- published: 16 Jan 2016
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Del, or nabla, is an operator used in mathematics, in particular, 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 its standard derivative as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), del may denote the gradient (locally steepest slope) of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations), the divergence of a vector field, or the curl (rotation) of a vector field, depending on the way it is applied.
Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, dot product, and cross product, respectively, of the del "operator" with the field. These formal products do not necessarily commute with other operators or products.
A pen is an enclosure for holding livestock. The term describes multiple types of enclosures that may confine one or many animals. Construction and terminology vary depending on the region of the world, purpose, animal species to be confined, local materials used, and cultural tradition. Pen or penning as a verb refers to the act of confining animals in an enclosure.
In Australia and New Zealand a pen is a small enclosure for livestock (especially sheep or cattle), which is part of a larger construction, e.g. calf pen, forcing pen (or yard) in sheep or cattle yards, or a sweating pen or catching pen in a shearing shed. In Australia, a paddock may encompass a large, fenced grazing area of many acres, not to be confused with the American English use of paddock as interchangeable with corral or pen, describing smaller, confined areas.
In the United States, the term pen usually describes small enclosures for holding sheep, goats, and pigs. A pen for cattle is also sometimes called a corral. Pens may be named by their purpose, such as a holding pen, used for short-term confinement. Groups of pens that are part of a larger complex may be called a stockyard, where a series of pens holds a large number of animals, or a feedlot, which is type of stockyard used to confine animals that are being fattened. A large pen for horses is called a paddock (Eastern USA) or a corral (Western USA), a borrowing from the Spanish language. In some places an exhibition arena may be called a show pen. A small pen for horses (no more than 15–20 feet on any side) is a pen if it lacks any roof or shelter, otherwise it is called a stall and is part of a stable. A large fenced grazing area of many acres is called a pasture, or, in some cases, rangeland.