1.
Magnetohydrodynamics (journal)
–
This involves both classical and emerging areas in the physics, thermodynamics, hydrodynamics, and electrodynamics of magnetic fluids. As of 2010, the editor-in-chief is Andrejs Cēbers of the Institute of Physics of the University of Latvia, since 2001 the journal has been published solely in English. The English and online edition were published by Kluwer Academic Publishers through volume 36, now the entire content is available by subscription directly from the journal’s website. The journal is abstracted and indexed in the Science Citation Index Expanded, as well as the Journal Citation Reports, and Inspec
2.
Electrical resistivity and conductivity
–
Electrical resistivity is an intrinsic property that quantifies how strongly a given material opposes the flow of electric current. A low resistivity indicates a material that allows the flow of electric current. Resistivity is commonly represented by the Greek letter ρ, the SI unit of electrical resistivity is the ohm-metre. Electrical conductivity or specific conductance is the reciprocal of electrical resistivity and it is commonly represented by the Greek letter σ, but κ or γ are also occasionally used. Its SI unit is siemens per metre and CGSE unit is reciprocal second, many resistors and conductors have a uniform cross section with a uniform flow of electric current, and are made of one material. All copper wires, irrespective of their shape and size, have approximately the same resistivity, every material has its own characteristic resistivity – for example, resistivity of rubber is far larger than coppers. If the pipes are the size and shape, the pipe full of sand has higher resistance to flow. Resistance, however, is not solely determined by the presence or absence of sand and it also depends on the length and width of the pipe, short or wide pipes have lower resistance than narrow or long pipes. The above equation can be transposed to get Pouillets law, R = ρ ℓ A, the resistance of a given material increases with length, but decreases with increasing cross-sectional area. From the above equations, resistivity has the SI unit ohm metre, the formula R = ρ ℓ / A can be used to intuitively understand the meaning of a resistivity value. For example, if A = 7000100000000000000♠1 m2 ℓ = 7000100000000000000♠1 m, Conductivity, σ, is defined as the inverse of resistivity, σ =1 ρ. Conductivity has SI units of siemens per metre, the above definition was specific to resistors or conductors with a uniform cross-section, where current flows uniformly through them. A more basic and general definition starts from the fact that a field inside a material makes electric current flow. Conductivity is the inverse, σ =1 ρ = J E, for example, rubber is a material with large ρ and small σ—because even a very large electric field in rubber makes almost no current flow through it. On the other hand, copper is a material with small ρ, according to elementary quantum mechanics, electrons in an atom do not take on arbitrary energy values. Rather, electrons only occupy discrete energy levels in an atom or crystal. When a large number of such allowed energy levels are spaced close together —i. e. have similar energies— we can talk about these energy levels together as an energy band. There can be many such bands in a material, depending on the atomic number
3.
Plasma (physics)
–
Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas. Yet unlike these three states of matter, plasma does not naturally exist on the Earth under normal surface conditions, the term was first introduced by chemist Irving Langmuir in the 1920s. However, true plasma production is from the separation of these ions and electrons that produces an electric field. Based on the environmental temperature and density either partially ionised or fully ionised forms of plasma may be produced. The positive charge in ions is achieved by stripping away electrons from atomic nuclei, the number of electrons removed is related to either the increase in temperature or the local density of other ionised matter. Plasma may be the most abundant form of matter in the universe, although this is currently tentative based on the existence. Plasma is mostly associated with the Sun and stars, extending to the rarefied intracluster medium, Plasma was first identified in a Crookes tube, and so described by Sir William Crookes in 1879. The nature of the Crookes tube cathode ray matter was identified by British physicist Sir J. J. The term plasma was coined by Irving Langmuir in 1928, perhaps because the glowing discharge molds itself to the shape of the Crookes tube and we shall use the name plasma to describe this region containing balanced charges of ions and electrons. Plasma is a neutral medium of unbound positive and negative particles. Although these particles are unbound, they are not ‘free’ in the sense of not experiencing forces, in turn this governs collective behavior with many degrees of variation. The average number of particles in the Debye sphere is given by the plasma parameter, bulk interactions, The Debye screening length is short compared to the physical size of the plasma. This criterion means that interactions in the bulk of the plasma are more important than those at its edges, when this criterion is satisfied, the plasma is quasineutral. Plasma frequency, The electron plasma frequency is compared to the electron-neutral collision frequency. When this condition is valid, electrostatic interactions dominate over the processes of ordinary gas kinetics, for plasma to exist, ionization is necessary. The term plasma density by itself refers to the electron density, that is. The degree of ionization of a plasma is the proportion of atoms that have lost or gained electrons, even a partially ionized gas in which as little as 1% of the particles are ionized can have the characteristics of a plasma. The degree of ionization, α, is defined as α = n i n i + n n, where n i is the number density of ions and n n is the number density of neutral atoms
4.
Brine
–
Brine is a solution of salt in water. In different contexts, brine may refer to salt solutions ranging from about 3. 5% up to about 26%, fahrenheit could reliably reproduce by freezing brine. By reference, seawater usually freezes at −2 °C, at 100 °C, saturated sodium chloride brine is about 28% salt by weight. At 0 °C, brine can only hold about 26% salt, the thermal conductivity of seawater is 0.6 W/mK at 25 °C. The majority of hydrogen produced through electrolysis is a side product in the production of chlorine. 2 NaCl +2 H2O →2 NaOH + H2 + Cl2 Brine is a fluid used in large refrigeration installations for the transport of thermal energy from place to place. The lowest freezing point obtainable for NaCl brine is −21.1 °C at 23. 3wt% NaCl and this is called the eutectic point. In lower temperatures, a solution can be used to de-ice or reduce freezing temperatures on roads. In cooking, brine is used for food brining and salting, in hydrology, brine is a form of salt water, namely, water with relatively high concentration of salt. Brine mining Saline water Brackish water
5.
Electrolyte
–
An electrolyte is a substance that produces an electrically conducting solution when dissolved in a polar solvent, such as water. The dissolved electrolyte separates into cations and anions, which disperse uniformly through the solvent, electrically, such a solution is neutral. The movement of anions and cations in opposite directions within the solution amounts to a current and this includes most soluble salts, acids, and bases. Some gases, such as chloride, under conditions of high temperature or low pressure can also function as electrolytes. Electrolyte solutions can also result from the dissolution of some biological and synthetic polymers, termed polyelectrolytes, a substance that dissociates into ions in solution acquires the capacity to conduct electricity. Sodium, potassium, chloride, calcium, magnesium, and phosphate are examples of electrolytes, in medicine, electrolyte replacement is needed when a patient has prolonged vomiting or diarrhea, and as a response to strenuous athletic activity. Commercial electrolyte solutions are available, particularly for children and athletes. Electrolyte monitoring is important in the treatment of anorexia and bulimia, the word electrolyte derives from the Greek lytós, meaning able to be untied or loosened. Arrheniuss explanation was that in forming a solution, the salt dissociates into charged particles, faradays belief had been that ions were produced in the process of electrolysis. Arrhenius proposed that, even in the absence of an electric current and he thus proposed that chemical reactions in solution were reactions between ions. For example, carbon dioxide gas dissolves in water to produce a solution that contains hydronium, carbonate, molten salts can also be electrolytes as, for example, when sodium chloride is molten, the liquid conducts electricity. An electrolyte in a solution may be described as concentrated if it has a concentration of ions. If a high proportion of the solute dissociates to form ions, the electrolyte is strong, if most of the solute does not dissociate. The properties of electrolytes may be exploited using electrolysis to extract constituent elements, in physiology, the primary ions of electrolytes are sodium, potassium, calcium, magnesium, chloride, hydrogen phosphate, and hydrogen carbonate. The electric charge symbols of plus and minus indicate that the substance is ionic in nature and has a distribution of electrons. Sodium is the main electrolyte found in extracellular fluid and potassium is the main intracellular electrolyte, all known higher lifeforms require a subtle and complex electrolyte balance between the intracellular and extracellular environments. In particular, the maintenance of precise osmotic gradients of electrolytes is important, such gradients affect and regulate the hydration of the body as well as blood pH, and are critical for nerve and muscle function. Various mechanisms exist in living species that keep the concentrations of different electrolytes under tight control, both muscle tissue and neurons are considered electric tissues of the body
6.
Magnetic field
–
A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
7.
Water
–
Water is a transparent and nearly colorless chemical substance that is the main constituent of Earths streams, lakes, and oceans, and the fluids of most living organisms. Its chemical formula is H2O, meaning that its molecule contains one oxygen, Water strictly refers to the liquid state of that substance, that prevails at standard ambient temperature and pressure, but it often refers also to its solid state or its gaseous state. It also occurs in nature as snow, glaciers, ice packs and icebergs, clouds, fog, dew, aquifers, Water covers 71% of the Earths surface. It is vital for all forms of life. Only 2. 5% of this water is freshwater, and 98. 8% of that water is in ice and groundwater. Less than 0. 3% of all freshwater is in rivers, lakes, and the atmosphere, a greater quantity of water is found in the earths interior. Water on Earth moves continually through the cycle of evaporation and transpiration, condensation, precipitation. Evaporation and transpiration contribute to the precipitation over land, large amounts of water are also chemically combined or adsorbed in hydrated minerals. Safe drinking water is essential to humans and other even though it provides no calories or organic nutrients. There is a correlation between access to safe water and gross domestic product per capita. However, some observers have estimated that by 2025 more than half of the population will be facing water-based vulnerability. A report, issued in November 2009, suggests that by 2030, in developing regions of the world. Water plays an important role in the world economy, approximately 70% of the freshwater used by humans goes to agriculture. Fishing in salt and fresh water bodies is a source of food for many parts of the world. Much of long-distance trade of commodities and manufactured products is transported by boats through seas, rivers, lakes, large quantities of water, ice, and steam are used for cooling and heating, in industry and homes. Water is an excellent solvent for a variety of chemical substances, as such it is widely used in industrial processes. Water is also central to many sports and other forms of entertainment, such as swimming, pleasure boating, boat racing, surfing, sport fishing, Water is a liquid at the temperatures and pressures that are most adequate for life. Specifically, at atmospheric pressure of 1 bar, water is a liquid between the temperatures of 273.15 K and 373.15 K
8.
Nobel Prize
–
The Nobel Prize is a set of annual international awards bestowed in a number of categories by Swedish and Norwegian institutions in recognition of academic, cultural, or scientific advances. The will of the Swedish inventor Alfred Nobel established the prizes in 1895, the prizes in Chemistry, Literature, Peace, Physics, and Physiology or Medicine were first awarded in 1901. Medals made before 1980 were struck in 23 carat gold, between 1901 and 2016, the Nobel Prizes and the Prize in Economic Sciences were awarded 579 times to 911 people and organisations. With some receiving the Nobel Prize more than once, this makes a total of 23 organisations, the prize ceremonies take place annually in Stockholm, Sweden. Each recipient, or laureate, receives a medal, a diploma. The Nobel Prize is widely regarded as the most prestigious award available in the fields of literature, medicine, physics, chemistry, peace, and economics. The prize is not awarded posthumously, however, if a person is awarded a prize and dies before receiving it, though the average number of laureates per prize increased substantially during the 20th century, a prize may not be shared among more than three people. Alfred Nobel was born on 21 October 1833 in Stockholm, Sweden and he was a chemist, engineer, and inventor. In 1894, Nobel purchased the Bofors iron and steel mill and this invention was a precursor to many smokeless military explosives, especially the British smokeless powder cordite. As a consequence of his patent claims, Nobel was eventually involved in a patent infringement lawsuit over cordite, Nobel amassed a fortune during his lifetime, with most of his wealth from his 355 inventions, of which dynamite is the most famous. In 1888, Nobel was astonished to read his own obituary, titled The merchant of death is dead, as it was Alfreds brother Ludvig who had died, the obituary was eight years premature. The article disconcerted Nobel and made him apprehensive about how he would be remembered and this inspired him to change his will. On 10 December 1896, Alfred Nobel died in his villa in San Remo, Italy, Nobel wrote several wills during his lifetime. He composed the last over a year before he died, signing it at the Swedish–Norwegian Club in Paris on 27 November 1895, Nobel bequeathed 94% of his total assets,31 million SEK, to establish the five Nobel Prizes. Because of skepticism surrounding the will, it was not until 26 April 1897 that it was approved by the Storting in Norway. The executors of Nobels will, Ragnar Sohlman and Rudolf Lilljequist, formed the Nobel Foundation to take care of Nobels fortune, Nobels instructions named a Norwegian Nobel Committee to award the Peace Prize, the members of whom were appointed shortly after the will was approved in April 1897. Soon thereafter, the other prize-awarding organisations were designated or established and these were Karolinska Institutet on 7 June, the Swedish Academy on 9 June, and the Royal Swedish Academy of Sciences on 11 June. The Nobel Foundation reached an agreement on guidelines for how the prizes should be awarded, and, in 1900, in 1905, the personal union between Sweden and Norway was dissolved
9.
Electromagnetic induction
–
Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor due to its dynamic interaction with a magnetic field. Michael Faraday is generally credited with the discovery of induction in 1831, Lenzs law describes the direction of the induced field. Faradays law was later generalized to become the Maxwell-Faraday equation, one of the four Maxwells equations in James Clerk Maxwells theory of electromagnetism, electromagnetic induction has found many applications in technology, including electrical components such as inductors and transformers, and devices such as electric motors and generators. Electromagnetic induction was first discovered by Michael Faraday, who made his discovery public in 1831 and it was discovered independently by Joseph Henry in 1832. In Faradays first experimental demonstration, he wrapped two wires around opposite sides of a ring or torus. He plugged one wire into a galvanometer, and watched it as he connected the wire to a battery. He saw a transient current, which he called a wave of electricity and this induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. Within two months, Faraday found several other manifestations of electromagnetic induction, Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, an exception was James Clerk Maxwell, who used Faradays ideas as the basis of his quantitative electromagnetic theory. Heavisides version is the form recognized today in the group of known as Maxwells equations. In 1834 Heinrich Lenz formulated the law named after him to describe the flux through the circuit, Lenzs law gives the direction of the induced EMF and current resulting from electromagnetic induction. Faradays law of induction makes use of the magnetic flux ΦB through a region of space enclosed by a wire loop. The magnetic flux is defined by an integral, Φ B = ∫ Σ B ⋅ d A. The dot product B·dA corresponds to an amount of magnetic flux. In more visual terms, the flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop. When the flux through the changes, Faradays law of induction says that the wire loop acquires an electromotive force. The direction of the force is given by Lenzs law which states that an induced current will flow in the direction that will oppose the change which produced it. This is due to the sign in the previous equation
10.
Fluid dynamics
–
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. They are expressed using the Reynolds transport theorem, in addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects, however, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of molecules is ignored. The unsimplified equations do not have a general solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve, some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation
11.
Maxwell's equations
–
Maxwells equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such as power generation, electric motors, wireless communication, cameras, televisions. Maxwells equations describe how electric and magnetic fields are generated by charges, currents, one important consequence of the equations is the demonstration of how fluctuating electric and magnetic fields can propagate at the speed of light. This electromagnetic radiation manifests itself in ways from radio waves to light. The equations have two major variants, the microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges, the macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a description of the electromagnetic response of materials. The term Maxwells equations is used for equivalent alternative formulations. The space-time formulations, are used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest. In many situations, though, deviations from Maxwells equations are immeasurably small, exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons. In the electric and magnetic field there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources, Gausss law describes how electric fields emanate from electric charges. Gausss law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles, the two homogeneous equations describe how the fields circulate around their respective sources. A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles, a version of this law was included in the original equations by Maxwell but, by convention, is no longer. The precise formulation of Maxwells equations depends on the definition of the quantities involved. Conventions differ with the systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently, the vector calculus formulation below has become standard. For formulations using tensor calculus or differential forms, see alternative formulations, for relativistically invariant formulations, see relativistic formulations
12.
Electromagnetism
–
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as fields, magnetic fields. The other three fundamental interactions are the interaction, the weak interaction, and gravitation. The word electromagnetism is a form of two Greek terms, ἤλεκτρον, ēlektron, amber, and μαγνῆτις λίθος magnētis lithos, which means magnesian stone. The electromagnetic force plays a role in determining the internal properties of most objects encountered in daily life. Ordinary matter takes its form as a result of forces between individual atoms and molecules in matter, and is a manifestation of the electromagnetic force. Electrons are bound by the force to atomic nuclei, and their orbital shapes. The electromagnetic force governs the processes involved in chemistry, which arise from interactions between the electrons of neighboring atoms, there are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential, although electromagnetism is considered one of the four fundamental forces, at high energy the weak force and electromagnetic force are unified as a single electroweak force. In the history of the universe, during the epoch the unified force broke into the two separate forces as the universe cooled. Originally, electricity and magnetism were considered to be two separate forces, Magnetic poles attract or repel one another in a manner similar to positive and negative charges and always exist as pairs, every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding magnetic field outside the wire. Its direction depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a field, or a magnet is moved towards or away from it. While preparing for a lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation. As he was setting up his materials, he noticed a compass needle deflected away from north when the electric current from the battery he was using was switched on. At the time of discovery, Ørsted did not suggest any explanation of the phenomenon. However, three later he began more intensive investigations
13.
Differential equation
–
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
14.
Numerical analysis
–
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
15.
Michael Faraday
–
Michael Faraday FRS was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism, although Faraday received little formal education, he was one of the most influential scientists in history. It was by his research on the field around a conductor carrying a direct current that Faraday established the basis for the concept of the electromagnetic field in physics. Faraday also established that magnetism could affect rays of light and that there was a relationship between the two phenomena. He similarly discovered the principles of induction and diamagnetism. His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, Faraday ultimately became the first and foremost Fullerian Professor of Chemistry at the Royal Institution of Great Britain, a lifetime position. James Clerk Maxwell took the work of Faraday and others and summarized it in a set of equations which is accepted as the basis of all theories of electromagnetic phenomena. The SI unit of capacitance is named in his honour, the farad, albert Einstein kept a picture of Faraday on his study wall, alongside pictures of Isaac Newton and James Clerk Maxwell. Faraday was born in Newington Butts, which is now part of the London Borough of Southwark but was then a part of Surrey. His family was not well off and his father, James, was a member of the Glassite sect of Christianity. James Faraday moved his wife and two children to London during the winter of 1790 from Outhgill in Westmorland, where he had been an apprentice to the village blacksmith, Michael was born in the autumn of that year. The young Michael Faraday, who was the third of four children, at the age of 14 he became an apprentice to George Riebau, a local bookbinder and bookseller in Blandford Street. During his seven-year apprenticeship Faraday read many books, including Isaac Wattss The Improvement of the Mind, at this time he also developed an interest in science, especially in electricity. Faraday was particularly inspired by the book Conversations on Chemistry by Jane Marcet, many of the tickets for these lectures were given to Faraday by William Dance, who was one of the founders of the Royal Philharmonic Society. Faraday subsequently sent Davy a 300-page book based on notes that he had taken during these lectures, davys reply was immediate, kind, and favourable. In 1813, when Davy damaged his eyesight in an accident with nitrogen trichloride, very soon Davy entrusted Faraday with the preparation of nitrogen trichloride samples, and they both were injured in an explosion of this very sensitive substance. In the class-based English society of the time, Faraday was not considered a gentleman, Faraday was forced to fill the role of valet as well as assistant throughout the trip. Davys wife, Jane Apreece, refused to treat Faraday as an equal, the trip did, however, give him access to the scientific elite of Europe and exposed him to a host of stimulating ideas
16.
Waterloo Bridge
–
Waterloo Bridge is a road and foot traffic bridge crossing the River Thames in London, between Blackfriars Bridge and Hungerford Bridge. Its name commemorates the victory of the British, the Dutch, thanks to its location at a strategic bend in the river, the views from the bridge are widely held to be the finest from any spot in London at ground level. The first bridge on the site was designed in 1809–10 by John Rennie for the Strand Bridge Company, before its opening it was known as the Strand Bridge. During the 1840s the bridge gained a reputation as a place for suicide attempts. In 1841 the American daredevil Samuel Gilbert Scott was killed performing a act in which he hung by a rope from a scaffold on the bridge. In 1844 Thomas Hood wrote the poem The Bridge of Sighs, paintings of the bridge were created by the French Impressionist Claude Monet and the English Romantic John Constable. The bridge was nationalised in 1878 and placed under the control of the Metropolitan Board of Works, Michael Faraday tried in 1832 to measure the potential difference between each side of the bridge caused by the ebbing salt water flowing through the Earths magnetic field. From 1884 serious problems were found in Rennies bridge piers, after scour from the river flow after Old London Bridge was demolished damaged their foundations. In 1925, a steel framework was built on top of the existing bridge. London County Council decided to demolish the bridge and replace it with a new designed by Sir Giles Gilbert Scott. The engineers were Ernest Buckton and John Cuerel of Rendel Palmer & Tritton, Scott, by his own admission, was no engineer and his design, with reinforced concrete beams under the footways, leaving the road to be supported by transverse slabs, was difficult to implement. The beams were shaped to look as much like arches as and they are clad in Portland stone, which cleans itself whenever it rains. To guard against the possibility of further subsidence from scour each pier was given a number of jacks that can be used to level the structure, the new bridge was partially opened on Tuesday 11 March 1942 and was completed in 1945. It is the only Thames bridge to have been damaged by German bombers during the Second World War, the building contractor was Peter Lind & Company Limited. It is frequently asserted that the force was largely female. Granite stones from the bridge were subsequently presented to various parts of the British world to further historic links in the British Commonwealth of Nations. The monument, built in 1945, is on Queens Wharf and it includes a bronze likeness of Paddy, a drinking fountain and drinking bowls below for dogs. The north end of the bridge passes above the Victoria Embankment where the road joins the Strand and this end housed the southern portal of the Kingsway Tramway Subway until the late 1950s
17.
Earth's magnetic field
–
Its magnitude at the Earths surface ranges from 25 to 65 microteslas. The North geomagnetic pole, located near Greenland in the hemisphere, is actually the south pole of the Earths magnetic field. Unlike a bar magnet, Earths magnetic field changes over time because it is generated by a geodynamo, however, at irregular intervals averaging several hundred thousand years, the Earths field reverses and the North and South Magnetic Poles relatively abruptly switch places. These reversals of the geomagnetic poles leave a record in rocks that are of value to paleomagnetists in calculating geomagnetic fields in the past, such information in turn is helpful in studying the motions of continents and ocean floors in the process of plate tectonics. The magnetosphere is the region above the ionosphere that is defined by the extent of the Earths magnetic field in space. The Earths magnetic field serves to deflect most of the solar wind, one stripping mechanism is for gas to be caught in bubbles of magnetic field, which are ripped off by solar winds. The study of past magnetic field of the Earth is known as paleomagnetism, reversals also provide the basis for magnetostratigraphy, a way of dating rocks and sediments. The field also magnetizes the crust, and magnetic anomalies can be used to search for deposits of metal ores, humans have used compasses for direction finding since the 11th century A. D. and for navigation since the 12th century. Although the magnetic declination does shift with time, this wandering is slow enough that a simple compass remains useful for navigation, using magnetoception various other organisms, ranging from some types of bacteria to pigeons, use the Earths magnetic field for orientation and navigation. At any location, the Earths magnetic field can be represented by a three-dimensional vector, a typical procedure for measuring its direction is to use a compass to determine the direction of magnetic North. Its angle relative to true North is the declination or variation, facing magnetic North, the angle the field makes with the horizontal is the inclination or magnetic dip. The intensity of the field is proportional to the force it exerts on a magnet, another common representation is in X, Y and Z coordinates. The intensity of the field is measured in gauss, but is generally reported in nanoteslas. A nanotesla is also referred to as a gamma, the tesla is the SI unit of the Magnetic field, B. The Earths field ranges between approximately 25,000 and 65,000 nT, by comparison, a strong refrigerator magnet has a field of about 10,000,000 nanoteslas. A map of intensity contours is called an isodynamic chart, as the World Magnetic Model shows, the intensity tends to decrease from the poles to the equator. A minimum intensity occurs in the South Atlantic Anomaly over South America while there are maxima over northern Canada, Siberia, the inclination is given by an angle that can assume values between -90° to 90°. In the northern hemisphere, the field points downwards and it is straight down at the North Magnetic Pole and rotates upwards as the latitude decreases until it is horizontal at the magnetic equator
18.
Lenz's law
–
Lenzs law is shown by the negative sign in Faradays law of induction, E = − ∂ Φ ∂ t, which indicates that the induced voltage and the change in magnetic flux have opposite signs. It is a law that specifies the direction of induced current. Lenzs law can be seen as analogous to Newtons third law in classic mechanics, for a rigorous mathematical treatment, see electromagnetic induction and Maxwells equations. If a change in the field of current i1 induces another electric current, i2. If these currents are in two coaxial circular conductors ℓ1 and ℓ2 respectively, and both are initially 0, then the currents i1 and i2 must counter-rotate, the opposing currents will repel each other as a result. Currents bound inside the atoms of strong magnets can create counter-rotating currents in a copper or aluminum pipe and this is shown by dropping the magnet through the pipe. The descent of the magnet inside the pipe is observably slower than when dropped outside the pipe, the induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the flux is increasing, the field acts in opposition to it. If it is decreasing, the field acts in the direction of the applied field to oppose the change. In electromagnetism, when charges move along electric field lines work is done on them, when net positive work is applied to a charge q1, it gains speed and momentum. The net work on q1 thereby generates a field whose strength is proportional to the speed increase of q1. This magnetic field can interact with a neighboring charge q2, passing on this momentum to it, the charge q2 can also act on q1 in a similar manner, by which it returns some of the momentum that it received from q1. This back-and-forth component of momentum contributes to magnetic inductance, the closer that q1 and q2 are, the greater the effect. When q2 is inside a conductive medium such as a thick slab made of copper or aluminum, the energy of q1 is not instantly consumed as heat generated by the current of q2 but is also stored in two opposing magnetic fields. Momentum must be conserved in the process, so if q1 is pushed in one direction, however, the situation becomes more complicated when the finite speed of electromagnetic wave propagation is introduced. Famous 19th century electrodynamicist James Clerk Maxwell called this the electromagnetic momentum, yet, such a treatment of fields may be necessary when Lenzs law is applied to opposite charges. It is normally assumed that the charges in question have the same sign, if they do not, such as a proton and an electron, the interaction is different. An electron generating a field would generate an EMF that causes a proton to accelerate in the same direction as the electron
19.
Topology
–
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
20.
Continuity equation
–
A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, continuity equations are a stronger, local form of conservation laws. For example, a version of the law of conservation of energy states that energy can neither be created nor destroyed—i. e. The total amount of energy is fixed and this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is conserved, Energy can neither be created nor destroyed. A continuity equation is the way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of charge at any point can only change by the amount of electric current flowing into or out of that point. In an everyday example, there is a continuity equation for the number of people alive, it has a term to account for people being born. Any continuity equation can be expressed in a form, which applies to any finite region. Continuity equations underlie more specific transport equations such as the equation, Boltzmann transport equation. Before we can write down the continuity equation, we must first define flux, the continuity equation is useful when there is some quantity q which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let ρ be the density of this property, i. e. the amount of q per unit volume. The way that this quantity q is flowing is described by its flux, the flux of q is a vector field, which we denote as j. Here are some examples and properties of flux, The dimension of flux is amount of q flowing per unit time, outside the pipe, where there is no water, the flux is zero. In a well-known example, the flux of electric charge is the current density. In a simple example, V could be a building, and q could be the number of people in the building, the surface S would consist of the walls, doors, roof, and foundation of the building. Terms that generate or remove q are referred to as a sources and this general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation, mathematically it is an automatic consequence of Maxwells equations, although charge conservation is more fundamental than Maxwells equations
21.
Cauchy momentum equation
–
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. U is the velocity vector field, which depends on time. Since this equation must hold for any volume, it must be true that the integrand is zero. The main step in deriving this equation is establishing that the derivative of the tensor is one of the forces that constitutes Fi. U ⊗ u is the dyad of the velocity, here j and s have same number of dimensions N as the flow speed and the body acceleration, while F, being a tensor, has N2. In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations, a significant feature of the Navier–Stokes equations is the presence of convective acceleration, the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent-acceleration, the acceleration of the flow field is a spatial effect. Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect, convective acceleration is present in most flows, but its dynamic effect is disregarded in creeping flow. Convective acceleration is represented by the nonlinear quantity u · ∇u, both interpretations give the same result, independent of the coordinate system — provided ∇ is interpreted as the covariant derivative. The convection term is written as u, where the advection operator u · ∇ is used. Usually this representation is preferred as it is simpler than the one in terms of the tensor derivative ∇u, here ∇u is the tensor derivative of the velocity vector, equal in Cartesian coordinates to the component-by-component gradient. Note that the gradient of a vector is being defined as mi = ∂m vi, the Lamb form has use in irrotational flow, where the curl of the velocity ω = ∇ × u is equal to zero. Therefore, this reduces to only u ⋅ ∇ u = ∇, the effect of stress in the continuum flow is represented by the ∇p and ∇ · τ terms, these are gradients of surface forces, analogous to stresses in a solid. Here ∇p is called the gradient and arises from the isotropic part of the Cauchy stress tensor. This part is given by normal stresses that turn up in almost all situations, the anisotropic part of the stress tensor gives rise to ∇ · τ, which conventionally describes viscous forces, for incompressible flow, this is only a shear effect. Thus, τ is the stress tensor, and the stress tensor is equal to, σ = − p 1 + τ where 1 is the identity matrix in the space considered. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, by assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations. The divergence of the stress tensor can be written as ∇ ⋅ σ = − ∇ p + ∇ ⋅ τ, the effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure
22.
Conservation of energy
–
In physics, the law of conservation of energy states that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can neither be created nor destroyed, rather, it transforms from one form to another, for instance, chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite. A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist and that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings. Ancient philosophers as far back as Thales of Miletus c.550 BCE had inklings of the conservation of some underlying substance of everything is made. However, there is no reason to identify this with what we know today as mass-energy. Empedocles wrote that in his system, composed of four roots, nothing comes to be or perishes, instead. In 1605, Simon Stevinus was able to solve a number of problems in statics based on the principle that perpetual motion was impossible. Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. The remarkable aspect of this observation is that the height that a moving body ascends to does not depend on the shape of the surface that the body is moving on. In 1669, Christian Huygens published his laws of collision, among the quantities he listed as being invariant before and after the collision of bodies were both the sum of their linear momentums as well as the sum of their kinetic energies. However, the difference between elastic and inelastic collision was not understood at the time and this led to the dispute among later researchers as to which of these conserved quantities was the more fundamental. In his Horologium Oscillatorium, he gave a much clearer statement regarding the height of ascent of a moving body, Huygens study of the dynamics of pendulum motion was based on a single principle, that the center of gravity of heavy objects cannot lift itself. The fact that energy is scalar, unlike linear momentum which is a vector. It was Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion. Using Huygens work on collision, Leibniz noticed that in mechanical systems. He called this quantity the vis viva or living force of the system, the principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Many physicists at that time, such as Newton, held that the conservation of momentum and it was later shown that both quantities are conserved simultaneously, given the proper conditions such as an elastic collision. In 1687, Isaac Newton published his Principia, which was organized around the concept of force and momentum
23.
Adiabatic process
–
In thermodynamics, an adiabatic process is one that occurs without transfer of heat or matter between a thermodynamic system and its surroundings. In an adiabatic process, energy is transferred only as work, the adiabatic process provides a rigorous conceptual basis for the theory used to expound the first law of thermodynamics, and as such it is a key concept in thermodynamics. The adiabatic flame temperature is the temperature that would be achieved by a if the process of combustion took place in the absence of heat loss to the surroundings. A process that does not involve the transfer of heat or matter into or out of a system, so that Q =0, is called an adiabatic process, the assumption that a process is adiabatic is a frequently made simplifying assumption. Even though the cylinders are not insulated and are quite conductive, the same can be said to be true for the expansion process of such a system. The assumption of adiabatic isolation of a system is a useful one, the behaviour of actual machines deviates from these idealizations, but the assumption of such perfect behaviour provide a useful first approximation of how the real world works. According to Laplace, when sound travels in a gas, there is no loss of heat in the medium and the propagation of sound is adiabatic. For this adiabatic process, the modulus of elasticity E = γP where γ is the ratio of specific heats at constant pressure and at constant volume, such a process is called an isentropic process and is said to be reversible. Fictively, if the process is reversed, the energy added as work can be recovered entirely as work done by the system, if the walls of a system are not adiabatic, and energy is transferred in as heat, entropy is transferred into the system with the heat. Such a process is neither adiabatic nor isentropic, having Q >0, naturally occurring adiabatic processes are irreversible. The transfer of energy as work into an isolated system can be imagined as being of two idealized extreme kinds. In one such kind, there is no entropy produced within the system, in nature, this ideal kind occurs only approximately, because it demands an infinitely slow process and no sources of dissipation. The other extreme kind of work is work, for which energy is added as work solely through friction or viscous dissipation within the system. The second law of thermodynamics observes that a process, of transfer of energy as work, always consists at least of isochoric work. Every natural process, adiabatic or not, is irreversible, with ΔS >0, the adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature, in contrast, free expansion is an isothermal process for an ideal gas. Adiabatic heating occurs when the pressure of a gas is increased from work done on it by its surroundings and this finds practical application in diesel engines which rely on the lack of quick heat dissipation during their compression stroke to elevate the fuel vapor temperature sufficiently to ignite it. Adiabatic heating occurs in the Earths atmosphere when an air mass descends, for example, in a wind, Foehn wind
24.
Isothermal process
–
An isothermal process is a change of a system, in which the temperature remains constant, ΔT =0. In contrast, a process is where a system exchanges no heat with its surroundings. In other words, in a process, the value ΔT =0 and therefore ΔU =0 but Q ≠0, while in an adiabatic process. Isothermal processes can occur in any kind of system that has some means of regulating the temperature, including highly structured machines, some parts of the cycles of some heat engines are carried out isothermally. In the thermodynamic analysis of reactions, it is usual to first analyze what happens under isothermal conditions. Phase changes, such as melting or evaporation, are also isothermal processes when, as is usually the case, isothermal processes are often used and a starting point in analyzing more complex, non-isothermal processes. Isothermal processes are of special interest for ideal gases and this is a consequence of Joules second law which states that the internal energy of a fixed amount of an ideal gas depends only on its temperature. Thus, in a process the internal energy of an ideal gas is constant. This is a result of the fact that in a gas there are no intermolecular forces. Note that this is only for ideal gases, the internal energy depends on pressure as well as on temperature for liquids, solids. In the isothermal compression of a gas there is work is done on the system to decrease the volume, doing work on the gas increases the internal energy and will tend to increase the temperature. To maintain the constant temperature energy must leave the system as heat, if the gas is ideal, the amount of energy entering the environment is equal to the work done on the gas, because internal energy does not change. For details of the calculations, see calculation of work, for an adiabatic process, in which no heat flows into or out of the gas because its container is well insulated, Q =0. If there is no work done, i. e. a free expansion. For an ideal gas, this means that the process is also isothermal, thus, specifying that a process is isothermal is not sufficient to specify a unique process. For the special case of a gas to which Boyles law applies, the value of the constant is nRT, where n is the number of moles of gas present and R is the ideal gas constant. In other words, the gas law pV = nRT applies. This means that p = n R T V = constant V holds, the family of curves generated by this equation is shown in the graph in Figure 1
25.
Density
–
The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume, ρ = m V, where ρ is the density, m is the mass, and V is the volume. In some cases, density is defined as its weight per unit volume. For a pure substance the density has the numerical value as its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity, osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser. Thus a relative density less than one means that the floats in water. The density of a material varies with temperature and pressure and this variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object, increasing the temperature of a substance decreases its density by increasing its volume. In most materials, heating the bottom of a results in convection of the heat from the bottom to the top. This causes it to rise relative to more dense unheated material, the reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is a property in that increasing the amount of a substance does not increase its density. Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated easily and compared with the mass, upon this discovery, he leapt from his bath and ran naked through the streets shouting, Eureka. As a result, the term eureka entered common parlance and is used today to indicate a moment of enlightenment, the story first appeared in written form in Vitruvius books of architecture, two centuries after it supposedly took place. Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time, from the equation for density, mass density has units of mass divided by volume. As there are units of mass and volume covering many different magnitudes there are a large number of units for mass density in use. The SI unit of kilogram per metre and the cgs unit of gram per cubic centimetre are probably the most commonly used units for density.1,000 kg/m3 equals 1 g/cm3. In industry, other larger or smaller units of mass and or volume are often more practical, see below for a list of some of the most common units of density
26.
Pressure
–
Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the relative to the ambient pressure. Various units are used to express pressure, Pressure may also be expressed in terms of standard atmospheric pressure, the atmosphere is equal to this pressure and the torr is defined as 1⁄760 of this. Manometric units such as the centimetre of water, millimetre of mercury, Pressure is the amount of force acting per unit area. The symbol for it is p or P, the IUPAC recommendation for pressure is a lower-case p. However, upper-case P is widely used. The usage of P vs p depends upon the field in one is working, on the nearby presence of other symbols for quantities such as power and momentum. Mathematically, p = F A where, p is the pressure, F is the normal force and it relates the vector surface element with the normal force acting on it. It is incorrect to say the pressure is directed in such or such direction, the pressure, as a scalar, has no direction. The force given by the relationship to the quantity has a direction. If we change the orientation of the element, the direction of the normal force changes accordingly. Pressure is distributed to solid boundaries or across arbitrary sections of normal to these boundaries or sections at every point. It is a parameter in thermodynamics, and it is conjugate to volume. The SI unit for pressure is the pascal, equal to one newton per square metre and this name for the unit was added in 1971, before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as pounds per square inch, the CGS unit of pressure is the barye, equal to 1 dyn·cm−2 or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre, but using the names kilogram, gram, kilogram-force, or gram-force as units of force is expressly forbidden in SI. The technical atmosphere is 1 kgf/cm2, since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to density and may be expressed in units such as joules per cubic metre. Similar pressures are given in kilopascals in most other fields, where the prefix is rarely used
27.
Electric field
–
An electric field is a vector field that associates to each point in space the Coulomb force that would be experienced per unit of electric charge, by an infinitesimal test charge at that point. Electric fields are created by electric charges and can be induced by time-varying magnetic fields, the electric field combines with the magnetic field to form the electromagnetic field. The electric field, E, at a point is defined as the force, F. A particle of charge q would be subject to a force F = q E and its SI units are newtons per coulomb or, equivalently, volts per metre, which in terms of SI base units are kg⋅m⋅s−3⋅A−1. Electric fields are caused by electric charges or varying magnetic fields, in the special case of a steady state, the Maxwell-Faraday inductive effect disappears. The resulting two equations, taken together, are equivalent to Coulombs law, written as E =14 π ε0 ∫ d r ′ ρ r − r ′ | r − r ′ |3 for a charge density ρ. Notice that ε0, the permittivity of vacuum, must be substituted if charges are considered in non-empty media, the equations of electromagnetism are best described in a continuous description. A charge q located at r 0 can be described mathematically as a charge density ρ = q δ, conversely, a charge distribution can be approximated by many small point charges. Electric fields satisfy the principle, because Maxwells equations are linear. This principle is useful to calculate the field created by point charges. Q n are stationary in space at r 1, r 2, in that case, Coulombs law fully describes the field. If a system is static, such that magnetic fields are not time-varying, then by Faradays law, in this case, one can define an electric potential, that is, a function Φ such that E = − ∇ Φ. This is analogous to the gravitational potential, Coulombs law, which describes the interaction of electric charges, F = q = q E is similar to Newtons law of universal gravitation, F = m = m g. This suggests similarities between the electric field E and the gravitational field g, or their associated potentials, mass is sometimes called gravitational charge because of that similarity. Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law, a uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to other and maintaining a voltage between them, it is only an approximation because of boundary effects. Assuming infinite planes, the magnitude of the electric field E is, electrodynamic fields are E-fields which do change with time, for instance when charges are in motion. The electric field cannot be described independently of the field in that case
28.
Gradient
–
In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a variable, for functions of several variables. The gradient is a function, as opposed to a derivative. If f is a differentiable, real-valued function of several variables, like the derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, the components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. The Jacobian is the generalization of the gradient for vector-valued functions of several variables, a further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by a field, T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, the magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point is H, the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%, if, instead, the road goes around the hill at an angle, then it will have a shallower slope. This observation can be stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a unit vector is equal to the directional derivative of H in the direction of that unit vector. The gradient of a function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator. The notation grad f is commonly used for the gradient. The gradient of f is defined as the vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is
29.
Curl (mathematics)
–
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector, the attributes of this vector characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the rule. If the vector represents the flow velocity of a moving fluid. A vector field whose curl is zero is called irrotational, the curl is a form of differentiation for vector fields. The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used for curl F and this is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. The name curl was first suggested by James Clerk Maxwell in 1871, the curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. As such, the curl operator maps continuously differentiable functions f, ℝ3 → ℝ3 to continuous functions g, in fact, it maps Ck functions in ℝ3 to Ck −1 functions in ℝ3. Implicitly, curl is defined by, ⋅ n ^ = d e f lim A →0 where ∮C F · dr is a line integral along the boundary of the area in question, and | A | is the magnitude of the area. Note that the equation for each component, k can be obtained by exchanging each occurrence of a subscript 1,2,3 in cyclic permutation, 1→2, 2→3, and 3→1. If are the Cartesian coordinates and are the coordinates, then h i =2 +2 +2 is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices,3,1,2 →1,2,3 →2,3,1. Suppose the vector field describes the velocity field of a fluid flow, if the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, such notation involving operators is common in physics and algebra. However, in coordinate systems, such as polar-toroidal coordinates. This expands as follows, i + j + k Although expressed in terms of coordinates, equivalently, = e k ε k l m ∇ l F m where ek are the coordinate vector fields. Equivalently, using the derivative, the curl can be expressed as, ∇ × F = ♯ Here ♭ and ♯ are the musical isomorphisms
30.
Lorentz force
–
In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with velocity v in the presence of an electric field E, the first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell. Hendrik Lorentz derived it a few years after Heaviside, the force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by, where × is the vector cross product. More explicitly stated, F = q in which r is the vector of the charged particle, t is time. The term qE is called the force, while the term qv × B is called the magnetic force. According to some definitions, the term Lorentz force refers specifically to the formula for the magnetic force and this article will not follow this nomenclature, In what follows, the term Lorentz force will refer only to the expression for the total force. The magnetic force component of the Lorentz force manifests itself as the force acts on a current-carrying wire in a magnetic field. In that context, it is called the Laplace force. For a continuous distribution in motion, the Lorentz force equation becomes. If both sides of this equation are divided by the volume of this piece of the charge distribution dV. Rather than the amount of charge and its velocity in electric and magnetic fields, see Covariant formulation of classical electromagnetism for more details. The above-mentioned formulae use SI units which are the most common among experimentalists, technicians, in cgs-Gaussian units, which are somewhat more common among theoretical physicists, one has instead F = q c g s. where c is the speed of light. Where ε0 is the permittivity and μ0 the vacuum permeability. In practice, the subscripts cgs and SI are always omitted, early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, however, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a balance, was able to definitively show through experiment that this was true. Soon after the discovery in 1820 by H. C, in all these descriptions, the force was always given in terms of the properties of the objects involved and the distances between them rather than in terms of electric and magnetic fields. J. J. Thomson was the first to attempt to derive from Maxwells field equations the electromagnetic forces on a charged object in terms of the objects properties
31.
Vector calculus identities
–
The gradient of a tensor field, A, of order n, is generally written as grad = ∇ A and is a tensor field of order n +1. In particular, if the field has order 0, ψ. The divergence of a field, A, of non-zero order n, is generally written as div = ∇ ⋅ A and is a contraction to a tensor field of order n −1. Specifically, the divergence of a vector is a scalar, specifically, for the outer product of two vectors, ∇ ⋅ = b + b. For a 3-dimensional vector field v, curl is also a 3-dimensional vector field, generally written as, ∇ × v or in Einstein notation as, ε i j k ∂ v k ∂ x j where ε is the Levi-Civita symbol. In Cartesian coordinates, the Laplacian of a function f is Δ f = ∇2 f = f = ∂2 f ∂ x 2 + ∂2 f ∂ y 2 + ∂2 f ∂ z 2. For a tensor field, A, the laplacian is generally written as, in Feynman subscript notation, ∇ B = A × + B where the notation ∇B means the subscripted gradient operates on only the factor B. A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed, the above identity is then expressed as, ∇ ˙ = A × + B where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, for the remainder of this article, Feynman subscript notation will be used where appropriate. Where JA denotes the Jacobian of A. Alternatively, using Feynman subscript notation, as a special case, when A = B,12 ∇ = J A T A = A + A ×. ∇ × = ∇ − ∇2 A Here, ∇2 is the vector Laplacian operating on the vector field A, exterior derivative Vector calculus Del in cylindrical and spherical coordinates Comparison of vector algebra and geometric algebra
32.
Heat capacity
–
Heat capacity or thermal capacity is a measurable physical quantity equal to the ratio of the heat added to an object to the resulting temperature change. The unit of capacity is joule per kelvin J K. Specific heat is the amount of heat needed to raise the temperature of one kilogram of mass by 1 kelvin, Heat capacity is an extensive property of matter, meaning it is proportional to the size of the system. The molar heat capacity is the capacity per unit amount of a pure substance. In some engineering contexts, the heat capacity is used. Other contributions can come from magnetic and electronic degrees of freedom in solids, for quantum mechanical reasons, at any given temperature, some of these degrees of freedom may be unavailable, or only partially available, to store thermal energy. In such cases, the capacity is a fraction of the maximum. As the temperature approaches zero, the heat capacity of a system approaches zero. Quantum theory can be used to predict the heat capacity of simple systems. In a previous theory of common in the early modern period, heat was thought to be a measurement of an invisible fluid. Bodies were capable of holding an amount of this fluid, hence the term heat capacity, named. Heat is no longer considered a fluid, but rather a transfer of disordered energy, nevertheless, at least in English, the term heat capacity survives. In some other languages, the thermal capacity is preferred. In the International System of Units, heat capacity has the unit joules per kelvin, if the temperature change is sufficiently small the heat capacity may be assumed to be constant, C = Q Δ T. Heat capacity is a property, meaning it depends on the extent or size of the physical system studied. A sample containing twice the amount of substance as another sample requires the transfer of twice the amount of heat to achieve the change in temperature. For many purposes it is convenient to report heat capacity as an intensive property. In practice, this is most often an expression of the property in relation to a unit of mass, in science and engineering, International standards now recommend that specific heat capacity always refer to division by mass
33.
Equation of state
–
In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describes the state of matter under a given set of physical conditions. It is an equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, the most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of the simplest equations of state for this purpose is the gas law. However, this becomes increasingly inaccurate at higher pressures and lower temperatures. Therefore, a number of more accurate equations of state have been developed for gases, at present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. Measurements of equation-of-state parameters, especially at pressures, can be made using lasers. In addition, there are equations of state describing solids. There are equations that model the interior of stars, including stars, dense matter. A related concept is the perfect fluid equation of state used in cosmology, in practical context, the equations of state are instrumental for PVT calculation in process engineering problems and especially in petroleum gas/liquid equilibrium calculations. A successful PVT model based on an equation of state can be helpful to determine the state of the flow regime. Boyles Law was perhaps the first expression of an equation of state, in 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the level in the short end of the tube and that in the long. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure, in mathematical form, this can be stated as, p V = c o n s t a n t. The above relationship has also attributed to Edme Mariotte and is sometimes referred to as Mariottes law. However, Mariottes work was not published until 1676, in 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80 kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, daltons Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone
![{\displaystyle {\frac {\omega }{k}}=\left[{\frac {1}{2}}(v_{A}^{2}+v_{s}^{2})\pm {\frac {1}{2}}{\sqrt {(v_{A}^{2}+v_{s}^{2})^{2}-4v_{s}^{2}v_{A}^{2}\cos ^{2}\theta }}\right]^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3eb43d5ddf7007337fd0ff6b3e6d367dcdeb9)
. Bibcode:2011NHESS..11.1091A. doi:10.5194/nhess-11-1091-2011.
