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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
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Key Term Reference
 area
 Appears in these related concepts: Approximate Integration, Planetary Motion According to Kepler and Newton, and Double Integrals Over Rectangles
 axis
 Appears in these related concepts: Area Between Curves, TwoComponent Forces, 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: Development of Nervous Tissue, Caulobacter Differentiation, and The Challenge of Competition
 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: Expected Value, Trigonometric Integrals, and Integration Using Tables and Computers
 limit
 Appears in these related concepts: Indeterminate Forms and L'Hôpital's Rule, Summing an Infinite Series, and Limits and Continuity
 line integral
 Appears in these related concepts: Vector Fields, Line Integrals, and Green's Theorem
 normal
 Appears in these related concepts: Arc Length and Curvature, Fluorescence and Phosphorescence, and Normal Forces
 scalar
 Appears in these related concepts: VectorValued Functions, Calculus of VectorValued Functions, and Multiplying Vectors by a Scalar
 vector
 Appears in these related concepts: Vectors in the Plane, Electric Field Lines: Multiple Charges, and Introduction to Memory Storage
 vector field
 Appears in these related concepts: Conservative Vector Fields, Fundamental Theorem for Line Integrals, and Surface Integrals of Vector Fields
 velocity
 Appears in these related concepts: Arc Length and Speed, Centripetial Acceleration, and Position, Displacement, Velocity, and Acceleration as Vectors
 volume
 Appears in these related concepts: Volumes, Cylindrical Shells, and Volume and Density
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Cite This Source
Source: Boundless. “Curl and Divergence.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 31 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/vectorcalculus23/curlanddivergence1712887/