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Gradient Theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. If is a differentiable function and a differentiable curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Integral
In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, although that is typically reserved for #Complex line integral, line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the Dot product, scalar product of the vector field with a Differential (infinitesimal), differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on interval (mathematics), intervals. Many simple formulae in physics, such as the definition of Work (physics), work as have natural continuous analogues in terms of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pearson Education International
Pearson Education, known since 2011 as simply Pearson, is the educational publishing and services subsidiary of the international corporation Pearson plc. The subsidiary was formed in 1998, when Pearson plc acquired Simon & Schuster's educational business and combined it with Pearson's existing education company Addison-Wesley Longman. Pearson Education was restyled as simply Pearson in 2011. In 2016, the diversified parent corporation Pearson plc rebranded to focus entirely on education publishing and services; further, as of 2023, Pearson Education is Pearson plc's main subsidiary. In 2019, Pearson Education began phasing out the prominence of its hard-copy textbooks in favor of digital textbooks, which cost the company far less, and can be updated frequently and easily. As of 2023, Pearson Education has testing/teaching centers in over 55 countries worldwide; the UK and the U.S. have the most centers. The headquarters of parent company Pearson plc are in London, England. P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. If is a differentiable function and a differentiable curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Potential Energy
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An ''object'' may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged ''objects''. The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields. Definition The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physics), work needed to move a test charge from a reference point to a specific point in a static electric field. The test charge used is small enough that disturbance to the field is unnoticeable, and its motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is Earth (electricity), earth or a point at infinity, although any point can be used. In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar (physics), scalar quantity denoted by or occasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is: It is a measure of how dense of an electric field is "permitted" to form in response to electric charges and relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law: F_\text = \frac \frac Here, ''q''1 and ''q''2 are the charges, ''r'' is the distance between their centres, and the value of the constant fraction 1/(4π''ε''0) is approximately . Likewise, ''ε''0 appears in Maxwell's equations, which describe the properties of electr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitude (mathematics), magnitude and Direction (geometry, geography), direction of a force are both important, force is a Euclidean vector, vector quantity. The SI unit of force is the newton (unit), newton (N), and force is often represented by the symbol . Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include Elasticity (physics), elastic, frictional, Normal force, contact or "normal" forces, and gravity, gravitational. The rotational version of force is torque, which produces angular acceleration, changes in the rotational speed of an object. In an extended body, each part applies forces on the adjacent pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb's Law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric force is conventionally called the ''electrostatic force'' or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the classical electromagnetism, theory of electromagnetism and maybe even its starting point, as it allowed meaningful discussions of the amount of electric charge in a particle. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point Electric charge, charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them. Coulomb discovered that bodies with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Work (physics)
In science, work is the energy transferred to or from an Physical object, object via the application of force along a Displacement (vector), displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the Product (mathematics), product of the force strength and the distance traveled. A force is said to do ''positive work'' if it has a component in the direction of the displacement of the point of application. A force does ''negative work'' if it has a component opposite to the direction of the displacement at the point of application of the force. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and unlike charges attract each other. An object with no net charge is referred to as neutral particle, electrically neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum mechanics, quantum effects. In an isolated system, the total charge stays the same - the amount of positive charge minus the amount of negative charge does not change over time. Electric charge is carried by subatomic particles. In ordinary matter, negative charge is carried by electrons, and positive charge is carried by the protons in the atomic nucleus, nuclei of atoms. If there are more electrons than protons in a piece of matter, it will have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Particle
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space. A point particle is an appropriate representation of any object whenever its size, shape, and structure are irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object. Point masses and point charges, discussed below, are two common cases. When a point particle has an additive property, such as mass or charge, it is often represented mathematically by a Dirac delta function. In classical mechanics there is usually no concept of rotation of point particles about their "center". In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
