exact

Work (abbreviated W) is the energy transferred by a force to a moving object. Work is a scalar quantity, but it can be positive or negative. Work is associated with a change in energy, but not all changes in energy can be readily analysed in terms of work. In addition, not all forces do work. For instance, a centripetal force in uniform circular motion does not transfer energy; the kinetic energy of the object undergoing the motion remains constant. Mechanical work - Definition. Note: Reade ...

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• Mechanical work - Definition
• Mechanical work - Units
• Mechanical work - Simpler formulae
• Mechanical work - Types of work
• Mechanical work - PV work
• Mechanical work - Mechanical energy
• Mechanical work - Conservation of mechanical energy

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The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor FA is exact if and only if A is projective. The functor GA(X) = HomA(X,A) is a contravariant left-exact functor; it is e ...

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Exact functor, Exact functor - Examples, Exact functor - Some facts

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Consider the following commutative diagram in any Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field, or in the category of groups). The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism. The four lemmas state: If m and p are epimorphisms and q is a monomorphism, then n is an epimorphism. If m and p are monomorphisms and l is an epimorphism, ...

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Five lemma, Five lemma - Statement, Five lemma - Proof

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Any product of (even infinitely many) injective modules is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules or infinite direct sums of injective modules need not be injective. In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of See also:

Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electrical work", an electric field does work on charged particles as they move through a medium. One mechanism of heat conduction is collisions between fast-moving atoms in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, it averages to nearly zero in bulk, so conduction is not considered to be mechanical work.

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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The SI derived unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one meter. The dimensionally equivalent newton-meter (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, and the liter-atmosphere. ...

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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Note: Readers not familiar with multivariate calculus or vectors, please see "Simpler formulae" below Work is defined as the following line integral: where: C is the path or curve traversed by the object; is the force vector; is the position vector. This formula readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero (viz. circu ...

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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In the simplest case, that of a body moving in a steady direction, and acted on by a force parallel to that direction, the work is given by the formula where F is the force and s is the distance traveled by the object. The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector quantities, and combined using the dot product: where φ is the ...

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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More formally, a left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions: If Q is a submodule of some other left R-module M, then there exists another submodule K of M such that M is the internal direct sum of Q and K, i.e. Q + K = M and Q ∩ K = {0}. If X is a submodule of the left R-module Y and g : X → Q ...

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Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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Trivially, the zero module {0} is injective. Every vector space Q is injective. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map g in ...

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Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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