Determinant
Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if f: R n → R n {\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}} is the linear map given by multiplication with a matrix A {\displaystyle A}, and S ⊂ R n {\displaystyle S\subset \mathbf {R} ^{n}} is any measurable subset, then the volume of f (S) {\displaystyle f(S)} is given by | det (A) | {\displaystyle |\det(A)|} times the volume of S {\displaystyle S}. More generally, if the linear map f: R n → R m {\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}} is represented by the m × n {\displaystyle m\times n} -matrix A {\displaystyle A}, then the n {\displaystyle n} - dimensional volume of f (S) {\displaystyle f(S)} is given by:

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices a, b, c, d {\displaystyle a,b,c,d}, 1 6 ⋅ | det (a − b, b − c, c − d) | {\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}, or any other combination of pairs of vertices that form a spanning tree over the vertices.

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For

the Jacobian matrix is the n × n matrix whose entries are given by the partial derivatives

Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of R (the domain of f), the integral over f (U) of some other function φ : R → R is given by

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.
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