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102-408: Volume is a measure of regions in three-dimensional space . It is often quantified numerically using SI derived units (such as the cubic metre and litre ) or by various imperial or US customary units (such as the gallon , quart , cubic inch ). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of

204-451: A σ {\displaystyle \sigma } -algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended real number line is called a measure if the following conditions hold: If at least one set E {\displaystyle E} has finite measure, then

306-429: A greatest element μ sf . {\displaystyle \mu _{\text{sf}}.} We say the semifinite part of μ {\displaystyle \mu } to mean the semifinite measure μ sf {\displaystyle \mu _{\text{sf}}} defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for

408-514: A least measure μ 0 − ∞ . {\displaystyle \mu _{0-\infty }.} Also, we have μ = μ sf + μ 0 − ∞ . {\displaystyle \mu =\mu _{\text{sf}}+\mu _{0-\infty }.} We say the 0 − ∞ {\displaystyle \mathbf {0-\infty } } part of μ {\displaystyle \mu } to mean

510-425: A region D in three-dimensional space is given by the triple or volume integral of the constant function f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} over the region. It is usually written as: ∭ D 1 d x d y d z . {\displaystyle \iiint _{D}1\,dx\,dy\,dz.} In cylindrical coordinates ,

612-433: A reservoir , the container's volume is modeled by shapes and calculated using mathematics. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m). The cubic metre is also a SI derived unit . Therefore, volume has

714-449: A unit dimension of L. The metric units of volume uses metric prefixes , strictly in powers of ten . When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm = 2.3 (cm) = 2.3 (0.01 m) = 0.0000023 m (five zeros). Commonly used prefixes for cubed length units are

816-420: A vacuum with gravity acting on it. Suppose that, when the rock is lowered into the water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea-floor. It is generally easier to lift an object through

918-475: A volume integral with the help of the Gauss theorem : where V is the measure of the volume in contact with the fluid, that is the volume of the submerged part of the body, since the fluid doesn't exert force on the part of the body which is outside of it. The magnitude of buoyancy force may be appreciated a bit more from the following argument. Consider any object of arbitrary shape and volume V surrounded by

1020-399: A basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz). Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate

1122-548: A countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [ k , k + 1 ] {\displaystyle [k,k+1]} for all integers k ; {\displaystyle k;} there are countably many such intervals, each has measure 1, and their union

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1224-403: A fluid or liquid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Archimedes' principle allows the buoyancy of any floating object partially or fully immersed in a fluid to be calculated. The downward force on the object is simply its weight. The upward, or buoyant, force on the object is that stated by Archimedes' principle above. Thus, the net force on the object

1326-610: A formula exists for the shape's boundary. Zero- , one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume. The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids , cylinders , frustum and cones . These math problems have been written in

1428-449: A liquid. The force the liquid exerts on an object within the liquid is equal to the weight of the liquid with a volume equal to that of the object. This force is applied in a direction opposite to gravitational force, that is of magnitude: where ρ f is the density of the fluid, V disp is the volume of the displaced body of liquid, and g is the gravitational acceleration at the location in question. If this volume of liquid

1530-485: A measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits , the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification . All these are linked in one way or another to

1632-655: A measure on A . {\displaystyle {\cal {A}}.} We say μ {\displaystyle \mu } is semifinite to mean that for all A ∈ μ pre { + ∞ } , {\displaystyle A\in \mu ^{\text{pre}}\{+\infty \},} P ( A ) ∩ μ pre ( R > 0 ) ≠ ∅ . {\displaystyle {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{>0})\neq \emptyset .} Semifinite measures generalize sigma-finite measures, in such

1734-1556: A measure. If E 1 {\displaystyle E_{1}} and E 2 {\displaystyle E_{2}} are measurable sets with E 1 ⊆ E 2 {\displaystyle E_{1}\subseteq E_{2}} then μ ( E 1 ) ≤ μ ( E 2 ) . {\displaystyle \mu (E_{1})\leq \mu (E_{2}).} For any countable sequence E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } of (not necessarily disjoint) measurable sets E n {\displaystyle E_{n}} in Σ : {\displaystyle \Sigma :} μ ( ⋃ i = 1 ∞ E i ) ≤ ∑ i = 1 ∞ μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are increasing (meaning that E 1 ⊆ E 2 ⊆ E 3 ⊆ … {\displaystyle E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots } ) then

1836-1712: A monotonically non-decreasing sequence converging to t . {\displaystyle t.} The monotonically non-increasing sequences { x ∈ X : f ( x ) > t n } {\displaystyle \{x\in X:f(x)>;t_{n}\}} of members of Σ {\displaystyle \Sigma } has at least one finitely μ {\displaystyle \mu } -measurable component, and { x ∈ X : f ( x ) ≥ t } = ⋂ n { x ∈ X : f ( x ) > t n } . {\displaystyle \{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.} Continuity from above guarantees that μ { x ∈ X : f ( x ) ≥ t } = lim t n ↑ t μ { x ∈ X : f ( x ) > t n } . {\displaystyle \mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.} The right-hand side lim t n ↑ t F ( t n ) {\displaystyle \lim _{t_{n}\uparrow t}F\left(t_{n}\right)} then equals F ( t ) = μ { x ∈ X : f ( x ) > t } {\displaystyle F(t)=\mu \{x\in X:f(x)>t\}} if t {\displaystyle t}

1938-440: A negative value, similar to length and area . Like all continuous monotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to Cavalieri's principle and to the infinitesimal calculus of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus

2040-941: A null set. One defines μ ( Y ) {\displaystyle \mu (Y)} to equal μ ( X ) . {\displaystyle \mu (X).} If f : X → [ 0 , + ∞ ] {\displaystyle f:X\to [0,+\infty ]} is ( Σ , B ( [ 0 , + ∞ ] ) ) {\displaystyle (\Sigma ,{\cal {B}}([0,+\infty ]))} -measurable, then μ { x ∈ X : f ( x ) ≥ t } = μ { x ∈ X : f ( x ) > t } {\displaystyle \mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}} for almost all t ∈ [ − ∞ , ∞ ] . {\displaystyle t\in [-\infty ,\infty ].} This property

2142-948: A sense, semifinite once its 0 − ∞ {\displaystyle 0-\infty } part (the wild part) is taken away. Theorem (Luther decomposition)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists a 0 − ∞ {\displaystyle 0-\infty } measure ξ {\displaystyle \xi } on A {\displaystyle {\cal {A}}} such that μ = ν + ξ {\displaystyle \mu =\nu +\xi } for some semifinite measure ν {\displaystyle \nu } on A . {\displaystyle {\cal {A}}.} In fact, among such measures ξ , {\displaystyle \xi ,} there exists

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2244-449: A special case of semifinite measures and a generalization of sigma-finite measures. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be a measure on A . {\displaystyle {\cal {A}}.} A measure

2346-437: A way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.) The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to μ . {\displaystyle \mu .} It can be shown there

2448-438: Is a greatest measure with these two properties: Theorem (semifinite part)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists, among semifinite measures on A {\displaystyle {\cal {A}}} that are less than or equal to μ , {\displaystyle \mu ,}

2550-488: Is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory ) are Radon measures . Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support . This approach

2652-1052: Is a point of continuity of F . {\displaystyle F.} Since F {\displaystyle F} is continuous almost everywhere, this completes the proof. Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I {\displaystyle I} and any set of nonnegative r i , i ∈ I {\displaystyle r_{i},i\in I} define: ∑ i ∈ I r i = sup { ∑ i ∈ J r i : | J | < ∞ , J ⊆ I } . {\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<;\infty ,J\subseteq I\right\rbrace .} That is, we define

2754-466: Is a unique t 0 ∈ { − ∞ } ∪ [ 0 , + ∞ ) {\displaystyle t_{0}\in \{-\infty \}\cup [0,+\infty )} such that F {\displaystyle F} is infinite to the left of t {\displaystyle t} (which can only happen when t 0 ≥ 0 {\displaystyle t_{0}\geq 0} ) and finite to

2856-524: Is achieved when these two weights (and thus forces) are equal. The equation to calculate the pressure inside a fluid in equilibrium is: where f is the force density exerted by some outer field on the fluid, and σ is the Cauchy stress tensor . In this case the stress tensor is proportional to the identity tensor: Here δ ij is the Kronecker delta . Using this the above equation becomes: Assuming

2958-474: Is called a measurable space , and the members of Σ {\displaystyle \Sigma } are called measurable sets . A triple ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called a measure space . A probability measure is a measure with total measure one – that is, μ ( X ) = 1. {\displaystyle \mu (X)=1.} A probability space

3060-429: Is common for measuring small volume of fluids or granular materials , by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients . Air displacement pipette is used in biology and biochemistry to measure volume of fluids at

3162-408: Is equal to the weight of the fluid displaced by the immersed part of the body(s). Consider a cuboid immersed in a fluid, its top and bottom faces orthogonal to the direction of gravity (assumed constant across the cube's stretch). The fluid will exert a normal force on each face, but only the normal forces on top and bottom will contribute to buoyancy. The pressure difference between the bottom and

Volume - Misplaced Pages Continue

3264-407: Is equal to the weight of the fluid displaced by the object, or the density ( ρ ) of the fluid multiplied by the submerged volume (V) times the gravity (g) We can express this relation in the equation: where F a {\displaystyle F_{a}} denotes the buoyant force applied onto the submerged object, ρ {\displaystyle \rho } denotes

3366-519: Is equivalent to the statement that the ideal of null sets is κ {\displaystyle \kappa } -complete. A measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called finite if μ ( X ) {\displaystyle \mu (X)} is a finite real number (rather than ∞ {\displaystyle \infty } ). Nonzero finite measures are analogous to probability measures in

3468-465: Is false without the assumption that at least one of the E n {\displaystyle E_{n}} has finite measure. For instance, for each n ∈ N , {\displaystyle n\in \mathbb {N} ,} let E n = [ n , ∞ ) ⊆ R , {\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} ,} which all have infinite Lebesgue measure, but

3570-414: Is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y {\displaystyle Y} which differ by a negligible set from a measurable set X , {\displaystyle X,} that is, such that the symmetric difference of X {\displaystyle X} and Y {\displaystyle Y} is contained in

3672-532: Is necessarily of finite variation , hence complex measures include finite signed measures but not, for example, the Lebesgue measure . Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure ; these are used in functional analysis for the spectral theorem . When it

3774-406: Is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination , while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure , also known as a content . This is the same as

3876-405: Is replaced by a solid body of exactly the same shape, the force the liquid exerts on it must be exactly the same as above. In other words, the "buoyancy force" on a submerged body is directed in the opposite direction to gravity and is equal in magnitude to The net force on the object must be zero if it is to be a situation of fluid statics such that Archimedes principle is applicable, and is thus

3978-466: Is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes . If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable ; examples of such sets include the Vitali set , and the non-measurable sets postulated by

4080-1000: Is semifinite then μ = μ sf . {\displaystyle \mu =\mu _{\text{sf}}.} Every 0 − ∞ {\displaystyle 0-\infty } measure that is not the zero measure is not semifinite. (Here, we say 0 − ∞ {\displaystyle 0-\infty } measure to mean a measure whose range lies in { 0 , + ∞ } {\displaystyle \{0,+\infty \}} : ( ∀ A ∈ A ) ( μ ( A ) ∈ { 0 , + ∞ } ) . {\displaystyle (\forall A\in {\cal {A}})(\mu (A)\in \{0,+\infty \}).} ) Below we give examples of 0 − ∞ {\displaystyle 0-\infty } measures that are not zero measures. Measures that are not semifinite are very wild when restricted to certain sets. Every measure is, in

4182-576: Is spatial distribution of mass (see for example, gravity potential ), or another non-negative extensive property , conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet. Let μ {\displaystyle \mu } be

Volume - Misplaced Pages Continue

4284-820: Is such that μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } then monotonicity implies μ { x ∈ X : f ( x ) ≥ t } = + ∞ , {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty ,} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as required. If μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } for all t {\displaystyle t} then we are done, so assume otherwise. Then there

4386-399: Is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures . Some important measures are listed here. Other 'named' measures used in various theories include: Borel measure , Jordan measure , ergodic measure , Gaussian measure , Baire measure , Radon measure , Young measure , and Loeb measure . In physics an example of a measure

4488-632: Is the volume element ; this formulation is useful when working with different coordinate systems , spaces and manifolds . The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches . However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as gourds , sheep or pig stomachs , and bladders . Later on, as metallurgy and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method

4590-424: Is the difference between the magnitudes of the buoyant force and its weight. If this net force is positive, the object rises; if negative, the object sinks; and if zero, the object is neutrally buoyant—that is, it remains in place without either rising or sinking. In simple words, Archimedes' principle states that, when a body is partially or completely immersed in a fluid, it experiences an apparent loss in weight that

4692-471: Is the entire real line. Alternatively, consider the real numbers with the counting measure , which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to

4794-425: Is the mass density of the fluid. Taking the pressure as zero at the surface, where z is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is So pressure increases with depth below the surface of a liquid, as z denotes the distance from the surface of the liquid into it. Any object with a non-zero vertical depth will have different pressures on its top and bottom, with

4896-401: Is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing .) Example: If you drop wood into water, buoyancy will keep it afloat. Example: A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by

4998-644: Is used in connection with Lebesgue integral . Both F ( t ) := μ { x ∈ X : f ( x ) > t } {\displaystyle F(t):=\mu \{x\in X:f(x)>t\}} and G ( t ) := μ { x ∈ X : f ( x ) ≥ t } {\displaystyle G(t):=\mu \{x\in X:f(x)\geq t\}} are monotonically non-increasing functions of t , {\displaystyle t,} so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to

5100-503: Is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = π ∫ a b | f ( x ) 2 − g ( x ) 2 | d x {\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx} where f ( x ) {\textstyle f(x)} and g ( x ) {\textstyle g(x)} are

5202-415: Is very small compared to most solids and liquids. For this reason, the weight of an object in air is approximately the same as its true weight in a vacuum. The buoyancy of air is neglected for most objects during a measurement in air because the error is usually insignificant (typically less than 0.1% except for objects of very low average density such as a balloon or light foam). A simplified explanation for

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5304-549: The Hausdorff paradox and the Banach–Tarski paradox . For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure , while such a function with values in the complex numbers is called a complex measure . Observe, however, that complex measure

5406-492: The Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be

5508-719: The Moscow Mathematical Papyrus (c. 1820 BCE). In the Reisner Papyrus , ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the cubit , palm , digit ) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). The last three books of Euclid's Elements , written in around 300 BCE, detailed

5610-512: The area of a circle . But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel , Henri Lebesgue , Nikolai Luzin , Johann Radon , Constantin Carathéodory , and Maurice Fréchet , among others. Let X {\displaystyle X} be a set and Σ {\displaystyle \Sigma }

5712-499: The axiom of choice . Contents remain useful in certain technical problems in geometric measure theory ; this is the theory of Banach measures . A charge is a generalization in both directions: it is a finitely additive, signed measure. (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R .) Archimedes%27 principle Archimedes' principle (also spelled Archimedes's principle ) states that

5814-410: The cube , cuboid and cylinder , they have an essentially the same volume calculation formula as one for the prism : the base of the shape multiplied by its height . The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution , by rotating a plane curve around a line on the same plane. The washer or disc integration method

5916-402: The density of the fluid, V {\displaystyle V} represents the volume of the displaced fluid and g {\displaystyle g} is the acceleration due to gravity . Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy. Suppose a rock's weight is measured as 10 newtons when suspended by a string in

6018-590: The imperial gallon was defined to be the volume occupied by ten pounds of water at 17 °C (62 °F). This definition was further refined until the United Kingdom's Weights and Measures Act 1985 , which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded

6120-675: The intersection of the sets E n {\displaystyle E_{n}} is measurable; furthermore, if at least one of the E n {\displaystyle E_{n}} has finite measure then μ ( ⋂ i = 1 ∞ E i ) = lim i → ∞ μ ( E i ) = inf i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).} This property

6222-607: The sester , amber , coomb , and seam . The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England . The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the London Pharmacopoeia (medicine compound catalog) adopted the Roman gallon or congius as

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6324-964: The union of the sets E n {\displaystyle E_{n}} is measurable and μ ( ⋃ i = 1 ∞ E i )   =   lim i → ∞ μ ( E i ) = sup i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are decreasing (meaning that E 1 ⊇ E 2 ⊇ E 3 ⊇ … {\displaystyle E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots } ) then

6426-439: The volume integral is ∭ D r d r d θ d z , {\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,} In spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and φ {\displaystyle \varphi } measured from the polar axis; see more on conventions ),

6528-566: The Lebesgue measure. If t < 0 {\displaystyle t<0} then { x ∈ X : f ( x ) ≥ t } = X = { x ∈ X : f ( x ) > t } , {\displaystyle \{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as desired. If t {\displaystyle t}

6630-445: The air and will drift in the same direction as the car's acceleration. When an object is immersed in a liquid, the liquid exerts an upward force, which is known as the buoyant force, that is proportional to the weight of the displaced liquid. The sum force acting on the object, then, is equal to the difference between the weight of the object ('down' force) and the weight of displaced liquid ('up' force). Equilibrium, or neutral buoyancy,

6732-401: The area of the bottom surface. Similarly, the downward force on the cube is the pressure on the top surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal top surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the top surface. As this is a cube,

6834-409: The buoyant force is equal to the weight of the displaced fluid. The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force (F b ) on an object

6936-406: The condition of non-negativity is dropped, and μ {\displaystyle \mu } takes on at most one of the values of ± ∞ , {\displaystyle \pm \infty ,} then μ {\displaystyle \mu } is called a signed measure . The pair ( X , Σ ) {\displaystyle (X,\Sigma )}

7038-434: The container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of fuel oil will not be able to contain the same 7,200 t (15,900,000 lb) of naphtha , due to naphtha's lower density and thus larger volume. For many shapes such as

7140-586: The container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy , the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume ). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas . Volumes of more complicated shapes can be calculated with integral calculus if

7242-429: The cubic millimetre (mm), cubic centimetre (cm), cubic decimetre (dm), cubic metre (m) and the cubic kilometre (km). The conversion between the prefix units are as follows: 1000 mm = 1 cm, 1000 cm = 1 dm, and 1000 dm = 1 m. The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm = 1000 cm = 0.001 m. For the litre unit, the commonly used prefixes are

7344-531: The exact formulas for calculating the volume of parallelepipeds , cones, pyramids , cylinders, and spheres . The formula were determined by prior mathematicians by using a primitive form of integration , by breaking the shapes into smaller and simpler pieces. A century later, Archimedes ( c.  287 – 212 BCE ) devised approximate volume formula of several shapes using the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes

7446-409: The floor of the fluid or rises to the surface and settles, Archimedes principle can be applied alone. For a floating object, only the submerged volume displaces water. For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor. In order for Archimedes' principle to be used alone, the object in question must be in equilibrium (the sum of

7548-637: The following hold: ⋃ α ∈ λ X α ∈ Σ {\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma } μ ( ⋃ α ∈ λ X α ) = ∑ α ∈ λ μ ( X α ) . {\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).} The second condition

7650-407: The forces on the object must be zero), therefore; and therefore showing that the depth to which a floating object will sink, and the volume of fluid it will displace, is independent of the gravitational field regardless of geographic location. It can be the case that forces other than just buoyancy and gravity come into play. This is the case if the object is restrained or if the object sinks to

7752-568: The golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a hydrostatic balance . Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle . In the Middle Ages , many units for measuring volume were made, such as

7854-472: The integration of the pressure over the contact area may be stated as follows: Consider a cube immersed in a fluid with the upper surface horizontal. The sides are identical in area, and have the same depth distribution, therefore they also have the same pressure distribution, and consequently the same total force resulting from hydrostatic pressure, exerted perpendicular to the plane of the surface of each side. There are two pairs of opposing sides, therefore

7956-422: The intersection is empty. A measurable set X {\displaystyle X} is called a null set if μ ( X ) = 0. {\displaystyle \mu (X)=0.} A subset of a null set is called a negligible set . A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set

8058-864: The measure μ 0 − ∞ {\displaystyle \mu _{0-\infty }} defined in the above theorem. Here is an explicit formula for μ 0 − ∞ {\displaystyle \mu _{0-\infty }} : μ 0 − ∞ = ( sup { μ ( B ) − μ sf ( B ) : B ∈ P ( A ) ∩ μ sf pre ( R ≥ 0 ) } ) A ∈ A . {\displaystyle \mu _{0-\infty }=(\sup\{\mu (B)-\mu _{\text{sf}}(B):B\in {\cal {P}}(A)\cap \mu _{\text{sf}}^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.} Localizable measures are

8160-595: The metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. The definition of the metre was redefined again in 1983 to use the speed of light and second (which is derived from the caesium standard ) and reworded for clarity in 2019 . As a measure of the Euclidean three-dimensional space , volume cannot be physically measured as

8262-593: The microscopic scale. Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories . There, volume of liquids is measured using graduated cylinders , pipettes and volumetric flasks . The largest of such calibrated containers are petroleum storage tanks , some can hold up to 1,000,000  bbl (160,000,000 L) of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in

8364-416: The millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Various other imperial or U.S. customary units of volume are also in use, including: Capacity is the maximum amount of material that a container can hold, measured in volume or weight . However, the contained volume does not need to fill towards

8466-504: The modern integral calculus, which remains in use in the 21st century. On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the stère  (1 m) for volume of firewood; the litre  (1 dm) for volumes of liquid; and the gramme , for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824,

8568-410: The outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function: Then: Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field. Let the z -axis point downward. In this case the field is gravity, so Φ = − ρ f gz where g is the gravitational acceleration, ρ f

8670-405: The plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2 π ∫ a b x | f ( x ) − g ( x ) | d x {\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx} The volume of

8772-418: The pressure on the bottom being greater. This difference in pressure causes the upward buoyancy force. The buoyancy force exerted on a body can now be calculated easily, since the internal pressure of the fluid is known. The force exerted on the body can be calculated by integrating the stress tensor over the surface of the body which is in contact with the fluid: The surface integral can be transformed into

8874-563: The requirement μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} is met automatically due to countable additivity: μ ( E ) = μ ( E ∪ ∅ ) = μ ( E ) + μ ( ∅ ) , {\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),} and therefore μ ( ∅ ) = 0. {\displaystyle \mu (\varnothing )=0.} If

8976-416: The resultant horizontal forces balance in both orthogonal directions, and the resultant force is zero. The upward force on the cube is the pressure on the bottom surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal bottom surface of the cube is the hydrostatic pressure at that depth multiplied by

9078-872: The right. Arguing as above, μ { x ∈ X : f ( x ) ≥ t } = + ∞ {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty } when t < t 0 . {\displaystyle t<t_{0}.} Similarly, if t 0 ≥ 0 {\displaystyle t_{0}\geq 0} and F ( t 0 ) = + ∞ {\displaystyle F\left(t_{0}\right)=+\infty } then F ( t 0 ) = G ( t 0 ) . {\displaystyle F\left(t_{0}\right)=G\left(t_{0}\right).} For t > t 0 , {\displaystyle t>t_{0},} let t n {\displaystyle t_{n}} be

9180-404: The semifinite part: Since μ sf {\displaystyle \mu _{\text{sf}}} is semifinite, it follows that if μ = μ sf {\displaystyle \mu =\mu _{\text{sf}}} then μ {\displaystyle \mu } is semifinite. It is also evident that if μ {\displaystyle \mu }

9282-421: The sense that any finite measure μ {\displaystyle \mu } is proportional to the probability measure 1 μ ( X ) μ . {\displaystyle {\frac {1}{\mu (X)}}\mu .} A measure μ {\displaystyle \mu } is called σ-finite if X {\displaystyle X} can be decomposed into

9384-409: The solid floor. An object which tends to float requires a tension restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a normal force of constraint N exerted upon it by the solid floor. The constraint force can be tension in a spring scale measuring its weight in the fluid, and is how apparent weight is defined. If the object would otherwise float,

9486-606: The sum of the r i {\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them. A measure μ {\displaystyle \mu } on Σ {\displaystyle \Sigma } is κ {\displaystyle \kappa } -additive if for any λ < κ {\displaystyle \lambda <\kappa } and any family of disjoint sets X α , α < λ {\displaystyle X_{\alpha },\alpha <\lambda }

9588-500: The sum of the buoyancy force and the object's weight If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise. An object whose weight exceeds its buoyancy tends to sink. Calculation of the upwards force on a submerged object during its accelerating period cannot be done by the Archimedes principle alone; it is necessary to consider dynamics of an object involving buoyancy. Once it fully sinks to

9690-653: The tension to restrain it fully submerged is: When a sinking object settles on the solid floor, it experiences a normal force of: Another possible formula for calculating buoyancy of an object is by finding the apparent weight of that particular object in the air (calculated in Newtons), and apparent weight of that object in the water (in Newtons). To find the force of buoyancy acting on the object when in air, using this particular information, this formula applies: The final result would be measured in Newtons. Air's density

9792-426: The top and bottom surfaces are identical in shape and area, and the pressure difference between the top and bottom of the cube is directly proportional to the depth difference, and the resultant force difference is exactly equal to the weight of the fluid that would occupy the volume of the cube in its absence. This means that the resultant upward force on the cube is equal to the weight of the fluid that would fit into

9894-417: The top face is directly proportional to the height (difference in depth of submersion). Multiplying the pressure difference by the area of a face gives a net force on the cuboid—the buoyancy—equaling in size the weight of the fluid displaced by the cuboid. By summing up sufficiently many arbitrarily small cuboids this reasoning may be extended to irregular shapes, and so, whatever the shape of the submerged body,

9996-413: The upward buoyant force that is exerted on a body immersed in a fluid , whether fully or partially, is equal to the weight of the fluid that the body displaces . Archimedes' principle is a law of physics fundamental to fluid mechanics . It was formulated by Archimedes of Syracuse . In On Floating Bodies , Archimedes suggested that (c. 246 BC): Any object, totally or partially immersed in

10098-564: The volume integral is ∭ D ρ 2 sin ⁡ φ d ρ d θ d φ . {\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .} A polygon mesh is a representation of the object's surface, using polygons . The volume mesh explicitly define its volume and surface properties. Measure (mathematics) The intuition behind this concept dates back to ancient Greece , when Archimedes tried to calculate

10200-399: The volume of any object. He devised Cavalieri's principle , which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat , John Wallis , Isaac Barrow , James Gregory , Isaac Newton , Gottfried Wilhelm Leibniz and Maria Gaetana Agnesi in the 17th and 18th centuries to form

10302-399: The water than it is to pull it out of the water. For a fully submerged object, Archimedes' principle can be reformulated as follows: then inserted into the quotient of weights, which has been expanded by the mutual volume yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volume is (This formula

10404-497: Was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the Middle East and India . Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge

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