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Curl

Curl or CURL may refer to:

Mathematics and physics

Curl in vector calculus

In vector calculus, the curl of a vector field \mathbf{F} in three-dimensional Euclidean space is a vector operator that measures the infinitesimal circulation or rotation of the field around a point. Denoted by \nabla \times \mathbf{F}, it quantifies the tendency of the field to swirl or rotate locally, with the direction of the resulting vector indicating the axis of rotation via the right-hand rule. The curl is expressed in Cartesian coordinates for a vector field \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} as: \nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}. This determinant form arises from the cross product of the del operator \nabla with \mathbf{F}. The operator is linear, satisfying \nabla \times (a\mathbf{F} + b\mathbf{G}) = a(\nabla \times \mathbf{F}) + b(\nabla \times \mathbf{G}) for scalars a, b and vector fields \mathbf{F}, \mathbf{G}. Key identities include the curl of a gradient being zero, \nabla \times (\nabla \phi) = \mathbf{0} for any scalar potential \phi, which implies that conservative fields have zero curl. Additionally, the divergence of the curl vanishes, \nabla \cdot (\nabla \times \mathbf{F}) = 0, indicating that curl fields are solenoidal. Stokes' theorem connects the curl to circulation, stating that the surface integral of \nabla \times \mathbf{F} over an oriented surface equals the line integral of \mathbf{F} around its boundary curve. The curl operator was formulated in the 1860s by James Clerk Maxwell and Peter Guthrie Tait using quaternions for electromagnetic theory, with Maxwell suggesting the term "curl" in his 1871 paper and further developing its application in his 1873 Treatise on Electricity and Magnetism, building on Hermann Grassmann's 19th-century work on . further refined and popularized the notation, including the term "curl," in the 1880s as part of developing modern vector analysis alongside . In other coordinate systems, the curl takes modified forms to account for the geometry. In cylindrical coordinates (r, \theta, z) for \mathbf{F} = F_r \mathbf{e}_r + F_\theta \mathbf{e}_\theta + F_z \mathbf{e}_z, \nabla \times \mathbf{F} = \frac{1}{r} \left( \frac{\partial F_z}{\partial \theta} - \frac{\partial (r F_\theta)}{\partial z} \right) \mathbf{e}_r + \left( \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r} \right) \mathbf{e}_\theta + \frac{1}{r} \left( \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right) \mathbf{e}_z. In spherical coordinates (r, \theta, \phi) for \mathbf{F} = F_r \mathbf{e}_r + F_\theta \mathbf{e}_\theta + F_\phi \mathbf{e}_\phi, \nabla \times \mathbf{F} = \frac{1}{r \sin \theta} \left( \frac{\partial (F_\phi \sin \theta)}{\partial \theta} - \frac{\partial F_\theta}{\partial \phi} \right) \mathbf{e}_r + \frac{1}{r} \left( \frac{1}{\sin \theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial (r F_\phi)}{\partial r} \right) \mathbf{e}_\theta + \frac{1}{r} \left( \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right) \mathbf{e}_\phi. These expressions facilitate computations in systems with rotational or radial symmetry.

Applications in physics

In , the curl operator plays a central role in , which unify and . Faraday's law states that the curl of the \mathbf{E} is equal to the negative time of the \mathbf{B}, expressed as \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}. This equation describes how a changing magnetic field induces a rotational electric field, fundamental to phenomena like in generators and transformers. Similarly, Ampère's law with Maxwell's correction gives \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, where \mu_0 is the permeability of free space, \mathbf{J} is the , and \epsilon_0 is the of free space; this relates the curl of the magnetic field to electric currents and the time-varying electric field, enabling the of electromagnetic waves. These formulations, developed in the , revolutionized physics by predicting as an electromagnetic wave. In , the curl of the velocity field \mathbf{v} defines \boldsymbol{\omega} = \nabla \times \mathbf{v}, which quantifies the local or swirling motion of fluid elements. The magnitude of \boldsymbol{\omega} measures the intensity of this , while its direction aligns with the axis of according to the , where curling the fingers in the direction of fluid circulation points the thumb along \boldsymbol{\omega}. For instance, in irrotational flows, \nabla \times \mathbf{v} = 0, implying no local spinning, as seen in potential flows around streamlined objects where friction is negligible. This concept became prominent in 20th-century , aiding analyses of , boundary layers, and aerodynamic in design. The Helmholtz decomposition further illustrates the curl's physical utility by expressing any sufficiently smooth vector field \mathbf{F} as the sum of an irrotational part -\nabla \phi (where \nabla \times (-\nabla \phi) = 0) and a solenoidal part \nabla \times \mathbf{A} (where \nabla \cdot (\nabla \times \mathbf{A}) = 0), with \phi a scalar potential and \mathbf{A} a vector potential determined by the divergence and curl of \mathbf{F}. This separation, rooted in 19th-century work, underpins applications in both electromagnetism—decomposing fields into electrostatic and magnetostatic components—and fluid dynamics, distinguishing rotational from compressional motions in complex flows like ocean currents or stellar interiors.

Computing

cURL command-line tool

cURL, often stylized as and pronounced "see URL", is an open-source command-line tool designed for transferring data using Uniform Resource Locators (). It operates as a client for interacting with servers across various network protocols, enabling users to send and receive data in scripts, command lines, or integrated applications. The tool is part of the and includes libcurl, a companion library that powers the command-line interface while also serving as a reusable component for . The development of began in late 1996 when , working on an for an Amiga-related channel, needed a simple utility to fetch data over HTTP. Initially named HttpGet, it evolved through renamings—first to urlget in 1997 with added FTP support, then to in March 1998. SSL support was integrated that same year, marking an early focus on secure transfers. The libcurl library was first released in August 2000 as version 7.1, allowing broader integration into applications. Key milestones include the switch to a permissive MIT-like license in January 2001, full HTTP/1.1 support in March 2001, in 2014, and in 2019. As of November 2025, the latest stable release is version 8.17.0, released on November 5, supporting over 25 protocols such as DICT, FILE, FTP, , GOPHER, GOPHERS, HTTP, , IMAP, IMAPS, LDAP, LDAPS, , POP3, POP3S, RTMP, RTMPS, RTSP, , , , SMBS, SMTP, , , TFTP, WS, and . cURL's key features include robust URL parsing, support for authentication methods like Basic, Digest, and , handling of cookies for session management, proxy configurations (including and HTTP proxies), and secure connections via SSL/TLS with multiple backend options such as and . It provides flexible command-line syntax, for example, curl -X GET https://[example.com](/page/Example.com) to perform an HTTP GET request, or curl -u user:pass -O ftp://ftp.[example.com](/page/Example.com)/file.txt to download a file via FTP with . These capabilities make it versatile for both simple data fetches and complex scripted operations. Common uses of encompass testing by sending custom HTTP requests, downloading or uploading files from remote servers, automating web interactions in scripts, and basic for data extraction. It serves as a lightweight alternative to tools like for non-interactive transfers and is frequently embedded in automation pipelines for or system administration tasks. cURL and libcurl are released under a permissive inspired by the , allowing free use, modification, and distribution for any purpose without royalties, while requiring preservation of copyright notices. The project is maintained by the curl project under the leadership of , with contributions from a global community. It enjoys widespread adoption, pre-installed in most major distributions such as , , and , and powering libcurl in over 20 billion devices including cars, televisions, routers, and mobile phones.

Curl programming language

Curl is a reflective, object-oriented programming language designed for developing interactive web applications, integrating markup for content presentation, styling, and procedural logic within a single syntax to facilitate seamless creation of rich internet applications. It blends elements of declarative markup languages like HTML and CSS with imperative programming capabilities, allowing developers to embed executable code directly alongside visual elements without requiring separate files or languages. Originally conceived to address the limitations of early web technologies by providing a "gentle slope" from simple document authoring to complex applications, Curl supports features such as strong typing with optional dynamic types, multiple inheritance, and first-class procedures. Development of Curl began in 1998 at the Institute of Technology's Laboratory for Computer Science, led by researchers including Steve Ward and influenced by languages such as , C++, Tcl/Tk, and , as part of a broader effort to unify web content creation. The project was commercialized through Curl Corporation, founded in , with the first public release occurring in 2000. In 2003, Curl Corporation was acquired by Sumisho Computer Systems Corporation (now ), a IT firm, which shifted focus toward enterprise applications, particularly in . Under SCSK, Curl has continued to evolve, with the latest release of the Curl Runtime Environment (RTE) version 8.0.15 occurring on October 16, 2025, maintaining support for modern operating systems including and later. As of November 2025, active development persists through SCSK, though adoption remains niche, primarily in enterprise settings for data-intensive web interfaces. Key features of Curl emphasize a unified approach to , , and presentation, enabling developers to write everything in one language executed via a or standalone RTE. For instance, dynamic content is generated using enclosed commands like {curl ...}, which can invoke procedures, handle events, or fetch data, reducing the need for multiple technologies. The language supports applets for interactive elements, threading for concurrency, and garbage collection for , making it suitable for responsive applications. Curl's extensibility allows custom operators and environments, promoting reusable components in object-oriented designs. Curl's combines markup-like declarations with procedural , using curly braces {} to denote expressions in text. For example, a simple "Hello, World!" display might use:
{Hello, World!}
To apply formatting, such as bold text, the integrates visual commands directly:
{bold Hello, World!}
This renders "Hello, World!" in bold font. For computational elements, arithmetic or logic is expressed inline, like {+ 10 4}, which evaluates to 14 and can be bound to updates. Object-oriented constructs include definitions, as in a basic button example:
{button
    label = "Click me",
    {on e-ButtonPressed do
        {alert "Button clicked!"}
    }
}
This creates an interactive that shows an on , demonstrating Curl's for event handling and components in a concise, integrated manner. Historically prominent in before 2010, particularly for financial and automotive applications in (e.g., Bank of Tokyo-Mitsubishi and ), Curl has been largely supplanted in general by , CSS3, and frameworks such as due to broader and native . Nonetheless, its single-language continues to offer advantages for legacy systems and specialized rich applications, with ongoing maintenance ensuring compatibility rather than widespread innovation. In contexts, tools like the command-line utility can complement Curl by handling URL-based data transfers for backend .

Sports and exercise

Curling

Curling is a played on , in which two four-player teams take turns sliding heavy stones toward a circular target area known as the , consisting of four concentric circles with the center point called the . The objective is to accumulate points by positioning stones closer to the than the opponent's stones at the end of each segment of play. Teammates use specialized brooms to sweep the in front of a moving stone, which warms the surface slightly to reduce and either extend the stone's travel distance by up to 2-3 meters or straighten its curving path. Each team comprises four players: the lead, who throws the first two stones; the second, who throws the next pair; the third, who delivers the following two; and the skip, the team captain who throws last and directs strategy from behind the house. A standard game consists of 10 ends, with each team delivering eight stones per end while alternating throws with the opponent; the team with the lowest-scoring stone in the preceding end throws first. Scoring occurs after all stones are thrown, awarding one point for each stone closer to the button than the opponent's nearest stone, with a maximum of eight points possible per end; if no stones lie within the house, the end is blank and scores nothing. The primary equipment includes granite curling stones, each weighing between 38 and 44 pounds (typically around 42 pounds or 19.1 kilograms), with a maximum circumference of 36 inches and a handle for gripping during delivery. Brooms, often made with synthetic bristles, are used for sweeping and must comply with World Curling Federation standards to avoid damaging the ice. Games are played on a rectangular ice sheet measuring 150 feet (45.72 meters) in length and up to 15 feet 7 inches (4.75 meters) in width, with houses positioned 12 feet (3.66 meters) in diameter at each end and a centerline dividing the playing area. Players wear specialized shoes with one grippy sole for stability and one slippery slider for the delivery foot. The sport originated in 16th-century , with the earliest written reference dating to 1540, and is considered one of the world's oldest team sports. It was formalized in the early , with the Grand Caledonian Curling Club adopting the first official rules in 1838 and receiving royal patronage to become the Royal Caledonian Curling Club in 1843. spread globally in the , particularly to , the , , , , and , leading to the formation of the International Curling Federation (now World Curling Federation) in 1966 to oversee championships. It debuted as a at the but achieved full medal status starting with the 1998 Games, boosting its international profile. Variants include wheelchair curling, introduced at the in 2002 and added to the Paralympics in 2006, which modifies delivery using a stick while retaining core rules; and , a faster-paced format with two players per team (one male, one female) throwing five stones each over eight ends, first contested at world level in 2008. The World Curling Federation promotes these and other disciplines, fostering global participation across 76 member nations as of 2025.

Curl exercises in weight training

Curl exercises in weight training are isolation movements that involve flexing specific joints using resistance from weights, primarily targeting the in the upper or the hamstrings in the lower legs. These exercises emphasize controlled flexion for arm curls or knee flexion for leg curls, promoting targeted without significant involvement of larger muscle groups. Common variations for biceps curls include the standing barbell curl, where a barbell is grasped with a shoulder-width underhand grip and raised until the forearms are vertical while keeping elbows at the sides; the dumbbell curl, performed alternately or simultaneously by rotating the forearm to supinate the wrist, which maximizes biceps brachii activation through peak contraction; the hammer curl, using a neutral grip to emphasize the brachialis and brachioradialis; and the concentration curl, executed seated with the elbow braced against the inner thigh for isolation. Supination during the concentric phase of biceps curls enhances recruitment of the biceps long head, contributing to greater overall arm flexion torque. For leg curls, key types are the lying leg curl on a machine, where one lies prone and flexes the knees to raise padded levers toward the buttocks while maintaining torso contact with the bench; the seated leg curl, performed with the back supported and lower legs secured under pads; and the Nordic curl, a bodyweight or assisted variation involving kneeling and controlled lowering of the torso by eccentrically contracting the hamstrings. Proper technique in curl exercises prioritizes a slow eccentric phase to minimize risk, such as lower back from in standing variations or hyperextension in lying leg curls, and involves like weights (dumbbells, barbells, EZ-bars) or machines for stability. For curls, elbows remain fixed at the sides without forward swinging, and full —from arms extended to forearms vertical—ensures optimal muscle loading. In leg curls, avoiding torso lift and focusing on knee flexion isolates the hamstrings while the gastrocnemius assists if ankles are dorsiflexed. These techniques support balanced development and reduce stress when performed with 8–12 repetitions per set at moderate loads. The primary benefits of curl exercises include promoting muscle hypertrophy and strength gains in the targeted areas, with concentration and preacher curls eliciting the highest electromyographic activity in the biceps brachii compared to other variations. For hamstrings, leg curls enhance eccentric strength, improving sprint performance and reducing strain injury rates—for instance, teams incorporating lying leg curls experienced significantly fewer hamstring injuries than those relying solely on compound lifts. Overall, these exercises are staples in bodybuilding routines for aesthetic arm and leg development, while also supporting functional pulling and knee stabilization. Curl exercises trace their origins to 19th-century movements in Europe, where figures like promoted dumbbell-based flexion work for building muscular strength and aesthetics, challenging prevailing views on exercise. By the early , such routines were integrated into global fitness via institutions like the and pioneers, evolving into standardized gym practices by the 1950s with the rise of machines and isolation-focused training.

Other uses

Hair curling

Hair curling is a hairstyling technique that involves shaping hair strands into coils, spirals, or waves to enhance texture and volume. This process can be temporary or semi-permanent, achieved through mechanical, thermal, or chemical means, and contrasts with naturally occurring curls determined by follicle geometry. Natural curls arise from genetic factors influencing the shape of hair follicles, where round follicles produce straight hair, oval ones yield waves, and flattened or C-shaped follicles create curls due to asymmetric keratin distribution during growth. Induced curling, however, relies on external methods to alter hair structure post-formation. Common approaches include heat styling with tools like curling irons or wands, typically set between 300°F and 400°F to temporarily reshape hydrogen bonds in the hair shaft, suitable for creating loose waves or defined coils. Wet sets use mechanical tension on damp hair, such as foam rollers or flexi-rods, allowing curls to form as the hair dries, often enhanced by setting lotions for hold. Chemical perms provide longer-lasting results by breaking and reforming disulfide bonds in keratin using reducing agents like ammonium thioglycolate, derived from thioglycolic acid, followed by neutralization to fix the curl pattern. Key tools for hair curling include curling irons, available in ceramic barrels that distribute heat evenly to minimize damage and frizz—ideal for fine or damaged hair—or titanium barrels that heat rapidly and retain high temperatures for thick hair but require caution to avoid overexposure. Barrel sizes vary: ½-inch for tight ringlets, 1-inch for versatile curls, and 1½- to 2-inch for beachy waves. Supporting products encompass curl-enhancing creams that provide moisture and definition to amplify natural or styled patterns, and blow-dryer diffusers that disperse airflow gently to preserve curl integrity without flattening. The practice dates to around 3000 BCE, where bronze curling tongs heated over fire styled wigs and hair into curls, often secured with beeswax-based gels for elaborate ceremonial looks. In 1909, German inventor Charles Nessler patented the first modern permanent wave machine in , using electrical heaters and brass rollers with a solution to set curls semi-permanently, revolutionizing techniques by 1914 with improved patents. By the early , chemical perms using thioglycolates became widespread, enabling durable curls without constant reheating. As of 2025, trends emphasize heatless methods like silk rod curlers or satin rollers, which wrap damp hair overnight to form soft via mechanical tension alone, reducing thermal damage while achieving voluminous styles such as tight curls or effortless beach . For care, applying heat protectants—formulations with silicones or polymers that coat the hair to slow —is essential before thermal styling, preventing protein denaturation and breakage, especially for types like loose or defined coils that benefit from lower temperatures around –350°F. Regular deep conditioning complements this to maintain elasticity, with curls classified by pattern tightness (e.g., 2A loose to 4C tight coils) guiding product selection for optimal health.

Material and biological curling

In manufacturing, paper curl arises primarily from moisture imbalances between the machine direction (MD) and cross direction (CD), leading to differential hygroscopic expansion where fibers swell unevenly due to varying orientation and composition. This phenomenon, rooted in the physics of 's response to changes, causes the sheet to bend toward the drier side as the wetter side expands more, a exacerbated during if moisture gradients persist across the thickness. Historical records indicate such curl issues emerged prominently in 19th-century with the advent of continuous machines, where uneven in early Fourdrinier processes amplified dimensional instabilities in mass-produced sheets. To prevent curl, industrial controls emphasize balanced techniques, such as symmetric moisture profiling via impingement dryers or steam boxes, which minimize expansion differentials and maintain uniform fiber saturation. Metal curling, in contrast, is an intentional forming process in fabrication used to create smooth, rounded edges on components, enhancing safety and structural integrity by reducing sharp burrs from prior operations like cutting or . This technique involves progressively rolling the metal edge into a cylindrical profile using dies or rollers, often applied to materials like or aluminum in automotive and appliance manufacturing. In biological contexts, leaf curl diseases manifest as pathological distortions in plants, driven by fungal, viral, or bacterial infections that disrupt cellular growth and turgor. For instance, peach leaf curl, caused by the fungus Taphrina deformans, infects buds during wet spring conditions, resulting in thickened, reddish, puckered leaves that yellow and drop prematurely, potentially defoliating trees and reducing fruit yield. Symptoms typically appear shortly after bud break, with the fungus overwintering in bark scales; prevention relies on dormant-season fungicide applications, such as copper-based sprays in late fall or early spring before bud swell. Similarly, tobacco leaf curl virus (Tobacco leaf curl virus, a begomovirus in the Geminiviridae family) induces stunted growth, twisted stems, and upward-curling leaves with enations, transmitted semipersistently by whitefly vectors (Bemisia tabaci). These viral infections, along with fungal pathogens like those causing celery anthracnose (leaf curl), infiltrate vascular tissues or mesophyll, leading to hormone imbalances and inhibited expansion. Documented since the early 1800s in phytopathology studies, such diseases highlight the role of microbial agents in altering plant morphology, with control strategies focusing on vector management and resistant cultivars. Additional examples include filament curling in , where thermoplastic materials like warp due to uneven cooling and thermal contraction after , causing layers to lift from the build plate if bed adhesion or temperature gradients are inadequate. Prevention involves optimizing print settings, such as enclosed chambers for uniform heating and adhesion aids like heated beds or brims.

People

Scientists and academics

The surname Curl is of English origin, derived as a nickname for someone with curly hair, from crul, crol, or curl meaning "curly," and is particularly associated with the region. It was relatively uncommon in the United States during the , with the 1880 census recording Curl families primarily in states like and , representing a small fraction of the total population. By 2010, it ranked around the 6,000th most common in the U.S., with approximately 5,000 bearers. One of the most prominent scientists bearing the surname is Robert Floyd Curl Jr. (1933–2022), an renowned for his contributions to and . Curl shared the 1996 with E. Smalley and Harold W. Kroto for their discovery of fullerenes, a new class of carbon molecules including (C60, or buckyballs), which they identified using vaporization techniques on in 1985. This breakthrough, conducted at where Curl served as the Kenneth S. Pitzer-Schlumberger Professor Emeritus of Natural Sciences, opened avenues for applications in , , and , leading to over 100,000 subsequent publications on fullerene derivatives. Earlier in his career, Curl pioneered supersonic jet , a method using pulsed jets to cool molecules for high-resolution studies of their structure and dynamics, as detailed in his work on infrared applications for transient species in . Post-Nobel, Curl engaged in discussions on the societal implications of , expressing concerns about commercialization pressures and emphasizing ethical considerations in scientific advancement, including the need for balanced oversight to prevent undue hype or risks. Another notable academic figure is John Curl (born 1940), an author, , and activist focused on cultures and economic systems. Curl has contributed to scholarship on through translations of pre-Columbian poetry from and other Native American languages, as well as advocacy for to honor Native histories and replace celebrations. His work extends to , exemplified by his book For All the People: Uncovering the Hidden History of , Movements, and in America (2009, revised 2012), which traces grassroots initiatives from mutual aid practices to modern worker-owned enterprises, drawing on over four decades of involvement in movements. Curl's interdisciplinary approach integrates historical analysis with activism, highlighting how models rooted in and communal traditions can address and . Leigh Ann Curl (born 1963) is an American orthopedic surgeon and specialist, notable as the first woman to serve as head team physician in the (NFL) for the Baltimore Ravens since 1998. She was elected president of the NFL Physicians Society in 2018 and received the 2024 Jerry "Hawk" Rhea Award as the NFL Physician of the Year. Curl's career includes pioneering work in arthroscopic and contributions to in sports medicine.

Artists and other professionals

Martina Gangle Curl (1906–1994) was an American painter and activist associated with the modernist movement. She studied at the and participated in the (WPA) during the , creating murals and community art. Curl was also involved in civil rights and advocacy. In the realm of sports and media, Bill Curl (c. 1941–2018) was a prominent American sports information director, serving from 1966 to 1973 and later handling for the Louisiana Superdome and , earning induction into the Louisiana Sports Hall of in 1985.

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