Lift (force) in the context of Euclidean vector

Terrestrial animals are animals that live predominantly or entirely on land (e.g., cats, chickens, ants, most spiders), as compared with aquatic animals (e.g., fish, whales, octopuses, lobsters, etc.), who live predominantly or entirely in bodies of water; and semiaquatic animals (e.g., crocodilians, seals, platypus and most amphibians), who inhabit coastal, riparian or wetland areas and rely on both aquatic and terrestrial habitats. While most insects (who constitute over half of all known species in the animal kingdom) are terrestrial, some groups, such as mosquitoes and dragonflies, spend their egg and larval stages in water but emerge as fully terrestrial adults (imagos) after completing metamorphosis.

Terrestrial animals conduct respiratory gas exchange directly with the atmosphere, typically via specialized respiratory organs known as lungs, or via cutaneous respiration across the skin. They have also evolved homeostatic features such as impermeable cuticles that can restrict fluid loss, temperature fluctuations and infection, and an excretory system that can filter out nitrogenous waste in the form of urea or uric acid, in contrast to the ammonia-based excretion of aquatic animals. Without the buoyancy of an aqueous environment to support their weight, they have evolved robust skeletons that can hold up their body shape, as well as powerful appendages known as legs or limbs to facilitate terrestrial locomotion, although some perform limbless locomotion using body surface projections such as scales and setae. Some terrestrial animals even have wings or membranes that act as airfoils to generate lift, allowing them to fly and/or glide as airborne animals.

A sail is a tensile structure, which is made from fabric or other membrane materials, that uses wind power to propel sailing craft, including sailing ships, sailboats, windsurfers, ice boats, and even sail-powered land vehicles. Sails may be made from a combination of woven materials—including canvas or polyester cloth, laminated membranes or bonded filaments, usually in a three- or four-sided shape.

A sail provides propulsive force via a combination of lift and drag, depending on its angle of attack, its angle with respect to the apparent wind. Apparent wind is the air velocity experienced on the moving craft and is the combined effect of the true wind velocity with the velocity of the sailing craft. Angle of attack is often constrained by the sailing craft's orientation to the wind or point of sail. On points of sail where it is possible to align the leading edge of the sail with the apparent wind, the sail may act as an airfoil, generating propulsive force as air passes along its surface, just as an airplane wing generates lift, which predominates over aerodynamic drag retarding forward motion. The more that the angle of attack diverges from the apparent wind as a sailing craft turns downwind, the more drag increases and lift decreases as propulsive forces, until a sail going downwind is predominated by drag forces. Sails are unable to generate propulsive force if they are aligned too closely to the wind.

A limb (from Old English lim, meaning "body part") is a jointed, muscled appendage of a tetrapod vertebrate animal used for weight-bearing, terrestrial locomotion and physical interaction with other objects. The distalmost portion of a limb is known as its extremity. The limbs' bony endoskeleton, known as the appendicular skeleton, is homologous among all tetrapods, who use their limbs for walking, running and jumping, swimming, climbing, grasping, touching and striking.

All tetrapods have four limbs that are organized into two bilaterally symmetrical pairs, with one pair at each end of the torso, which phylogenetically correspond to the four paired fins (pectoral and pelvic fins) of their fish (sarcopterygian) ancestors. The cranial pair (i.e. closer to the head) of limbs are known as the forelimbs or front legs, and the caudal pair (i.e. closer to the tail or coccyx) are the hindlimbs or back legs. In animals with a more erect bipedal posture (mainly hominid primates, particularly humans), the forelimbs and hindlimbs are often called upper and lower limbs, respectively. The fore-/upper limbs are connected to the thoracic cage via the pectoral/shoulder girdles, and the hind-/lower limbs are connected to the pelvis via the hip joints. Many animals, especially the arboreal species, have prehensile forelimbs adapted for grasping and climbing, while some (mostly primates) can also use hindlimbs for grasping. Some animals (birds and bats) have expanded forelimbs (and sometimes hindlimbs as well) with specialized feathers or membranes to achieve lift and fly. Aquatic and semiaquatic tetrapods usually have limb features (such as webbings) adapted to better provide propulsion in water, while marine mammals and sea turtles have convergently evolved flattened, paddle-like limbs known as flippers.

Bird wings are paired forelimbs in birds, which evolved specialized feathers to generate lift and thrust and allow the birds to fly.

Terrestrial flightless birds have reduced wings or none at all (for example, moa). In aquatic flightless birds (penguins), wings can serve as flippers.

Sailing into the wind is a sailing expression that refers to a sail boat's ability to move forward despite heading toward, but not directly into, the wind. A sailboat cannot sail directly into the wind; the closest it can point is called close hauled, typically at an angle of about 45 degrees to the wind.

This maneuver is possible due to the interaction between the sails and the keel. The sail generates lift (similar to an airplane wing), and the keel resists lateral movement through the water, converting the side force from the wind into forward motion. Although the sail force acts partly sideways, the keel’s counteracting force keeps the boat on course and propels it forward.

An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or the dynamic lift of an airfoil, or, in a few cases, direct downward thrust from its engines. Common examples of aircraft include airplanes, drones, rotorcraft (including helicopters), airships (including blimps), gliders, paramotors, and hot air balloons. Part 1 (Definitions and Abbreviations) of Subchapter A of Chapter I of Title 14 of the U. S. Code of Federal Regulations states that aircraft "means a device that is used or intended to be used for flight in the air."

The human activity that surrounds aircraft is called aviation. The science of aviation, including designing and building aircraft, is called aeronautics. Crewed aircraft are flown by an onboard pilot, whereas unmanned aerial vehicles may be remotely controlled or self-controlled by onboard computers. Aircraft may be classified by different criteria, such as lift type, aircraft propulsion (if any), usage and others.

A helicopter is a type of rotorcraft in which lift and thrust are supplied by horizontally spinning rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forward, backward and laterally. These attributes allow helicopters to be used in congested or isolated areas where fixed-wing aircraft and many forms of short take-off and landing (STOL) or short take-off and vertical landing (STOVL) aircraft cannot perform without a runway.

The Focke-Wulf Fw 61 was the first successful, practical, and fully controllable helicopter in 1936, while in 1942, the Sikorsky R-4 became the first helicopter to reach full-scale production. Starting in 1939 and through 1943, Igor Sikorsky worked on the development of the VS-300, which over four iterations, became the basis for modern helicopters with a single main rotor and a single tail rotor.

Fins are moving appendages protruding from the body of fish that interact with water to generate thrust and lift, which help the fish swim. Apart from the tail or caudal fin, fish fins have no direct articulations with the axial skeleton and are attached to the core only via muscles and ligaments.

Fish fins are distinctive anatomical features with varying internal structures among different clades: in ray-finned fish (Actinopterygii), fins are mainly composed of spreading bony spines or "rays" covered by a thin stretch of scaleless skin, resembling a folding fan; in lobe-finned fish (Sarcopterygii) such as coelacanths and lungfish, fins are short rays based around a muscular central bud internally supported by a jointed appendicular skeleton; in cartilaginous fish (Chondrichthyes) and jawless fish (Agnatha), fins are fleshy "flippers" supported by a cartilaginous skeleton. The limbs of tetrapods, a mostly terrestrial clade evolved from freshwater lobe-finned fish, are homologous to the pectoral and pelvic fins of all jawed fish.

A wing is a type of fin that produces both lift and drag while moving through air. Wings are defined by two shape characteristics, an airfoil section and a planform. Wing efficiency is expressed as lift-to-drag ratio, which compares the benefit of lift with the air resistance of a given wing shape, as it flies. Aerodynamics includes the study of wing performance in air.

Equivalent foils that move through water are found on hydrofoil power vessels and foiling sailboats that lift out of the water at speed and on submarines that use diving planes to point the boat upwards or downwards, while running submerged. The study of foil performance in water is a subfield of Hydrodynamics.

The patagium (pl.: patagia) is a membranous body part that assists an animal in obtaining lift when gliding or flying. The structure is found in extant and extinct groups of flying and gliding animals including bats, theropod dinosaurs (including birds and some dromaeosaurs), pterosaurs, gliding mammals, some flying lizards, and flying frogs. The patagium that stretches between an animal's hind limbs is called the uropatagium (especially in bats) or the interfemoral membrane.

An airfoil (American English) or aerofoil (British English) is a streamlined body that is capable of generating significantly more lift than drag. Wings, sails and propeller blades are examples of airfoils. Foils of similar function designed with water as the working fluid are called hydrofoils.

When oriented at a suitable angle, a solid body moving through a fluid deflects the oncoming fluid (for fixed-wing aircraft, a downward force), resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as aerodynamic force and can be resolved into two components: lift (perpendicular to the remote freestream velocity) and drag (parallel to the freestream velocity).

In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the generation of lift on wings.

Mathematically, the vorticity ${\displaystyle {\boldsymbol {\omega }}}$ is the curl of the flow velocity ${\displaystyle \mathbf {v} }$:

In physics, circulation is the line integral of a vector field around a closed curve embedded in the field. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.

In aerodynamics, it finds applications in the calculation of lift, for which circulation was first used independently by Frederick Lanchester, Ludwig Prandtl, Martin Kutta and Nikolay Zhukovsky. It is usually denoted by Γ (uppercase gamma).

A golf club is a club used to hit a golf ball in a game of golf. Each club is composed of a shaft with a grip and a club head. Woods are mainly used for long-distance fairway or tee shots; irons, the most versatile class, are used for a variety of shots; hybrids that combine design elements of woods and irons are becoming increasingly popular; putters are used mainly on the green to roll the ball into the hole. A set of clubs is limited by the rules of golf to a maximum of 14 golf clubs, and while there are traditional combinations sold at retail as matched sets, players are free to use any combination of legal clubs.

The most significant difference between clubs of the same type is loft, or the angle between the club's face and the vertical plane. It is loft that is the primary determinant of the ascending trajectory of the golf ball, with the tangential angle of the club head's swing arc at impact being a secondary and relatively minor consideration (though these small changes in swing angle can nevertheless have a significant influence on launch angle when using low-lofted clubs). The impact of the club compresses the ball, while grooves on the club face give the ball backspin. Together, the compression and backspin create lift. The majority of woods and irons are labeled with a number; higher numbers usually indicate shorter shafts and higher lofts, which give the ball a higher and shorter trajectory.

A parachute is a device designed to slow an object's descent through an atmosphere by creating drag or aerodynamic lift. It is primarily used to safely support people exiting aircraft at height, but also serves various purposes like slowing cargo, aiding in space capsule recovery on Earth, landing spacecraft on other planets, and stabilizing vehicles or objects. Modern parachutes are typically made from durable fabrics like nylon and come in various shapes, such as dome-shaped, rectangular, and inverted domes, depending on their specific function.

The concept of the parachute dates back to ancient attempts at flight. In AD 852, Armen Firman, in Córdoba, Spain, made the first recorded jump with a large cloak to slow his fall. Renaissance figures like Francesco di Giorgio Martini and Leonardo da Vinci later sketched designs resembling modern parachutes, but it was not until the 18th century that the first successful jumps occurred. French Louis-Sébastien Lenormand made the first public jump in 1783, and he coined the term "parachute" in 1785. In the following years, key advancements were made by figures like André Garnerin, with parachutes becoming lighter, more reliable, and easier to deploy.

An aircraft catapult is an acceleration device used to help fixed-wing aircraft reach liftoff speed (V_LOF) faster during takeoff, typically when trying to take off from a very short runway, as otherwise the aircraft engines alone cannot get the aircraft to sufficient airspeed quickly enough for the wings to generate the lift needed to sustain flight. Launching via catapults enables aircraft that typically are only capable of conventional takeoffs, especially heavier aircraft with significant payloads, to perform short takeoffs from the roll distances of light aircraft. Catapults are usually used on the deck of a ship — such as the flight deck of an aircraft carrier — as a form of assisted takeoff for navalised aircraft, but can also be installed on land-based runways, although this is rare.

Historically it was most common for seaplanes (which have pontoons instead of wheeled landing gears and thus cannot utilize runways) to be catapulted from ships onto nearby water for takeoff, allowing them to conduct aerial reconnaissance missions and be crane-hoisted back on board during retrieval, although by the late First World War their roles are largely supplanted by the more versatile biplanes that can take off and land on carrier decks unassisted. During the Second World War before the advent of escort carriers, monoplane fighter aircraft (notably the Hawker Hurricane) would sometimes be catapulted from "catapult-equipped merchant" (CAM) vessels for one-way sorties to repel enemy aircraft harassing shipping lanes, forcing the returning pilot to either divert to a land-based airstrip, jump out by parachute, or ditch in the water near the convoy and wait for rescue. By the time fleet carriers became the norm in WW2, catapult launches have become largely unnecessary and carrier-based fighter-bombers would routinely perform self-powered takeoffs and landings off and onto carrier decks, especially during the naval aviation-dominated Pacific War between the United States and the Empire of Japan. However, escalating arms races during the Cold War accelerated the adoption of the heavier jet aircraft for naval operations, thus motivating the development of new catapult systems, especially after the popularization of angled flight decks further limited the practical distance available as takeoff runways. Nowadays, jet aircraft can launch from aircraft carriers via either catapults or ski-jump deck, and perform optics-assisted landing onto the same ship with help from decelerative arresting gears.

A fin is a thin appendage or component attached to a larger body or structure. Fins typically function as foils that produce lift or thrust, or provide the ability to steer or stabilize motion while traveling in water, air, or other fluids. Fins are also used to increase surface areas for heat transfer purposes, or simply as ornamentation.

Fins first evolved on fish as a means of locomotion. Fish fins are used to generate thrust and control the subsequent motion. Fish and other aquatic animals, such as cetaceans, actively propel and steer themselves with pectoral and tail fins. As they swim, they use other fins, such as dorsal and anal fins, to achieve stability and refine their maneuvering.

A fixed-wing aircraft is a heavier-than-air aircraft, such as an airplane, which is capable of flight using aerodynamic lift. Fixed-wing aircraft are distinct from rotary-wing aircraft (in which a rotor mounted on a spinning shaft generates lift), and ornithopters (in which the wings oscillate to generate lift). The wings of a fixed-wing aircraft are not necessarily rigid; kites, hang gliders, variable-sweep wing aircraft, and airplanes that use wing morphing are all classified as fixed wing.

Gliding fixed-wing aircraft, including free-flying gliders and tethered kites, can use moving air to gain altitude. Powered fixed-wing aircraft (airplanes) that gain forward thrust from an engine include powered paragliders, powered hang gliders and ground effect vehicles. Most fixed-wing aircraft are operated by a pilot, but some are unmanned or controlled remotely or are completely autonomous (no remote pilot).