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

Determine the direction and the magnitude of the curl

Define curl and divergence vector operators
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 F, denoted by curl F or ∇ × F, is defined at a point in terms of its projection onto various lines through the point. If n is any unit vector, the projection of the curl of F onto n is defined to be the limiting value of a closed line integral in a plane orthogonal to n as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
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 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. [1] (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 F at a point p is defined as the limit of the net flow of F across the smooth boundary of a threedimensional region V divided by the volume of V as V shrinks to p. Formally,
Key Term Reference
 area
 Appears in these related concepts: Graphing on Computers and Calculators, Numerical Integration, and Hyperbolic Functions
 axis
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Area Between Curves, 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: Curve Sketching, Derivatives of Exponential Functions, and Integration By Parts
 differential
 Appears in these related concepts: Triple Integrals in Cylindrical Coordinates, Thermal Stresses, and Compensation Differentials
 differentiation
 Appears in these related concepts: The Many Goals of Persuasion, Defining Monopoly, and Socioemotional Development in Adolescence
 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, Arc Length and Surface Area, and Arc Length and Speed
 integral
 Appears in these related concepts: Tangent and Velocity Problems, Indefinite Integrals and the Net Change Theorem, and Volumes
 limit
 Appears in these related concepts: Limits and Continuity, Finding Limits Algebraically, and Horizontal Asymptotes and Limits at Infinity
 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: Unit Vectors and Multiplication by a Scalar, Introduction to Scalars and Vectors, and Addition and Subtraction; Scalar Multiplication
 vector
 Appears in these related concepts: Infectious Disease Transmission, Superposition of Electric Potential, and Introduction to Memory Storage
 vector field
 Appears in these related concepts: Surface Integrals of Vector Fields, Electric Fields and Conductors, and Stress and Strain
 velocity
 Appears in these related concepts: Centripetial Acceleration, Force, and Applications and ProblemSolving
 volume
 Appears in these related concepts: Volume and Density, Shape and Volume, and Line
Sources
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Cite This Source
Source: Boundless. “Curl and Divergence.” Boundless Calculus. Boundless, 12 Aug. 2015. Retrieved 25 Nov. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/vectorcalculus23/curlanddivergence1712887/