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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:
Both the authoring (HTML-level) and programming constructs of Curl can be extended in user code. The language is designed so Curl applications can be compiled to native code of the client machine by a just-in-time compiler and run at high speed. Curl applets can also be written so that they can run off-line when disconnected from a network ...
curl was first released in 1996. [9] It was originally named httpget and then became urlget before adopting the current name of curl [10] [11] The original author and lead developer is the Swedish developer Daniel Stenberg, who created curl because he wanted to automate the fetching of currency exchange rates for IRC users.
D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist.
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.
Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions. The vector field F = − z 1 + z 2 i + 1 1 + z 2 j {\displaystyle \mathbf {F} =-{\frac {z}{\sqrt {1+z^{2}}}}\mathbf {i} +{\frac {1}{\sqrt {1+z^{2}}}}\mathbf {j} } is a multiple of the standard contact structure − z i + j , and furnishes an ...
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If a vector field F with zero divergence is defined on a ball in R 3, then there exists some vector field G on the ball with F = curl G. For regions in R 3 more topologically complicated than this, the latter statement might be false (see Poincaré lemma ).