In mathematics, a plane curve is a curve in a Euclidean plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a real Euclidean plane R and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation {{{1}}} where ¿ : R ¿ R is a smooth function, and the partial derivatives ¿¿/¿x and ¿¿/¿y are never both 0.
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Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems. Since work is defined as a force acting through a distance (a length of space), energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length. The total energy contained in an object is identified with its mass, and energy (like mass), cannot be created or destroyed.
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Elastic energy
Elastic energy is the potential mechanical energy stored in the configuration of a material or physical system as work is performed to distort its volume or shape. Elasticity theory primarily develops an analytical understanding of the mechanics of solid bodies and materials. The essence of elasticity is reversibility. Forces applied to an elastic material transfer energy into the material which, upon yielding that energy to its surroundings, can recover its original shape.
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Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.
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