Curl and Divergence

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator ($\nabla$). The four most important differential operators are gradient ($\nabla f$), curl ($\nabla \times \mathbf{F}$), divergence ($\nabla \cdot \mathbf{F}$), and Laplacian ($\nabla^2 f = \nabla \cdot \nabla f$) .

Four Most Important Differential Operators

Gradient, curl, divergence, and Laplacian are four most important differential operators.

Curl

The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that field is represented by a vector. The attributes of this vector—length and direction—characterize the rotation at that point.

The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

The curl of a vector field $\mathbf{F}$, denoted by $\nabla \times \mathbf{F}$, is defined at a point in terms of its projection onto various lines through the point. If $\hat{\mathbf{n}}$ is any unit vector, the projection of the curl of $\mathbf{F}$ onto $\hat{\mathbf{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{\mathbf{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

Curl is defined by:

$\displaystyle{(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_{C} \mathbf{F} \cdot d\mathbf{r}\right)}$

Divergence

Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

In physical terms, the divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. It is a local measure of its "outgoingness"—the extent to which there is more exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point, then there must be a source or sink at that position. (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink, and so on.)

More rigorously, the divergence of a vector field $\mathbf{F}$ at a point $p$ is defined as the limit of the net flow of $\mathbf{F}$ across the smooth boundary of a three-dimensional region $V$ divided by the volume of $V$ as $V$ shrinks to $p$. Formally:

$\displaystyle{\nabla \cdot \mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} \frac{\mathbf{F}\cdot\mathbf{n}}{\left|V\right| } dS}$

where $\left|V\right|$ is the volume of $V$, $S(V)$ is the boundary of $V$, and the integral is a surface integral with n being the outward unit normal to that surface.