Curl and Divergence
The four most important differential operators are gradient, curl, divergence, and Laplacian.
Learning Objective

Calculate the direction and the magnitude of the curl, and the magnitude of the divergence
Key Points
 The curl is a vector operator that describes the infinitesimal rotation of a threedimensional vector field.
 The direction of the curl is the axis of rotation, as determined by the righthand rule, and the magnitude of the curl is the magnitude of rotation.
 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.
Term

gradient
of a function
$y = f(x)$ or the graph of such a function, the rate of change of$y$ with respect to$x$ ; that is, the amount by which$y$ changes for a certain (often unit) change in$x$
Full Text
Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (
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 3dimensional 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 righthand 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
Curl is defined by:
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 threedimensional 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
where
Key Term Reference
 area
 Appears in these related concepts: Double Integrals Over Rectangles, Introduction to Circles, and Pedagogical Problem Solving Tasks
 axis
 Appears in these related concepts: Area Between Curves, Regional Terms and Axes, and Components of a Vector
 curl
 Appears in these related concepts: Parametric Surfaces and Surface Integrals, Stokes' Theorem, and Conservative and Nonconservative Forces
 curve
 Appears in these related concepts: Integration By Parts, Arc Length and Surface Area, and Arc Length and Speed
 differential
 Appears in these related concepts: Triple Integrals in Cylindrical Coordinates, Thermal Stresses, and Compensation Differentials
 differentiation
 Appears in these related concepts: WBC Formation, Socioemotional Development in Adolescence, and Considering the Environment
 divergence
 Appears in these related concepts: Functions of Several Variables, The Divergence Theorem, and Growing Global Inequality
 fluid
 Appears in these related concepts: Pumps and the Heart, Physics and Engineering: Fluid Pressure and Force, and Drag
 flux
 Appears in these related concepts: Applications to Economics and Biology, Faraday's Law of Induction and Lenz' Law, and Maxwell's Equations
 infinitesimal
 Appears in these related concepts: Differentials, Area and Distances, and The Fundamental Theorem of Calculus
 integral
 Appears in these related concepts: Indefinite Integrals and the Net Change Theorem, The Correction Factor, and Expected Value
 limit
 Appears in these related concepts: Summing an Infinite Series, Horizontal Asymptotes and Limits at Infinity, and Indeterminate Forms and L'Hôpital's Rule
 line integral
 Appears in these related concepts: Vector Fields, Conservative Vector Fields, and Green's Theorem
 normal
 Appears in these related concepts: Vectors in the Plane, Arc Length and Curvature, and Normal Forces
 scalar
 Appears in these related concepts: VectorValued Functions, Superposition of Electric Potential, and Addition and Subtraction; Scalar Multiplication
 vector
 Appears in these related concepts: Multiplying Vectors by a Scalar, Series and Sigma Notation, and Translations
 vector field
 Appears in these related concepts: Stress and Strain, Surface Integrals of Vector Fields, and Electric Field from a Point Charge
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
 volume
 Appears in these related concepts: Volumes, Volume and Density, and Shape and Volume
Sources
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