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Clebsch Representation
In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field \boldsymbol(\boldsymbol) is: \boldsymbol = \boldsymbol \varphi + \psi\, \boldsymbol \chi, where the scalar fields \varphi(\boldsymbol), \psi(\boldsymbol) and \chi(\boldsymbol) are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and \boldsymbol is the gradient operator. Background In fluid dynamics and plasma physics, the Clebsch representation provides a means to overcome the difficulties to describe an inviscid flow with non-zero vorticity – in the Eulerian reference frame – using Lagrangian mechanics and Hamiltonian mechanics. At the critical point of such functionals the result is the Euler equations, a set of equations describing the fluid flow. Note that the mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Euler Equations (fluid Dynamics)
In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is divergence-free. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vector Calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, '' Vector Analysis'', though earlier mathematicians such as Isaac Newton pioneered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vector Calculus Identity
The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: : \operatorname(f) = \nabla f = \begin\displaystyle \frac,\ \frac,\ \frac \end f = \frac \mathbf + \frac \mathbf + \frac \mathbf where i, j, k are the standard unit vectors for the ''x'', ''y'', ''z''-axes. More generally, for a function of ''n'' variables \psi(x_1, \ldots, x_n), also called a scalar field, the gradient is the vector field: \nabla\psi = \begin\displaystyle\frac, \ldots, \frac \end\psi = \frac \mathbf_1 + \dots + \frac\mathbf_n where \mathbf_ \, (i=1,2,..., n) are mutually orthogonal unit vectors. As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change. For a vector field \mathbf = \left(A_1, \ldots, A_n\right), also called a tensor field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Helmholtz Decomposition
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz. Definition For a vector field \mathbf \in C^1(V, \mathbb^n) defined on a domain V \subseteq \mathbb^n, a Helmholtz decomposition is a pair of vector fields \mathbf \in C^1(V, \mathbb^n) and \mathbf \in C^1(V, \mathbb^n) such that: \begin \mathbf(\mathbf) &= \mathbf(\mathbf) + \mathbf(\mathbf), \\ \mathbf(\mathbf) &= - \nabla \Phi(\mathbf), \\ \nabla \cdot \mathbf(\mathbf) &= 0. \end Here, \Phi \in C^2(V, \mathbb) is a scalar potential, \nabla \Phi is its gradient, and \nabla \cdot \mathbf is the divergence of the vector fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Solenoidal
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf = 0. A common way of expressing this property is to say that the field has no sources or sinks.This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017. Properties The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where d\mathbf is the outward normal to each surface el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Constraint (mathematics)
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set. Example The following is a simple optimization problem: :\min f(\mathbf x) = x_1^2+x_2^4 subject to :x_1 \ge 1 and :x_2 = 1, where \mathbf x denotes the vector (''x''1, ''x''2). In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions. Without the constraints, the solution would be (0,0), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bounded Function
In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists a real number M such that :, f(x), \le M for all x in X. A function that is ''not'' bounded is said to be unbounded. If f is real-valued and f(x) \leq A for all x in X, then the function is said to be bounded (from) above by A. If f(x) \geq B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below. An important special case is a bounded sequence, where ''X'' is taken to be the set \mathbb N of natural numbers. Thus a sequence f = (a_0, a_1, a_2, \ldots) is bounded if there exists a real number M such that :, a_n, \le M for every natural number n. The set of all bounded sequences forms the sequence space l^\infty. The definition of boundedness can be generalized to functions f: X \rightarrow Y taking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Luke's Variational Principle
In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media. Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. Originally appeared in '' Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki'' 9(2): 86–94, 1968. This is often used when modelling the spectral density evolution of the free-surface in a sea state, sometimes called wave turbulence. Both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
