Volume

a) The volume of a three-dimensional body is a numerical characteristic of the body; in the simplest case, when the body can be decomposed into a finite set of unit cubes (i.e. cubes with edges of unit length), it is equal to the number of these cubes. Volumes of three-dimensional bodies (i.e. sets in three-dimensional Euclidean space) for which volume can be defined have properties analogous to those of areas of plane figures (cf. Area): 1) volume is non-negative; 2) volume is additive: If two bodies $ P $ and $ Q $ with no common interior points have volumes $ v ( P) $ and $ v ( Q) $, then the volume of their union is the sum of their volumes, $ v ( P \cup Q ) = v ( P) + v ( Q) $; 3) volume is invariant with respect to displacements: If the volumes of bodies $ P $ and $ Q $ are defined and the bodies are congruent, then $ v ( P) = v ( Q) $; and 4) the volume of the unit cube is equal to one. These properties imply that volume is monotone: If the volumes $ v ( P) $ and $ v ( Q) $ are defined for bodies $ P $ and $ Q $ and $ P \subset Q $, then $ v ( P) \leq v ( Q) $, and the volumes of two similar bodies are proportional to the cube of the factor of the corresponding similarity.

Moreover, there is a unique non-negative function on the set of all polyhedra satisfying properties 1)–4). However, while in the plane any two equal (having equal areas) polygons have equal shape, that is, each of them can be decomposed into polygons which can be put together to form the other, the analogous property is no longer true in three-dimensional space: There exist equal (having equal volume) polyhedra which are not equi-decomposable (see Equal content and equal shape, figures of).

The concept of volume can be extended, while still preserving properties 1)–4), by means of limiting processes from the set of polyhedra to wider classes of bodies, for example, bodies with piecewise-smooth boundaries, in particular, spheres, spherical shells, spherical segments and sectors, cylinders, cones, etc. If the boundary of a bounded body $ G $ is a piecewise-smooth surface, then the volume $ v ( G) $ can be defined as follows. Consider all polyhedra $ P $ contained in $ G $, and all polyhedra $ Q $ containing $ G $: $ P \subset G \subset Q $. Then the following equation holds:

$$ \sup _ {P \subset Q } v ( P) = \inf _ {Q \supset G } v ( Q) . $$

This common value is called the volume $ v ( G) $ of $ G $. As for plane figures, computation of the volume of the body can also be done by decomposing it into parts whose volumes are already known or can be calculated more simply. In this connection, Cavalieri's principle is valid for volumes: If two bodies whose volumes are defined are intersected by every plane parallel to a given plane in plane figures having equal area, then they have equal volume. Integral calculus provides a general method of computing volume, by reducing it to the computation of the corresponding multiple integrals. Integral calculus can also be used to justify Cavalieri's principle.

b) The extension of the concept of volume to a still wider class of subsets of three-dimensional Euclidean space leads to the concepts of Jordan content and cubeable sets (cf. Jordan measure). All bodies mentioned above are cubeable sets, and so their Jordan content is defined.

If $ M ^ {3} $ is a three-dimensional continuously-differentiable manifold with Riemannian metric

$$ d s ^ {2} = \ \sum _ {i , j = 1 } ^ { 3 } g _ {ij} d x ^ {i} d x ^ {j} $$

and $ g = \mathop{\rm det} \| g _ {ij} \| $, then the volume of a set $ E $ in $ M ^ {3} $ is defined as the value of the integral

$$ {\int\limits \int\limits \int\limits } _ { E } | g | d x ^ {1} d x ^ {2} d x ^ {3} . $$

The volume thus defined is invariant with respect to the choice of local coordinates in the given manifold.

c) The definition of the volume of a three-dimensional body can be generalized to a subset of any $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $; $ n $- dimensional volume, or content, is a set-function satisfying conditions 1)–4), with cube understood to mean $ n $- dimensional cube. The computation of the volume of a set in $ n $- dimensional space reduces to the computation of an $ n $- fold integral.

If $ E $ is the $ n $- dimensional parallelotope spanned by vectors $ \mathbf a _ {1} \dots \mathbf a _ {n} $, then

$$ v ( E) = \ \sqrt {| \mathop{\rm det} \| \mathbf a _ {i} , \mathbf a _ {j} \| | } $$

(the expression under the root sign is the absolute value of the Gram determinant of the vectors $ \mathbf a _ {1} \dots \mathbf a _ {n} $).

The volume of a subset $ E $ of an $ n $- dimensional Riemannian manifold with metric

$$ d s ^ {2} = \ \sum _ {i , j = 1 } ^ { n } g _ {ij} d x ^ {i} d x ^ {j} $$

is the integral

$$ {\int\limits \dots \int\limits } _ { E } | g | d x ^ {1} \dots d x ^ {n} , $$

where $ g = \mathop{\rm det} \| g _ {ij} \| $.

d) The concept of volume can be considered in affine spaces, as well as in Euclidean spaces. Let $ E $ be a subset of the $ n $- dimensional affine space $ \mathbf R ^ {n} $ and let an affine coordinate system $ ( x ^ {1} \dots x ^ {n} ) $ be given in $ \mathbf R ^ {n} $. Let

$$ V _ {x} ( E) = {\int\limits \dots \int\limits } _ { E } d x ^ {1} \dots d x ^ {n} . $$

If $ ( y ^ {1} \dots y ^ {n} ) $ is another coordinate system and $ y ^ {i} = a _ {j} ^ {i} x ^ {j} + b ^ {i} $, then

$$ V _ {y} ( E) = \ | \mathop{\rm det} \| a _ {j} ^ {i} \| | V _ {x} ( E) , $$

that is, the integral $ V _ {x} ( E) $ is a relative invariant of constant sign of weight $ - 1 $. The relative invariant $ V _ {x} ( E) $ is the affine volume of the set $ E $. The affine volume is unchanged under parallel displacement, and if $ E = E _ {1} \cup E _ {2} $, $ E _ {1} \cap E _ {2} = \emptyset $ and $ V _ {x} ( E _ {1} ) $, $ V _ {x} ( E _ {2} ) $ are defined, then $ V _ {x} ( E) = V _ {x} ( E _ {1} ) + V _ {x} ( E _ {2} ) $. Affine volume has the following properties, which are invariant under an affine change of coordinates: $ \alpha $) if the volumes of two bodies are equal in one system of coordinates, then they are equal in any other system; $ \beta $) if the volume of a set is equal to the sum of the volumes of two other sets in one coordinate system, then this is true in any other system; and $ \gamma $) the ratio of the volumes of two sets is invariant under affine transformations of coordinates.

Let $ e _ {1} \dots e _ {n} $ be the coordinate vectors of the system of coordinates $ x _ {1} \dots x _ {n} $ in $ \mathbf R ^ {n} $, and let the origin be at some fixed point $ 0 \in \mathbf R ^ {n} $. If $ \mathbf a _ {i} = a _ {i} ^ {j} \mathbf e _ {j} $, $ i= 1 \dots n $, is a system of vectors with origin at $ 0 $ and $ E $ is the $ n $- dimensional parallelotope spanned by $ \mathbf a _ {1} \dots \mathbf a _ {n} $, then

$$ V _ {x} ( E) = | \mathop{\rm det} \| a _ {i} ^ {j} \| | . $$

The quantity $ \mathop{\rm det} \| a _ {i} ^ {j} \| $ is called the oriented volume of $ E $ with respect to the chosen basis $ \mathbf e _ {1} \dots \mathbf e _ {n} $.

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