mathematics//vector calculus//gradient, divergence and curl

Which local properties of a field the different uses of nabla describe. \(\nabla\) is a **differential operator**, not the name of a single operation; in Cartesian coordinates its components are spatial partial derivatives.

Which local properties of a field the different uses of nabla describe. ∇\nabla∇ is a differential operator, not the name of a single operation; in Cartesian coordinates its components are spatial partial derivatives.

The gradient of a scalar field TTT is ∇T\nabla T∇T: the direction of fastest increase in the chosen Euclidean geometry. If TTT is temperature, it has units of temperature per length. The divergence of a vector field vvv is a scalar, ∇⋅v=∑i∂vi/∂xi\nabla\cdot v=\sum_i\partial v_i/\partial x_i∇⋅v=∑i​∂vi​/∂xi​. The curl in three dimensions is the vector ∇×v\nabla\times v∇×v: it measures oriented local circulation. Different objects that share an initial symbol.

Divergence: net expansion, not attraction in every direction. If vvv is a fluid velocity, ∇⋅v\nabla\cdot v∇⋅v is the local expansion or contraction of material volume. In incompressible flow ∇⋅v=0\nabla\cdot v=0∇⋅v=0: the fluid can move, deform and rotate without changing that volume. Positive divergence does not necessarily mean mass creation. Without sources, continuity is

∂ρ∂t+∇⋅(ρv)=0,\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho v)=0,∂t∂ρ​+∇⋅(ρv)=0,

and expansion can come with a drop in density. The interpretation depends on whether the field is a velocity, a mass flux or something else. In a dynamical system x˙=f(x)\dot x=f(x)x˙=f(x), ∇⋅f=tr⁡J\nabla\cdot f=\operatorname{tr}J∇⋅f=trJ. The example f(x,y)=(x,−2y)f(x,y)=(x,-2y)f(x,y)=(x,−2y) has divergence −1-1−1 but the origin is a saddle: it contracts area while expanding in one direction. Stability is not deduced from the trace alone (equilibrium and stability).

Curl: declare the orientation. In the xyxyxy plane,

(∇×v)z=∂vy∂x−∂vx∂y.(\nabla\times v)_z=\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}.(∇×v)z​=∂x∂vy​​−∂y∂vx​​.

With right-handed Cartesian axes, xxx to the right and yyy up, a positive sign is counterclockwise circulation seen from +z+z+z, as in the OpenStax reference below. The field v=(−y,x)v=(-y,x)v=(−y,x) has curl 222; v=(y,0)v=(y,0)v=(y,0) has curl −1-1−1 although its vectors are parallel. The curl of a phase portrait must not be read automatically as physical rotation: its coordinates may be position and velocity, with different units (phase portrait).

Continuous Laplacian and network. ΔT=∇⋅∇T\Delta T=\nabla\cdot\nabla TΔT=∇⋅∇T. With diffusive flux j=−k∇Tj=-k\nabla Tj=−k∇T and constant kkk, a local balance produces a term proportional to ΔT\Delta TΔT: that is the diffusion equation. Discretizing space yields the graph Laplacian, with the sign convention explained there.

The Jacobian of linearization gathers all the sensitivities of a field; the Lyapunov function uses a gradient to compute change along a trajectory.

Orientation reference: OpenStax, Calculus Volume 3, section 6.5, Divergence and Curl