The meter is defined to be a unit of proper length, but the SI definition does not specify the metric tensor to be used in determining it.

In addition, this article discusses the definition for the more general case of functions between two metric spaces.

The current definition, sometimes known as the metric carat, was adopted in 1907 at the Fourth General Conference on Weights and Measures, and soon afterward in many countries around the world.

This definition, however, was inadequate for the precision requirements of modern technology, and the metric kilogram was redefined in terms of a man-made platinum-iridium bar known as the international prototype kilogram.

This turns the seminormed space into a pseudometric space ( notice this is weaker than a metric ) and allows the definition of notions such as continuity and convergence.

An alternative and disputable definition of QoS, used especially in application layer services such as telephony and streaming video, is requirements on a metric that reflects or predicts the subjectively experienced quality.

The constant below in, ft · lbf / min, changes with the definition of the horsepower ; for example, using metric horsepower, it becomes ~ 32, 550.

Caesium clocks are the most accurate commercially produced time and frequency standards, and serve as the primary standard for the definition of the second in SI ( the metric system ).

This definition generalises to any subset S of a metric space X.

It does not define the time of day, as this is defined by various time scales, which may be based upon the metric definition of the second.

However, theories of analysis can be applied to any space of mathematical objects that has a definition of nearness ( a topological space ) or, more specifically, distance ( a metric space ).

This definition generalizes to any subset S of a metric space X.

With this definition of length, every connected Riemannian manifold M becomes a metric space ( and even a length metric space ) in a natural fashion: the distance d ( x, y ) between the points x and y of M is defined as

So far a world line ( and the concept of tangent vectors ) is defined in spacetime even without a definition of a metric.

The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space.

One can show that the topology induced by the metric is the same as the product topology in the above definition.

This quadratic form is interpreted in physics as the metric tensor of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime.

Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i. e. is a Levi-Civita connection.

The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

From this definition is derived the metric equivalencies for inches, feet, and miles ; as well as area measures ; and measures of volume.

This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above,

The " remarkable ", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R < sup > 3 </ sup > certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the inner metric of the surface without any further reference to the ambient space: it is an intrinsic invariant.

A relatively weak gravitational field, such as the Earth's or the sun's, is represented by a metric that is approximately, but not exactly, Euclidean.

The metric system is very similar to the units of measurement used in Mesopotamia about five millennia previously: for example, a metre is almost exactly 1 Mesopotamian step, a litre is almost exactly 1 Mesopotamian bowl, and a metric pound is almost exactly 1 Mesopotamian pound or 1 / 2 kilograms.

Surjective bilipschitz functions are exactly the isomorphisms of metric spaces.

Further minor variations exist where European or Japanese manufacturers round a U measurement up or down to some closer convenient metric equivalent ; for example the common 5U modules are exactly 8. 75 " ( 222. 25mm ), but non-American manufacturers may prefer 220mm or 230mm.

Einstein's first attempt at modeling relied on a cosmological constant that was finely tuned to exactly balance out matter curvature and provide a framework for an infinite and unchanging spacetime metric in which the objects of the universe were embedded.

A long ton then is exactly 1, 016, 046. 9088 grams, which is approximately 1, 016 kg – conveniently close to a metric tonne.

It is a non-GAAP metric that is measured exactly as stated.

By setting the phase of the complex coordinate to zero, one obtains exactly one-fourth of the Fisher information metric, exactly as above.

A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general mathematical usage.

The metric dönüm is particularly useful in hydrological calculations as 1 dönüm times 1 mm ( a unit commonly used for measuring precipitation ) equals exactly one cubic meter.

Pharmacists have since moved to metric measurements, with a drop being rounded to exactly 0. 05 mL ( that is, 20 drops per milliliter ).

The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the space ( or spacetime ) is curved at each point.

Note that the hyperbolic area of E is exactly 1, when computed using the canonical Poincaré metric.

Historically in France, under the mesures usuelles system ( intermediary between traditional French units and metric units ), a metric foot was exactly a third of a metre ( mm ).

In later times, with the adoption of the metric system, the value came to denote exactly 50 kg, at least in Germany ; in Austria and Switzerland the term is now in use for a measure of 100 kg.

This is exactly the special form of the metric proposed in Nordström's second theory, so the entire content of this theory can be even more elegantly summarized in the following two equations:

* Lambda is a non-SI metric unit of measure of volume ( symbol λ ) equal to one microliter ( 1 μL ) or one cubic millimeter ( 1 mm < sup > 3 </ sup >).

The litre ( American spelling: liter ; SI symbol L or l ) is a non-SI metric system unit of volume equal to 1 cubic decimetre ( dm < sup > 3 </ sup >), 1, 000 cubic centimetres ( cm < sup > 3 </ sup >) or 1 / 1, 000 cubic metre.

The original French metric system used the litre as a base unit, and it has been used in several subsequent versions of the metric system and is accepted for use with the SI, although not an official SI unit — the SI unit of volume is the cubic metre ( m < sup > 3 </ sup >).

However, the amount of silver coated glass worldwide is hundreds of millions of square meters per year leading to silver consumption on the order of 10 cubic meters or 100 metric tons / year.

Starship volumes are measured in " displacement tons " ( also " D-tons " or just " tons "), equal to the volume of a metric ton of liquid hydrogen, or about 14 cubic meters.

In the original version of the metric system the base units could be derived from a specified length ( the metre ) and the weight of a specified volume ( of a cubic metre ) of pure water.

where is the mass-energy density ( kilograms per cubic meter ), is the hydrostatic pressure ( pascals ), is the fluid's four velocity, and is the reciprocal of the metric tensor.

For instance, it takes 1, 300 cubic meters of water on average to produce one metric tonne of wheat.

However, in markets accustomed to cubic inches, the actual displacement measurements of an engine are still given by many manufacturers in these units, usually along with metric value ; e. g. the 6. 1-litre HEMI's published displacement is 370. 0 CID / 6, 059 cc.

The stere or stère is a unit of volume in the original metric system equal to one cubic metre.

In the metric system, firewood is normally sold by the cubic metre or stere ( 1 m³ = ~ 0. 276 cords ).

A typical large refinery processing 40, 000 metric tonnes of crude oil per day ( per day ) circulates about 80, 000 cubic metres of water per hour through its cooling tower system.

In the case of such cars, the use of cubic inches, rather than the metric cubic centimeters, can be considered vestigial.

As of 2009, India's total proven oil reserves stood at 775 million metric tonnes while gas reserves stood at 1074 billion cubic metres.

One metric ton of water, one cubic meter, can store 334 million joules ( MJ ) or 317, 000 BTUs ( 93kWh or 26. 4 ton-hours ).

In countries using the SI metric system of unit, the term " normal cubic metre " ( Nm < sup > 3 </ sup >) is very often used to denote gas volumes at some normalized or standard condition.

If the metric system is being used, a conversion factor of 6. 289808 can be used to convert cubic metres to stock tank barrels.