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The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3. It can be defined in several ways, to be mentioned below:
C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.
Table with the del operator in cartesian, cylindrical and spherical coordinates Operation Cartesian coordinates (x, y, z) Cylindrical coordinates (ρ, φ, z) Spherical coordinates (r, θ, φ), where θ is the polar angle and φ is the azimuthal angle α; Vector field A
Del operator, represented by the nabla symbol. Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus.
Rotation operator may refer to: An operator that specifies a rotation (mathematics) Three-dimensional rotation operator; Rot (operator) aka Curl, a differential operator in mathematics; Rotation operator (quantum mechanics)
Vector operators include: Gradient is a vector operator that operates on a scalar field, producing a vector field. Divergence is a vector operator that operates on a vector field, producing a scalar field. Curl is a vector operator that operates on a vector field, producing a vector field. Defined in terms of del:
Curl, (with operator symbol ) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point. As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with tensor calculus as well as vector calculus.
Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.