Vec may mean Mathematics vec A , the Vectorization mathematics vectorization of a matrix A . Vec denotes the category of vector spaces over the reals. Other Venetian language V neta , language code. Vecuronium , a muscle relaxant. vec, a sentient moravec robot from the Orion s Arm Universe Project see also Moravec robot See also VEC disambiguation Disambig ... more details
VEC is a three letter acronym which may refer to VEC M1 Veh culo de Exploraci n de Caballer a , a Spanish Army wheeled reconnaissance vehicle armed with a 25mm Bushmaster cannon. Vellore Engineering College , now known as Vellore Institute of Technology Victorian Electoral Commission The Irish Vocational Education Committee Volunteer Examiner Coordinator , an organization that has been approved by the Federal Communications Commission for the administration of amateur radio license examinations in the United States of America Veran polis Esporte Clube Recreativo e Cultural , a Brazilian football soccer club Vermont Electric Cooperative , United States Ventura Amtrak station , California, United States Amtrak station code VEC Virginia Employment Commission Virtual Experience Company , a former subsidiary of Blitz Games Studios See also Vec disambiguation disambig de VEC it VEC pt VEC ... more details
, then the vectors math vec Q h math and math vec F h math can be found via explicit integration, and the dynamics ... over dt partial L over partial vec omega partial L over partial vec omega times vec omega partial L over partial vec v times vec v, quad d over dt partial L over partial vec v partial L over partial vec v times vec omega, math math L vec omega, vec v 1 over 2 A vec omega, vec omega B vec omega, vec v 1 over 2 C vec v, vec v vec k, vec omega vec l, vec v . math Their first integrals read math J 0 left partial L over partial vec omega , vec omega right left partial L over partial vec v , vec v right L, quad J 1 left partial L over partial vec omega , partial L over partial vec v right , quad J 2 left partial L over partial vec v , partial L over partial vec v right math . Further integration produces ... more details
Orphan date November 2006 In Newtonian mechanics , many of the quantities in linear motion and rotational motion are analogous, in that they act the same way in many equations. Note that the vector quantities in rotational motion are actually pseudovector s which point along the axis of rotation according to the right hand rule . border colspan 2 Linear quantities colspan 2 Rotational quantities math vec s math displacement vector displacement math vec theta math angular displacement ref In some ways, angular displacement should not be considered a vector, because addition of angular displacements unlike vectors is not commutative, since rotation is not commutative in 3 or more dimensions. ref math vec v math velocity math vec omega math angular velocity math vec a math acceleration math vec alpha math angular acceleration math m math mass math I math moment of inertia math vec p math momentum math vec L math angular momentum math vec F math force math vec tau math torque border Linear motion Rotational motion math vec v frac d vec s dt math math vec omega frac d vec theta dt math math vec a frac d vec v dt math math vec alpha frac d vec omega dt math math vec p m vec v math math vec L I vec omega math math vec F m vec a math math vec tau I vec alpha math math vec F frac d vec p dt math math vec tau frac d vec L dt math math dW vec F cdot d vec s math math dW vec tau cdot d vec theta math math E frac1 2 mv 2 math math E frac1 2 I omega 2 math Footnotes div class references small references div br Category Classical mechanics Category Fundamental physics concepts ... more details
, but can be e.g. arc length of the curve. Derivatives of Position Velocity math vec v frac d vec r dt math Acceleration math vec a frac d vec v dt frac d 2 vec r dt 2 math Jolt Jerk Surge Lurch math vec j frac d vec a dt frac d 2 vec v dt 2 frac d 3 vec r dt 3 math Snap Jounce math vec s frac d vec j dt frac d 2 vec a dt 2 frac d 3 vec v dt 3 frac d 4 vec r dt 4 math Crackle Trounce math vec c frac d vec s dt frac d 2 vec j dt 2 frac d 3 vec a dt 3 frac d 4 vec v dt 4 frac d 5 vec r dt 5 math Pop Pounce math vec p frac d vec c dt frac d 2 vec s dt 2 frac d 3 vec j dt 3 frac d 4 vec a dt 4 frac d 5 vec v dt 5 frac d 6 vec r dt 6 math Lock math vec l frac d vec p dt frac d 2 vec c dt 2 frac d 3 vec s dt 3 frac d 4 vec j dt 4 frac d 5 vec a dt 5 frac d 6 vec v dt 6 frac d 7 vec r dt 7 math Drop math vec d frac d vec l dt frac d 2 vec p dt 2 frac d 3 vec c dt 3 frac d 4 vec s dt 4 frac d 5 vec j dt 5 frac d 6 vec a dt 6 frac d 7 vec v dt 7 frac d 8 vec r dt 8 math Where math vec r math is the position vector, math vec v math is the velocity vector, math vec a math is the acceleration vector, math vec j math is the jerk vector, math vec s math is the snap vector, math vec c math is the crackle vector, math vec p math is the pop vector, math vec l math is the lock vector, and math vec d math is the drop vector. Citation needed date January 2012 Relationship to displacement vectors ... more details
form, math vec r vec a cdot vec n 0 , math a plane is given by a normal vector math vec n math as well as an arbitrary position vector math vec a math of a point math A in E math . The direction of math vec n math is chosen to satisfy the following inequality math vec a cdot vec n geq 0 , math By dividing the normal vector math vec n math by its Euclidean vector Length of a vector Magnitude math vec n math , we obtain the unit or normalized normal vector math vec n 0 vec n over vec n , math and the above equation can be rewritten as math vec r vec a cdot vec n 0 0. , math Substituting math d vec a cdot vec n 0 0 , math we obtain the Hesse normal form math vec r cdot vec n 0 d 0. , math .... Because math vec r cdot vec n 0 d math holds for every point in the plane, it is also true at point Q the point where the vector from the origin meets the plane E , with math vec r vec r s math , per the definition of the Scalar product math d vec r s cdot vec n 0 vec r s cdot vec n 0 cdot cos 0 circ vec r s cdot 1 vec r s . , math The magnitude math vec r s math of math vec r s math is the shortest ... more details
the evolution of the vorticity math vec omega math of a fluid element as it moves around. The vorticity ... torques and line forces, the momentum conservation equation gives, math frac D vec V D t frac partial vec V partial t vec V cdot vec nabla vec V frac 1 rho vec nabla p rho vec B frac vec nabla ... vec omega vec nabla times vec V math . Taking curl of momentum equation yields the desired equation. The following identities are useful in derivation of the equation, math vec V cdot vec nabla vec V vec nabla tfrac 1 2 vec V cdot vec V vec V times vec omega math math vec nabla times vec V times vec omega vec omega vec nabla cdot vec V vec omega cdot vec nabla vec V vec V cdot vec nabla vec omega math math vec nabla times vec nabla phi 0 math , where math phi math is a scalar. math vec nabla cdot vec omega 0 math ref In its general vector form it may be expressed as follows, math begin align frac D vec omega Dt & frac partial vec omega partial t vec V cdot vec nabla vec omega & vec omega cdot vec nabla vec V vec omega vec nabla cdot vec V frac 1 rho 2 vec nabla rho times vec nabla p vec nabla times left frac vec nabla cdot underline underline tau rho right vec nabla times vec B end align math where, math vec V math is the velocity vector, math rho math is the density, math p math is the pressure, math underline underline tau math is the viscous stress tensor and math vec B math is the body ... math tfrac D vec omega Dt tfrac partial vec omega partial t vec V cdot vec nabla vec omega math is the Substantive derivative material derivative of the vorticity vector math vec omega math . It describes ... partial vec omega partial t math the unsteady term or due to the motion of the fluid particle as it moves from one point to another, math vec V cdot vec nabla vec omega math the convection term . The first term on the RHS of the vorticity equation, math vec omega cdot vec nabla vec V math , describes the stretching or tilting of vorticity due to the velocity gradients. Note that math vec nabla vec ... more details
, math frac partial f partial t vec v cdot frac partial f partial vec x frac vec F m cdot frac partial f partial vec v 0, math and adapted it to the case of a plasma, leading to the systems .... Such description uses distribution function s math f e vec r , vec p ,t math and math f i vec r , vec p ,t math for electron s and positive plasma ion s. The distribution function math f alpha vec r , vec p ,t math for species math alpha math describes the number of particles of the species math alpha math having approximately the momentum math vec p math near the position vector position math vec r math at time math t math . Instead of the Boltzmann equation, the following system of equations ... partial f e partial t vec v e cdot nabla f e e Bigl vec E frac 1 c vec v times vec B Bigr cdot frac partial f e partial vec p 0 math math frac partial f i partial t vec v i cdot nabla f i e Bigl vec E frac 1 c vec v times vec B Bigr cdot frac partial f i partial vec p 0 math math nabla times vec B frac 4 pi vec j c frac 1 c frac partial vec E partial t , quad nabla times vec E frac 1 c frac partial vec B partial t math math nabla cdot vec E 4 pi rho, quad nabla cdot vec B 0 math math rho e int f i f e d 3 vec p , quad vec j e int f i f e vec v d 3 vec p , quad vec v alpha frac vec p m alpha ... , math m alpha math the mass of the electron and ion respectively, math vec E vec r ,t math and math vec B vec r ,t math represent collective self consistent electromagnetic field created in the point math vec r math at time moment math t math by all plasma particles. The essential difference of this system ... and ions math f e vec r , vec p ,t math and math f i vec r , vec p ,t math . The Vlasov Poisson ... zero magnetic field limit math frac partial f alpha partial t vec v cdot frac partial f alpha partial vec x frac q alpha vec E m alpha cdot frac partial f alpha partial vec v 0, math and Poisson s equation for self consistent electric field in CGS units math nabla cdot vec E nabla ... more details
quantum mechanics, the probability current math vec j math of the wave function math Psi math in one ... x right , math in three dimensions, this generalizes to math vec j frac hbar 2mi left Psi vec nabla Psi Psi vec nabla Psi right , math where math hbar math is the reduced Planck constant , m is the reduced ... , momentum space is possible . The 3 d form in terms of the real and imaginary parts are math vec j frac hbar m mathrm Im left Psi vec nabla Psi right mathrm Re left Psi frac hbar im vec nabla Psi right ... and electromagnetism math frac partial rho partial t vec nabla cdot vec j 0 math where ... Psi 2 partial t right mathrm d V int V left vec nabla cdot vec j right mathrm d V 0 math then the divergence ... partial partial t int V Psi 2 mathrm d V int S vec j cdot mathrm d vec S 0 math where the V is any volume ... 145546 9 ref math T left frac vec j mathrm trans vec j mathrm inc right , qquad R left frac vec j mathrm ref vec j mathrm inc right , math or equivalently math T frac vec j mathrm trans cdot vec n vec j mathrm inc cdot vec n , qquad R frac vec j mathrm ref cdot vec n vec j mathrm inc cdot vec n , math ... relation, a statement of probability conservation math vec j mathrm trans vec j mathrm ref vec j mathrm inc . math Examples Plane wave For the three dimensional plane wave math Psi A e i vec k cdot vec r e i omega t math the associated probability current is math vec j frac hbar 2mi A 2 left e i vec k cdot vec r vec nabla e i vec k cdot vec r e i vec k cdot vec r vec nabla e i vec k cdot vec r right A 2 frac hbar vec k m . math This is just the square of the amplitude of the wave times the particle s velocity, math vec v frac vec p m frac hbar vec k m math . Note that the probability current ... particle the Hamiltonian operator is math hat H frac 1 2m left vec hat p frac q c vec A right 2 q phi math where math vec hat p i hbar vec nabla math is the 3 d momentum operator , q is the electric charge of the particle. math phi phi vec x ,t math is the scalar potential , in this case ... more details
The Einstein Infeld Hoffmann equations of motion , jointly derived by Albert Einstein , Leopold Infeld and Banesh Hoffmann , are the differential equation differential equations of motion describing the approximate dynamics physics dynamics of a system of point like masses due to their mutual gravitational interactions, including general relativity general relativistic effects. It uses a first order post Newtonian expansion and thus is valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak. Given a system of N bodies, labelled by indices A 1, ..., N , the barycentric acceleration vector of body A is given by math begin align vec a A & sum B not A frac G m B vec n BA r AB 2 & quad frac 1 c 2 sum B not A frac G m B vec n BA r AB 2 left v A 2 2v B 2 4 vec v A cdot vec v B frac 3 2 vec n AB cdot vec v B 2 right. & qquad left. 4 sum C not A frac G m C r AC sum C not B frac G m C r BC frac 1 2 vec x B vec x A cdot vec a B right & quad frac 1 c 2 sum B not A frac G m B r AB 2 left vec n AB cdot 4 vec v A 3 vec v B right vec v A vec v B & quad frac 7 2c 2 sum B not A frac G m B vec a B r AB end align math where math vec x A math is the barycentric position vector of body A math vec v A d vec x A dt math is the barycentric velocity vector of body A math vec a A d 2 vec x A dt 2 math is the barycentric acceleration vector of body A math r AB vec x A vec x B math is the coordinate distance between bodies A and B math vec n AB vec x A vec x B r AB math is the unit vector pointing from body B to body A math m A math is the mass of body A. math c math is the speed of light math G math is the gravitational constant . The coordinates used here are harmonic coordinate condition harmonic . The first term on the right hand side is the Newtonian gravitational acceleration at A in the limit as c &rarr &infin , one recovers Newton s law ... more details
The Abstract Rewriting Machine ARM is a virtual machine which implements term rewriting for minimal term rewriting systems. Minimal term rewriting systems are left linear term rewriting system s in which each rule takes on one of six forms blockquote Continuation math f vec x , vec y , vec z rightarrow g vec x ,h vec y , vec z math Return math f x rightarrow x math Match math f vec x ,g vec y , vec z rightarrow h vec x , vec y , vec z math Add math f vec x , vec z rightarrow g vec x ,y, vec z rm for y in vec x cup vec z math Delete math f vec x , vec y , vec z rightarrow g vec x , vec z math Ident math f vec x rightarrow g vec x math blockquote Each of these six forms is mapped in ARM to one or a few processor instructions on most contemporary micro processors. Accordingly, minimal term rewriting is achieved at tens to hundreds of clock cycles per reduction step millions of reduction steps per second. ARM implements general term rewriting, in that every single sorted unconditional left linear term rewriting system can be transformed compiled into a minimal term rewriting system that gives rise to the same normal form relation. An overview with references to this compilation process for innermost rewriting, as well as a detailed overview of ARM, can be found in http portal.acm.org citation.cfm?id 291903&dl GUIDE&coll &CFID 15151515&CFTOKEN 6184618 Within ARM s reach compilation of left linear rewrite systems via minimal rewrite systems . A description for lazy non innermost rewriting can be found in http portal.acm.org citation.cfm?id 345102&dl ACM&coll ACM&CFID 15151515&CFTOKEN 6184618 Lazy rewriting on eager machinery . A documented implementation of ARM with the term rewriting language Epic is available http www.babelfish.nl epicarm.html here . Note that site and software are no longer being actively maintained. References J rgen Giesl and Aart Middeldorp, Transformation Techniques for Context Sensitive Rewrite Systems , Aachener Informatik Berichte, 2002 revised ... more details
Used predominantly in geometry , solid state physics , and mineralogy , particularly in describing crystal structure , a primitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell. The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller. A crystal can be categorized by its lattice and the atoms that lie in a primitive cell the basis . A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations. Primitive translation vectors are used to define a crystal translation vector, math vec T math , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors math vec a 1 math , math vec a 2 math , and math vec a 3 math are primitive if the atoms look the same from any lattice points using integers math u 1 math , math u 2 math , and math u 3 math . math vec T u 1 vec a 1 u 2 vec a 2 u 3 vec a 3 math The primitive cell is defined by the primitive axes vectors math vec a 1 math , math vec a 2 math , and math vec a 3 math . The volume, math V c math , of the primitive cell is given by the parallelepiped from the above axes as, math V c vec a 1 cdot vec a 2 times vec a 3 . , math Category Condensed matter physics Category Crystallography Category Mineralogy ... more details
The inverse Faraday effect is the effect opposite to the Faraday effect . A static magnetization math vec M 0 math is induced by an external oscillating electrical field with the frequency math omega math , which can be achieved with a high intensity laser pulse for example. The induced magnetization is proportional to the vector product of math vec E math and math vec E math math vec M 0 propto vec E omega times vec E omega math From this equation we see that the circularly polarized light with the frequency math omega math should induce a magnetization along the wave vector math vec k math . Because math vec E math is in the vector product , left and right handed Polarization waves polarization waves should induce magnetization of opposite signs. The induced magnetization is comparable to the saturated magnetization of the media. References R. Hertel, Microscopic theory of the inverse Faraday effect , http arxiv.org abs cond mat 0509060 2005 A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov and Th. Rasing, Ultrafast non thermal control of magnetization by instantaneous photomagnetic pulses , Nature 435, 655 657 2005 Category Optical phenomena Category Electric and magnetic fields in matter Category Article Feedback 5 sr ... more details
Hamiltonian fluid mechanics is the application of Hamiltonian mechanics Hamiltonian methods to fluid mechanics . This formalism can only apply to non dissipative fluids. Irrotational barotropic flow Take the simple example of a barotropic , inviscid vorticity free fluid. Then, the conjugate fields are the mass density field &rho and the velocity potential &phi . The Poisson bracket is given by math varphi vec x , rho vec y delta d vec x vec y math and the Hamiltonian by math mathcal H int mathrm d d x left frac 1 2 rho vec nabla varphi 2 e rho right , math where e is the internal energy density, as a function of &rho . For this barotropic flow, the internal energy is related to the pressure p by math e frac 1 rho p , math where an apostrophe , denotes differentiation with respect to &rho . This Hamiltonian structure gives rise to the following two equations of motion math begin align frac partial rho partial t & frac delta mathcal H delta varphi vec nabla cdot rho vec v , frac partial varphi partial t & frac delta mathcal H delta rho frac 1 2 vec v cdot vec v e , end align math where math vec v stackrel mathrm def nabla varphi math is the velocity and is vorticity free . The second equation leads to the Euler equations math frac partial vec v partial t vec v cdot nabla vec v e nabla rho frac 1 rho nabla p math after exploiting the fact that the vorticity is zero math vec nabla times vec v vec 0 . math See also Luke s variational principle References cite journal journal Annual Review of Fluid Mechanics volume 20 pages 225 256 year 1988 doi 10.1146 annurev.fl.20.010188.001301 title Hamiltonian Fluid Mechanics author R. Salmon bibcode 1988AnRFM..20..225S cite journal doi 10.1016 S0065 2687 08 60429 X title Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics author T. G. Shepherd year 1990 journal Advances in Geophysics volume 32 pages 287 338 Category Fluid dynamics Category Hamiltonian mechanics Category Dynamical systems ... more details
Current algebra is a mathematical framework in quantum field theory where the fields form a Lie algebra under their commutation relations. For instance, in a non Abelian Yang Mills symmetry, where &rho is the charge density, math rho a vec x , rho b vec y if ab c delta vec x vec y rho c vec x math where f are the structure constants of the Lie algebra. If space is a one dimensional circle, there may exist Group extension Central extension central extensions . See also affine Lie algebra Virasoro algebra References Sam B. Treiman Roman Jackiw David J. Gross , Lectures on current algebra and its applications . Princeton Series in Physics. Princeton University Press, Princeton, N.J., 1972. x 362 pp. Category Quantum field theory Category Lie algebras quantum stub algebra stub ar ... more details
The Poisson Boltzmann equation is a differential equation that describes electrostatic interactions between molecules in ionic solution s. It is the mathematical base for the Gouy Chapman double layer interfacial theory first proposed by Louis Georges Gouy Gouy in 1910 and complemented by Chapman in 1913. The equation is important in the fields of molecular dynamics and biophysics because it can be used in modeling implicit solvation , an approximation of the effects of solvent on the structures and interactions of proteins , DNA , RNA , and other molecules in solutions of different ionic strength . It is often difficult to solve the Poisson Boltzmann equation for complex systems, but several computer programs have been created to solve it numerical analysis numerically . The equation can be written as in cgs math vec nabla cdot left epsilon vec r vec nabla Psi vec r right 4 pi rho f vec r 4 pi sum i c i infty z i q lambda vec r exp left frac z i q Psi vec r k B T right math or in SI units or MKS system of units mks math vec nabla cdot left epsilon vec r vec nabla Psi vec r right rho f vec r sum i c i infty z i q lambda vec r exp left frac z i q Psi vec r k B T right math where math vec nabla cdot math is the divergence operator, math epsilon vec r math represents the position dependent dielectric, math vec nabla Psi vec r math represents the gradient of the electrostatic potential, math rho f vec r math represents the charge density of the solute, math c i infty math represents the concentration of the ion i at a distance of infinity from the solute, math z i math is the charge of the ion, q is the charge of a proton, math k B math is the Boltzmann constant , T is the temperature , and math lambda vec r math is a factor for the position dependent accessibility of position r to the ions in solution. If the potential is not large compared to kT, the equation can be linearization linearized to be solved more efficiently, leading to the Debye H ckel equation . ref name ... more details
Unreferenced date November 2006 A displacement field is an assignment of displacement vector displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation mechanics deformation on a rigid body. Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function math vec R 0 Omega rightarrow P math where math vec R 0 math is a placement vector math Omega math are all the points of the body math P math are all the points in the space in which the body is present Most often it is a state of the body in which no forces are applied. Then given any other state of this body in which coordinates of all its points are described as math vec R 1 math the displacement field is the difference between two body states math vec u vec R 1 vec R 0 math where math vec u math is a displacement field, which for each point of the body specifies a displacement vector displacement vector . See also Stress mechanics Stress DEFAULTSORT Displacement Field Mechanics Category Continuum mechanics Category Materials science pl Przemieszczenie mechanika ... more details
vector ,area ABDC vec r t times vec r t Delta t . math Hence, math operatorname vector ,area ABC vec r t times vec r t Delta t over 2 Delta vec A . math The areal velocity vector is math frac d vec A d t lim Delta t rightarrow 0 Delta vec A over Delta t lim Delta t rightarrow 0 vec r t times vec r t Delta t over 2 Delta t math math lim Delta t rightarrow 0 vec r t times vec r t vec r , t Delta t over 2 Delta t math math lim Delta t rightarrow 0 vec r t times vec r , t over 2 left Delta t over Delta t right math math vec r t times vec r , t over 2 . math But, math vec r , t math is the velocity vector math vec v t math of the moving particle, so that math frac d vec A d t vec r times vec v ... momentum of the particle is math vec L vec r times m vec v , math and hence math vec L 2 m frac d vec A d t math . The direction of the angular momentum vector L is always the same as that of the areal ... more details
math z f x,y , quad vec r x,y x, y, f x,y . math surface of revolution Surfaces of revolution ... x &le b is rotated about the z axis then the resulting surface has a parametrization math vec r ... geometry cylinder of radius R about x axis has the following parametric representation math vec ... by math vec r theta, phi cos theta sin phi, sin theta sin phi, cos phi , quad 0 leq theta .... For example, the coordinate z plane can be parametrized as math vec r u,v au bv,cu dv, 0 math ... integral integration . Notation Let the parametric surface be given by the equation math vec r vec r u,v , math where math vec r math is a vector valued function of the parameters u , v and the parameters ... to the parameters are usually denoted math vec r u math and math vec r v, math and similarly for the higher derivatives, math vec r uu , vec r uv , vec r vv . math In vector calculus , the parameters ... math frac partial vec r partial s , frac partial vec r partial t , frac partial 2 vec r partial s 2 , frac partial 2 vec r partial s partial t , frac partial 2 vec r partial t 2 . math Tangent ... math vec r u, vec r v math are linearly independent. The tangent plane at a regular point is the affine ... combination of math vec r u math and math vec r v. math The cross product of these vectors is a normal ... surface at a regular point math vec n frac vec r u times vec r v left vec r u times vec r v right . math In general, there are two choices of the unit normal vector to a surface at a given ... area can be calculated by integrating the length of the normal vector math vec r u times vec ... vec r u times vec r v right du dv. math Although this formula provides a closed expression for the surface ... is used to calculate distances and angles. For a parametrized surface math vec r vec r u,v , math its coefficients can be computed as follows math E vec r u cdot vec r u, quad F vec r u cdot vec r v, quad G vec r v cdot vec r v. math Arc length of parametrised curves on the surface S , the angle ... more details
Image VEC Zaragoza.jpg thumb 250px Spanish Army s VEC in Spanish Armed Forces s day 2008 parade in Zaragoza . The Pegaso VEC M1 is a Spain Spanish military cavalry reconnaissance vehicle. It started service in the Spanish Army in 1980 as BMR 625 VEC aka Pegaso 3562 and all of them were upgraded in late 90 s to the M1 version. The vehicle was developed and produced by Pegaso, now Iveco , as a derivative of the well known Pegaso BMR . It is a 6x6 , currently powered by a 315 Horsepower hp Scania AB Scania DS9 diesel 6 cylinder engine, disposed in the rear right side of the hull, which replaced the original Pegaso 306 hp engine. It mounts an automatic 25 mm chain gun M242 Bushmaster into a two man turret and a coaxial 7.62 mm MG3S machine gun. Six electrically fired smoke grenade launchers are located on the sides of the turret, three on the left side and three on the right. It had amphibious ability, as two hydrojets for displacement in water were an optional equipment. The crew in composed by five men the commander, the gunner, the driver and two scouts. Operators ESP Employed the VEC in combat in the Yugoslav Wars Balkans , Lebanon , and in the Iraq War , where they were favoured by their crews and command because of their good all round capabilities, mechanical reliability, armor and firepower. However, the VEC achieved a bad reputation being prone to overturn. External references http armyreco.ifrance.com europe espagne vehicules a roues vecvec espagne description.htm Army Recognition page on VEC M1 in French http www.portierramaryaire.com foro viewtopic.php?p 1242&sid c563bfe3b7a587391cac0fec3c3e14c5 VEC and BMR data in Spanish Modern IFV and APC DEFAULTSORT Vec M1 Category Wheeled reconnaissance vehicles Category Armoured fighting vehicles of Spain es VEC it VEC M1 ja VEC ... more details
Context date October 2009 In plasma physics , a Taylor state is the minimum energy state of a plasma physics plasma satisfying the constraint of conserving magnetic helicity . ref cite book author Paul M. Bellan year 2000 title Spheromaks A Practical Application of Magnetohydrodynamic dynamos and plasma self organization pages 71 79 id ISBN 1 86094 141 9 ref Derivation Consider a closed, simply connected, flux conserving, perfectly conducting surface math S math surrounding a plasma with negligible thermal energy math beta rightarrow 0 math . Since math vec B . vec ds 0 math on math S math . This implies that math vec A 0 math . As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies math delta vec B . vec ds 0 math and math delta vec A 0 math on math S math . We formulate a variational problem of minimizing the plasma energy math W int d 3rB 2 2 mu circ math while conserving magnetic helicity math K int d 3r vec A . vec B math . The variational problem is math delta W lambda delta K 0 math . After some algebra this leads to the following constraint for the minimum energy state Need to expand on the derivation, may be explain the terms vector potential etc. math nabla times vec B lambda vec B math . See also John Bryan Taylor References Reflist Category Plasma physics physics stub ... more details
The KPZ equation named after its creators Mehran Kardar , Giorgio Parisi , and Vi Cheng Zhang is a non linear stochastic partial differential equation . It describes the temporal change of the height math h vec x,t math at place math vec x math and time math t math . It is given by math frac partial h vec x,t partial t nu nabla 2 h frac lambda 2 left nabla h right 2 eta vec x,t , math where math eta vec x,t math is White noise white Gaussian noise with average math langle eta vec x,t rangle 0 math and second moment math langle eta vec x,t eta vec x ,t rangle 2D delta d vec x vec x delta t t math . math nu math , math lambda math , and math D math are parameters of the model and math d math is the dimension. By use of renormalization group techniques it can be shown that the KPZ equation is the field theory of many surface growth models, such as the Eden growth model Eden model , ballistic deposition, and the SOS model. Sources references After listing your sources please cite them using inline citations and place them after the information they cite. Please see http en.wikipedia.org wiki Wikipedia REFB for instructions on how to add citations. Mehran Kardar , Giorgio Parisi , and Yi Cheng Zhang, Dynamic Scaling of Growing Interfaces , Physical Review Letters, Vol. 56 , 889 892 1986 . http prl.aps.org abstract PRL v56 i9 p889 1 APS A. L.Barab si and H.E.Stanley, Fractal concepts in surface growth Cambridge University Press, 1995 noinclude Category Statistical mechanics noinclude es Ecuaci n de Kardar Parisi Zhang ... more details
representing Array data structure vectors with a class code Vec code . It is natural to want to overload code operator code and code operator code so you could write code Vec x alpha u v code where code alpha code is a scalar and code u code and code v code are code Vec code s. A naive implementation would have code operator code and code operator code return code Vec code s. However, then the above ... evaluation so the expression code Vec x alpha u v code essentially generates at compile time a new code Vec code constructor taking a scalar and two code Vec code s as follows using the curiously recurring ... Vec class class Vec public VecExpression Vec container type data public reference operator size ... data.size Vec size type n data n Construct a given size Construct from any VecExpression template typename E Vec VecExpression E const& vec E const& v vec data.resize v.size for size type i 0 i v.size ... E1, E2 E1 const& u E2 const& v public typedef Vec size type size type typedef Vec value type ... v.size size type size const return v.size value type operator Vec size type i const return u i v i ... VecScaled double alpha, VecExpression E const& v alpha alpha , v v Vec size type size const return v.size Vec value type operator Vec size type i const return alpha v i Now we can overload operators ..., the expression code alpha u v code is of type source lang cpp VecScaled VecDifference Vec,Vec source so calling code Vec x alpha u v code calls the constructor that takes a code VecExpression VecScaled VecDifference Vec,Vec code . Each line of the code for code loop then expands from source lang cpp ... more details
step are defined as follows Collision step math f i t vec x ,t delta t f i vec x ,t frac 1 tau f f i eq f i , math Streaming step math f i vec x vec e i delta t,t delta t f i t vec x ,t delta t , math ... expansion and the collision operator. math f i vec x vec e i delta t,t delta t f i vec x ,t frac 1 tau f f i eq f i math For simplicity, write math f i vec x ,t math as math f i , math . The slightly .... math frac part f i part t vec e i nabla f i K left frac 1 2 vec e i vec e i nabla nabla f i vec e ... math rho vec u sum i f i eq vec e i math math 0 sum i f i k math math 0 sum i f i k vec e i math ... are nearly derived. For order, math K 0 , math math frac part f i eq part t 1 vec e i nabla 1 f i eq ... t 2 vec e i nabla f i 1 frac 1 2 vec e i vec e i nabla nabla f i eq vec e i cdot nabla frac part f i ... t 2 left 1 frac 2 tau right left frac part f i 1 part t 1 vec e i nabla 1 f i 1 right frac f i 2 ... rho vec u 0 math math frac part rho vec u part t nabla cdot Pi 0 math The momentum flux tensor, math Pi , math , has the following form then. math Pi xy sum i vec e ix vec e iy left f i eq left 1 frac 1 2 tau right f i 1 right , math Where math vec e ix vec e iy math is shorthand for the square of the sum of all the components of math vec e i math i.e. math left sum x vec e ix right 2 sum x sum y vec e ix vec e iy math and the equilibrium particle distribution with second order in order to be comparable to the Navier Stokes equation is math f i eq omega i rho left 1 frac 3 vec e i vec u c 2 frac 9 vec e i vec u 2 2c 4 frac 3 vec u 2 2c 2 right math . The equilibirum distribution is only valid ... tensor leads to math Pi xy 0 sum i vec e ix vec e iy f i eq p delta xy rho u x u y , math math Pi xy 1 left 1 frac 1 2 tau right sum i vec e ix vec e iy f i 1 nu left nabla x left rho vec u y right nabla y left rho vec u x right right , math Finally, the Navier Stokes equation is recovered under the assumption that density variation is small. math rho left frac part vec u x part t nabla y cdot ... more details
In mathematics , especially in linear algebra and matrix theory , the commutation matrix is used for transforming the vectorized form of a matrix mathematics matrix into the vectorized form of its transpose . Specifically, the commutation matrix K sup m,n sup is the mn mn matrix which, for any m n matrix A , transforms vec A into vec A sup T sup K sup m,n sup vec A vec A sup T sup . Here vec A is the mn 1 column vector obtain by stacking the columns of A on top of one another vec A A sub 1,1 sub , ..., A sub m,1 sub , A sub 1,2 sub , ..., A sub m,2 sub , ..., A sub 1,n sub , ..., A sub m,n sub sup T sup where A A sub i,j sub . The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product for every m n matrix A and every r q matrix B , K sup r,m sup A math otimes math B K sup n,q sup B math otimes math A . References Jan R. Magnus and Heinz Neudecker 1988 , Matrix Differential Calculus with Applications in Statistics and Econometrics , Wiley. Category Linear algebra Category Matrices Linear algebra stub sl Komutacijska matrika ... more details
Encyclopedia
top
