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In thermodynamics, the **volume** of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's **volume** per unit of mass. **Volume** is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, **volume** is related to the pressure and temperature of an ideal gas by the ideal gas law.

In calculus, a branch of mathematics, the **volume** of a region D in R 3 is given by a triple integral of the constant function f(x,y,z)=1 and is usually written as:

and the **volume** integral in spherical coordinates (using the convention for angles with \theta as the azimuth and \varphi measured from the polar axis; see more on conventions) has the form

The **volume** integral in cylindrical coordinates is

The percentage by **volume** (vol%) is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and mole fraction (percentage by moles, mol%) are others.

In the case of a mixture of ethanol and water, which are miscible in all proportions, the designation of solvent and solute is arbitrary. The **volume** of such a mixture is slightly less than the sum of the volumes of the components. Thus, by the above definition, the term "40% alcohol by volume" refers to a mixture of 40 **volume** units of ethanol with enough water to make a final **volume** of 100 units, rather than a mixture of 40 units of ethanol with 60 units of water.

**Volume** percent is the concentration of a certain solute, measured by volume, in a solution. It has as a denominator the **volume** of the mixture itself, as usual for expressions of concentration, rather than the total of all the individual component's volumes prior to mixing:

The specific **volume** of a substance is equal to the reciprocal of its mass density. Specific **volume** may be expressed in, , , or. :

The partial **volume** of a particular gas is the **volume** which the gas would have if it alone occupied the volume, with unchanged pressure and temperature, and is useful in gas mixtures, e.g. air, to focus on one particular gas component, e.g. oxygen.

**Volume** percent is usually used when the solution is made by mixing two fluids, such as liquids or gases. However, percentages are only additive for ideal gases.

Specific **volume** (\nu) is the **volume** occupied by a unit of mass of a material. In many cases the specific **volume** is a useful quantity to determine because, as an intensive property, it can be used to determine the complete state of a system in conjunction with another independent intensive variable. The specific **volume** also allows systems to be studied without reference to an exact operating volume, which may not be known (nor significant) at some stages of analysis.

Specific **volume** may also refer to molar volume.

Image segmentation is a manual or automatic procedure that can be used to section out large portions of the **volume** that one considers uninteresting before rendering, the amount of calculations that have to be made by ray casting or texture blending can be significantly reduced. This reduction can be as much as from O(n) to O(log n) for n sequentially indexed voxels. **Volume** segmentation also has significant performance benefits for other ray tracing algorithms. **Volume** segmentation can subsequently be used to highlight structures of interest.

In differential geometry, a branch of mathematics, a **volume** form on a differentiable manifold is a differential form of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a **volume** form if and only if it is orientable. An orientable manifold has infinitely many **volume** forms, since multiplying a **volume** form by a non-vanishing function yields another **volume** form. On non-orientable manifolds, one may instead define the weaker notion of a density. Integrating the **volume** form gives the **volume** of the manifold according to that form.

An oriented pseudo-Riemannian manifold has a natural **volume** form. In local coordinates, it can be expressed as :

A **volume** label is the name given to a specific **volume** in a filesystem. In the FAT filesystem, the **volume** label was traditionally restricted to 11 characters (reflecting the 8.3 restrictions, but not divided into name and extension fields) even when long file name was enabled, stored as an entry within a disk's root directory with a special volume-label attribute bit set, and also copied to an 11-byte field within the Extended BIOS Parameter Block of the disk's boot sector. The label is always stored as uppercase in FAT and VFAT filesystems, and cannot contain special characters that are also disallowed for regular filenames. In the NTFS filesystem, the length of its **volume** label is restricted to 32 characters, and can include lowercase characters and even Unicode. The `label` command is used to change the label in DOS, Windows, and OS/2. For GUI systems like Windows Explorer, can be pressed while the **volume** is highlighted, or a right-click on the name will bring up a context menu that allows it to be renamed, either of which is the same process as for renaming a file. Changing the label in Windows will also change the **volume** creation timestamp to the current date and time for FAT filesystems. NTFS partitions have the System **Volume** Information directory, whose creation timestamp is set when Windows creates the partition, or when it first recognizes a repartitioning (the creation of a new volume) by a separate disk utility.

In contrast to the label, the **volume** serial number is generally unique and is not normally changed by the user, and thus acts as a more consistent and reliable identifier of when a **volume** has been changed (as when a disk is removed and another inserted). Disk formatting changes the serial number, but relabeling does not. The `vol` command can be used from the command line to display the current label and serial number of a volume.

Logical **volume** management - Hybrid **volume**

A hybrid **volume** is any **volume** that intentionally and opaquely makes use of two separate physical volumes. For instance, a workload may consist of random seeks so an SSD may be used to permanently store frequently used or recently written data, while using higher-capacity rotational magnetic media for long-term storage of rarely needed data. On Linux, bcache or dm-cache may be used for this purpose, while Fusion Drive may be used on OS X. ZFS also implements this functionality at the file system level, by allowing administrators to configure multi-level read/write caching.

Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural **volume** form. In local coordinates, it can be expressed as :where the dx^i are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, |g| is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

The **volume** form is denoted variously by