywhere and its curl is zero. For the swirl F = a(−y, x), ∇×F = 2a ẑ and its divergence is zero — a neat pair of opposites.