Norway
Norway the Kingdom of Norway, is a Nordic country in Northern Europe whose territory comprises the western and northernmost portion of the Scandinavian Peninsula. The Antarctic Peter I Island and the sub-Antarctic Bouvet Island are dependent territories and thus not considered part of the kingdom. Norway lays claim to a section of Antarctica known as Queen Maud Land. Norway has a total area of 385,207 square kilometres and a population of 5,312,300; the country shares a long eastern border with Sweden. Norway is bordered by Finland and Russia to the north-east, the Skagerrak strait to the south, with Denmark on the other side. Norway has an extensive coastline, facing the Barents Sea. Harald V of the House of Glücksburg is the current King of Norway. Erna Solberg has been prime minister since 2013. A unitary sovereign state with a constitutional monarchy, Norway divides state power between the parliament, the cabinet and the supreme court, as determined by the 1814 constitution; the kingdom was established in 872 as a merger of a large number of petty kingdoms and has existed continuously for 1,147 years.
From 1537 to 1814, Norway was a part of the Kingdom of Denmark-Norway, from 1814 to 1905, it was in a personal union with the Kingdom of Sweden. Norway was neutral during the First World War. Norway remained neutral until April 1940 when the country was invaded and occupied by Germany until the end of Second World War. Norway has both administrative and political subdivisions on two levels: counties and municipalities; the Sámi people have a certain amount of self-determination and influence over traditional territories through the Sámi Parliament and the Finnmark Act. Norway maintains close ties with both the United States. Norway is a founding member of the United Nations, NATO, the European Free Trade Association, the Council of Europe, the Antarctic Treaty, the Nordic Council. Norway maintains the Nordic welfare model with universal health care and a comprehensive social security system, its values are rooted in egalitarian ideals; the Norwegian state has large ownership positions in key industrial sectors, having extensive reserves of petroleum, natural gas, lumber and fresh water.
The petroleum industry accounts for around a quarter of the country's gross domestic product. On a per-capita basis, Norway is the world's largest producer of oil and natural gas outside of the Middle East; the country has the fourth-highest per capita income in the world on the World IMF lists. On the CIA's GDP per capita list which includes autonomous territories and regions, Norway ranks as number eleven, it has the world's largest sovereign wealth fund, with a value of US$1 trillion. Norway has had the highest Human Development Index ranking in the world since 2009, a position held between 2001 and 2006, it had the highest inequality-adjusted ranking until 2018 when Iceland moved to the top of the list. Norway ranked first on the World Happiness Report for 2017 and ranks first on the OECD Better Life Index, the Index of Public Integrity, the Democracy Index. Norway has one of the lowest crime rates in the world. Norway has two official names: Norge in Noreg in Nynorsk; the English name Norway comes from the Old English word Norþweg mentioned in 880, meaning "northern way" or "way leading to the north", how the Anglo-Saxons referred to the coastline of Atlantic Norway similar to scientific consensus about the origin of the Norwegian language name.
The Anglo-Saxons of Britain referred to the kingdom of Norway in 880 as Norðmanna land. There is some disagreement about whether the native name of Norway had the same etymology as the English form. According to the traditional dominant view, the first component was norðr, a cognate of English north, so the full name was Norðr vegr, "the way northwards", referring to the sailing route along the Norwegian coast, contrasting with suðrvegar "southern way" for, austrvegr "eastern way" for the Baltic. In the translation of Orosius for Alfred, the name is Norðweg, while in younger Old English sources the ð is gone. In the 10th century many Norsemen settled in Northern France, according to the sagas, in the area, called Normandy from norðmann, although not a Norwegian possession. In France normanni or northmanni referred to people of Sweden or Denmark; until around 1800 inhabitants of Western Norway where referred to as nordmenn while inhabitants of Eastern Norway where referred to as austmenn. According to another theory, the first component was a word nór, meaning "narrow" or "northern", referring to the inner-archipelago sailing route through the land.
The interpretation as "northern", as reflected in the English and Latin forms of the name, would have been due to folk etymology. This latter view originated with philologist Niels Halvorsen Trønnes in 1847; the form Nore is still used in placenames such as the village of Nore and lake Norefjorden in Buskerud county, still has the same meaning. Among other arguments in favour of the theor
Sine wave
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave, it is named after the function sine. It occurs in pure and applied mathematics, as well as physics, signal processing and many other fields, its most basic form as a function of time is: y = A sin = A sin where: A, the peak deviation of the function from zero. F, ordinary frequency, the number of oscillations that occur each second of time. Ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second φ, specifies where in its cycle the oscillation is at t = 0. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds. A negative value represents a delay, a positive value represents an advance; the sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform; this property makes it acoustically unique.
In general, the function may have: a spatial variable x that represents the position on the dimension on which the wave propagates, a characteristic parameter k called wave number, which represents the proportionality between the angular frequency ω and the linear speed ν. The wavenumber is related to the angular frequency by:. K = ω v = 2 π f v = 2 π λ where λ is the wavelength, f is the frequency, v is the linear speed; this equation gives a sine wave for a single dimension. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed; this wave pattern occurs in nature, including wind waves, sound waves, light waves. A cosine wave is said to be sinusoidal, because cos = sin , a sine wave with a phase-shift of π/2 radians.
Because of this head start, it is said that the cosine function leads the sine function or the sine lags the cosine. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics. To the human ear, a sound, made of more than one sine wave will have perceptible harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, the reason why the same musical note played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform, including square waves. Fourier used it as an analytical tool in the study of waves and heat flow, it is used in signal processing and the statistical analysis of time series.
Since sine waves propagate without changing form in distributed linear systems, they are used to analyze wave propagation. Sine waves traveling in two directions in space can be represented as u = A sin When two waves having the same amplitude and frequency, traveling in opposite directions, superpose each other a standing wave pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed end
Frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency; the period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals, radio waves, light. For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics and radio, frequency is denoted by a Latin letter f or by the Greek letter ν or ν; the relation between the frequency and the period T of a repeating event or oscillation is given by f = 1 T.
The SI derived unit of frequency is the hertz, named after the German physicist Heinrich Hertz. One hertz means. If a TV has a refresh rate of 1 hertz the TV's screen will change its picture once a second. A previous name for this unit was cycles per second; the SI unit for period is the second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz. As a matter of convenience and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are described by their frequency instead of period; these used conversions are listed below: Angular frequency denoted by the Greek letter ω, is defined as the rate of change of angular displacement, θ, or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument to the sine function: y = sin = sin = sin d θ d t = ω = 2 π f Angular frequency is measured in radians per second but, for discrete-time signals, can be expressed as radians per sampling interval, a dimensionless quantity.
Angular frequency is larger than regular frequency by a factor of 2π. Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.: y = sin = sin d θ d x = k Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has an inverse relationship to the wavelength, λ. In dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ. In the special case of electromagnetic waves moving through a vacuum v = c, where c is the speed of light in a vacuum, this expression becomes: f = c λ; when waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can done in the following ways, Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period dividing the count by the length of the time period.
For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz If the number of counts is not large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count; this is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T
Conservative vector field
In vector calculus, a conservative vector field is a vector field, the gradient of some function. Conservative vector fields have the property that the line integral is path independent, i.e. the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is irrotational. An irrotational vector field is conservative provided that the domain is connected. Conservative vector fields appear in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path, so it is possible to define a potential energy, independent of the actual path taken. In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure.
Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements d R that don't have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy; this is. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance. M. C. Escher's painting Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground as one moves along the staircase.
It is rotational in that one can keep getting higher or keep getting lower while going around in circles. It is non-conservative in that one can return to one's starting point while ascending more than one descends or vice versa. On a real staircase, the height above the ground is a scalar potential field: If one returns to the same place, one goes upward as much as one goes downward, its gradient is irrotational. The situation depicted in the painting is impossible. A vector field v: U → R n, where U is an open subset of R n, is said to be conservative if and only if there exists a C 1 scalar field φ on U such that v = ∇ φ. Here, ∇ φ denotes the gradient of φ; when the equation above holds, φ is called a scalar potential for v. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. A key property of a conservative vector field v is that its integral along a path depends only on the endpoints of that path, not the particular route taken.
Suppose that P is a rectifiable path in U with initial point A and terminal point B. If v = ∇ φ for some C 1 scalar field φ so that v is a conservative vector field the gradient theorem states that ∫ P v ⋅ d r = φ − φ; this holds as the fundamental theorem of calculus. An equivalent formulation of this is that ∮ C v ⋅ d r = 0 for every rectifiable simple closed path C in U; the converse of this statement is true: If the circulation of v around every rectifiable simple closed path in U is 0 v is a conservative vector field. Let n =
Atmosphere of Earth
The atmosphere of Earth is the layer of gases known as air, that surrounds the planet Earth and is retained by Earth's gravity. The atmosphere of Earth protects life on Earth by creating pressure allowing for liquid water to exist on the Earth's surface, absorbing ultraviolet solar radiation, warming the surface through heat retention, reducing temperature extremes between day and night. By volume, dry air contains 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, small amounts of other gases. Air contains a variable amount of water vapor, on average around 1% at sea level, 0.4% over the entire atmosphere. Air content and atmospheric pressure vary at different layers, air suitable for use in photosynthesis by terrestrial plants and breathing of terrestrial animals is found only in Earth's troposphere and in artificial atmospheres; the atmosphere has a mass of about 5.15×1018 kg, three quarters of, within about 11 km of the surface. The atmosphere becomes thinner and thinner with increasing altitude, with no definite boundary between the atmosphere and outer space.
The Kármán line, at 100 km, or 1.57% of Earth's radius, is used as the border between the atmosphere and outer space. Atmospheric effects become noticeable during atmospheric reentry of spacecraft at an altitude of around 120 km. Several layers can be distinguished in the atmosphere, based on characteristics such as temperature and composition; the study of Earth's atmosphere and its processes is called atmospheric science. Early pioneers in the field include Richard Assmann; the three major constituents of Earth's atmosphere are nitrogen and argon. Water vapor accounts for 0.25% of the atmosphere by mass. The concentration of water vapor varies from around 10 ppm by volume in the coldest portions of the atmosphere to as much as 5% by volume in hot, humid air masses, concentrations of other atmospheric gases are quoted in terms of dry air; the remaining gases are referred to as trace gases, among which are the greenhouse gases, principally carbon dioxide, nitrous oxide, ozone. Filtered air includes trace amounts of many other chemical compounds.
Many substances of natural origin may be present in locally and seasonally variable small amounts as aerosols in an unfiltered air sample, including dust of mineral and organic composition and spores, sea spray, volcanic ash. Various industrial pollutants may be present as gases or aerosols, such as chlorine, fluorine compounds and elemental mercury vapor. Sulfur compounds such as hydrogen sulfide and sulfur dioxide may be derived from natural sources or from industrial air pollution; the relative concentration of gases remains constant until about 10,000 m. In general, air pressure and density decrease with altitude in the atmosphere. However, temperature has a more complicated profile with altitude, may remain constant or increase with altitude in some regions; because the general pattern of the temperature/altitude profile is constant and measurable by means of instrumented balloon soundings, the temperature behavior provides a useful metric to distinguish atmospheric layers. In this way, Earth's atmosphere can be divided into five main layers.
Excluding the exosphere, the atmosphere has four primary layers, which are the troposphere, stratosphere and thermosphere. From highest to lowest, the five main layers are: Exosphere: 700 to 10,000 km Thermosphere: 80 to 700 km Mesosphere: 50 to 80 km Stratosphere: 12 to 50 km Troposphere: 0 to 12 km The exosphere is the outermost layer of Earth's atmosphere, it extends from the exobase, located at the top of the thermosphere at an altitude of about 700 km above sea level, to about 10,000 km where it merges into the solar wind. This layer is composed of low densities of hydrogen and several heavier molecules including nitrogen and carbon dioxide closer to the exobase; the atoms and molecules are so far apart that they can travel hundreds of kilometers without colliding with one another. Thus, the exosphere no longer behaves like a gas, the particles escape into space; these free-moving particles follow ballistic trajectories and may migrate in and out of the magnetosphere or the solar wind. The exosphere is located too far above Earth for any meteorological phenomena to be possible.
However, the aurora borealis and aurora australis sometimes occur in the lower part of the exosphere, where they overlap into the thermosphere. The exosphere contains most of the satellites orbiting Earth; the thermosphere is the second-highest layer of Earth's atmosphere. It extends from the mesopause at an altitude of about 80 km up to the thermopause at an altitude range of 500–1000 km; the height of the thermopause varies due to changes in solar activity. Because the thermopause lies at the lower boundary of the exosphere, it is referred to as the exobase; the lower part of the thermosphere, from 80 to 550 kilometres above Earth's surface, contains the ionosphere. The temperature of the thermosphere increases with height. Unlike the stratosphere beneath it, wherein a temperature inversion is due to the absorption of radiation by ozone, the inversion in the t
Richard Feynman
Richard Phillips Feynman was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, jointly with Julian Schwinger and Shin'ichirō Tomonaga, received the Nobel Prize in Physics in 1965. Feynman developed a used pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time, he assisted in the development of the atomic bomb during World War II and became known to a wide public in the 1980s as a member of the Rogers Commission, the panel that investigated the Space Shuttle Challenger disaster.
Along with his work in theoretical physics, Feynman has been credited with pioneering the field of quantum computing and introducing the concept of nanotechnology. He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology. Feynman was a keen popularizer of physics through both books and lectures including a 1959 talk on top-down nanotechnology called There's Plenty of Room at the Bottom and the three-volume publication of his undergraduate lectures, The Feynman Lectures on Physics. Feynman became known through his semi-autobiographical books Surely You're Joking, Mr. Feynman! and What Do You Care What Other People Think? and books written about him such as Tuva or Bust! by Ralph Leighton and the biography Genius: The Life and Science of Richard Feynman by James Gleick. Feynman was born on May 11, 1918, in Queens, New York City, to Lucille née Phillips, a homemaker, Melville Arthur Feynman, a sales manager from Minsk in Belarus. Both were Lithuanian Jews.
Feynman was a late talker, did not speak until after his third birthday. As an adult he spoke with a New York accent strong enough to be perceived as an affectation or exaggeration—so much so that his friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like a "bum"; the young Feynman was influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, and, always ready to teach Feynman something new. From his mother, he gained the sense of humor; as a child, he had a talent for engineering, maintained an experimental laboratory in his home, delighted in repairing radios. When he was in grade school, he created a home burglar alarm system while his parents were out for the day running errands; when Richard was five his mother gave birth to a younger brother, Henry Phillips, who died at age four weeks. Four years Richard's sister Joan was born and the family moved to Far Rockaway, Queens. Though separated by nine years and Richard were close, they both shared a curiosity about the world.
Though their mother thought women lacked the capacity to understand such things, Richard encouraged Joan's interest in astronomy, Joan became an astrophysicist. Feynman's parents were not religious, by his youth, Feynman described himself as an "avowed atheist". Many years in a letter to Tina Levitan, declining a request for information for her book on Jewish Nobel Prize winners, he stated, "To select, for approbation the peculiar elements that come from some Jewish heredity is to open the door to all kinds of nonsense on racial theory", adding, "at thirteen I was not only converted to other religious views, but I stopped believing that the Jewish people are in any way'the chosen people'". In his life, during a visit to the Jewish Theological Seminary, he encountered the Talmud for the first time and remarked that it contained a medieval kind of reasoning and was a wonderful book. Feynman attended Far Rockaway High School, a school in Far Rockaway, attended by fellow Nobel laureates Burton Richter and Baruch Samuel Blumberg.
Upon starting high school, Feynman was promoted into a higher math class. A high-school-administered IQ test estimated his IQ at 125—high, but "merely respectable" according to biographer James Gleick, his sister Joan did better. Years he declined to join Mensa International, saying that his IQ was too low. Physicist Steve Hsu stated of the test: I suspect that this test emphasized verbal, as opposed to mathematical, ability. Feynman received the highest score in the United States by a large margin on the notoriously difficult Putnam mathematics competition exam... He had the highest scores on record on the math/physics graduate admission exams at Princeton... Feynman's cognitive abilities might have been a bit lopsided... I recall looking at excerpts from a notebook Feynman kept while an undergraduate... contained a number of misspellings and grammatical errors. I doubt Feynman cared much about such things; when Feynman was 15, he taught himself trigonometry, advanced algebra, infinite series, analytic geometry, both differential and integral calculus.
Before entering college, he was experimenting with and deriving mathematical topics such as the half-derivative using his own notation. He created special symbols for logarithm, sine and tangent functions so they did not look like three variables multiplied together, for the derivative, to remove the temptation of canceling out the d's. A member
Density
The density, or more the volumetric mass density, of a substance is its mass per unit volume. The symbol most used for density is ρ, although the Latin letter D can be used. Mathematically, density is defined as mass divided by volume: ρ = m V where ρ is the density, m is the mass, V is the volume. In some cases, density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more called specific weight. For a pure substance the density has the same numerical value as its mass concentration. Different materials have different densities, density may be relevant to buoyancy and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser. To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. the ratio of the density of the material to that of a standard material water.
Thus a relative density less than one means. The density of a material varies with pressure; this variation is small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object and thus increases its density. Increasing the temperature of a substance decreases its density by increasing its volume. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid; this causes it to rise relative to more dense unheated material. The reciprocal of the density of a substance is called its specific volume, a term sometimes used in thermodynamics. Density is an intensive property in that increasing the amount of a substance does not increase its density. In a well-known but apocryphal tale, Archimedes was given the task of determining whether King Hiero's goldsmith was embezzling gold during the manufacture of a golden wreath dedicated to the gods and replacing it with another, cheaper alloy.
Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated and compared with the mass. Baffled, Archimedes is said to have taken an immersion bath and observed from the rise of the water upon entering that he could calculate the volume of the gold wreath through the displacement of the water. Upon this discovery, he leapt from his bath and ran naked through the streets shouting, "Eureka! 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 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 many 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 cubic metre and the cgs unit of gram per cubic centimetre are the most used units for density. One g/cm3 is equal to one thousand kg/m3. One cubic centimetre is equal to one millilitre. In industry, other larger or smaller units of mass and or volume are more practical and US customary units may be used. See below for a list of some of the most common units of density. A number of techniques as well as standards exist for the measurement of density of materials; such techniques include the use of a hydrometer, Hydrostatic balance, immersed body method, air comparison pycnometer, oscillating densitometer, as well as pour and tap. However, each individual method or technique measures different types of density, therefore it is necessary to have an understanding of the type of density being measured as well as the type of material in question; the density at all points of a homogeneous object equals its total mass divided by its total volume. The mass is measured with a scale or balance.
To determine the density of a liquid or a gas, a hydrometer, a dasymeter or a Coriolis flow meter may be used, respectively. Hydrostatic weighing uses the displacement of water due to a submerged object to determine the density of the object. If the body is not homogeneous its density varies between different regions of the object. In that case the density around any given location is determined by calculating the density of a small volume around that location. In the limit of an infinitesimal volume the density of an inhomogeneous object at a point becomes: ρ = d m / d V, where d V is an elementary volume at position r; the mass of the body t
![\scriptstyle \sqrt[4]{g\sigma/\rho}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fba378198fe7494e9310dfecd81b655747a78c)
![V_{{\mathrm {st}}}=\sigma \iint dx\,dy\;\left[{\sqrt {1+\left({\frac {\partial \eta }{\partial x}}\right)^{2}+\left({\frac {\partial \eta }{\partial y}}\right)^{2}}}-1\right]\approx {\frac {1}{2}}\sigma \iint dx\,dy\;\left[\left({\frac {\partial \eta }{\partial x}}\right)^{2}+\left({\frac {\partial \eta }{\partial y}}\right)^{2}\right],](https://wikimedia.org/api/rest_v1/media/math/render/svg/4708c19acff7de64c97332d51cc2923b7a48cf00)
![{\displaystyle T={\frac {1}{2}}\iint dx\,dy\;\left[\int _{-\infty }^{\eta }dz\;\rho \,\left|\mathbf {\nabla } \Phi \right|^{2}+\int _{\eta }^{+\infty }dz\;\rho '\,\left|\mathbf {\nabla } \Phi '\right|^{2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a02fa05cd0b2280b73fdb1e4638e92e09f13f7e)
![T\approx {\frac {1}{2}}\iint dx\,dy\;\left[\rho \,\Phi \,{\frac {\partial \Phi }{\partial z}}\;-\;\rho '\,\Phi '\,{\frac {\partial \Phi '}{\partial z}}\right]_{{{\text{at }}z=0}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6819b4e24e6b0c3c6746fdb276edde7d22414411)
![L={\frac {1}{4}}\left[(\rho +\rho '){\frac {\omega ^{2}}{|k|}}-(\rho -\rho ')g-\sigma k^{2}\right]a^{2}\lambda .](https://wikimedia.org/api/rest_v1/media/math/render/svg/b98c5da92c6d3474ee71b2a4f60dd9c2c8210b1f)
![{\bar {E}}={\frac {1}{2}}\,\left[(\rho -\rho ')\,g+\sigma k^{2}\right]\,a^{2}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca523bb7fe7083ef2fe607f96709f4d0bada97db)
