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

The gradient, the divergence and the curl are the different uses of nabla, and each describes a different local property of a field. The symbol \(\nabla\) is a **differential operator** rather than the name of a single operation; in Cartesian coordinates its components are spatial partial derivatives.

The gradient, the divergence and the curl are the different uses of nabla, and each describes a different local property of a field. The symbol ∇\nabla∇ is a differential operator rather than 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. Its components are the partial derivatives of the field, one per input, and for a loss over parameters its negative is the direction gradient descent steps along, like a hiker in fog who feels only the slope under the boots. 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. They are different objects that share an initial symbol.

Divergence measures net expansion, which is different from 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.

So 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).

The curl is read only once its orientation is declared. In the xyxyxy plane, it is

(∇×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).

fieldv = (y, 0) div v0 + 0 = 0 curl v0 − 1 = −1 probe at (0.8, 0.6): outflow per area0.00 The divergence is what flows out of a small circle divided by its area; the curl is what flows around it divided by its area. A probe circle dragged over the field reads both.

Drag the probe circle over each field: in the saddle the arrows bend but the paddle never turns, in the shear it turns although every arrow is parallel, and over the bump the outflow changes sign from the peak to the far slopes.

The continuous Laplacian is the divergence of the gradient, ΔT=∇⋅∇T\Delta T=\nabla\cdot\nabla TΔT=∇⋅∇T, and it has a network counterpart. 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.

The orientation convention follows OpenStax, Calculus Volume 3, section 6.5, Divergence and Curl