Editing Determinant (section)

=== 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 <math>f : \mathbf R^n \to \mathbf R^n</math> is the linear map given by multiplication with a matrix <math>A</math>, and <math>S \subset \mathbf R^n</math> is any [[Lebesgue measure|measurable]] [[subset]], then the volume of <math>f(S)</math> is given by <math>|\det(A)|</math> times the volume of <math>S</math>.<ref>{{harvnb|Lang|1985|loc=§VII.6, Theorem 6.10}}</ref> More generally, if the linear map <math>f : \mathbf R^n \to \mathbf R^m</math> is represented by the <math>m \times n</math> matrix <math>A</math>, then the <math>n</math>-[[dimension]]al volume of <math>f(S)</math> is given by:
:<math>\operatorname{volume}(f(S)) = \sqrt{\det\left(A^\textsf{T} A\right)} \operatorname{volume}(S).</math>

By calculating the volume of the [[tetrahedron]] bounded by four points, they can be used to identify [[skew line]]s. The volume of any tetrahedron, given its [[vertex (geometry)|vertices]] <math>a, b, c, d</math>, <math>\frac 1 6 \cdot |\det(a-b,b-c,c-d)|</math>, or any other combination of pairs of vertices that form a [[spanning tree]] over the vertices.

[[File:Jacobian_determinant_and_distortion.svg|350px|thumb|right|A nonlinear map <math>f \colon \mathbf{R}^{2} \to \mathbf{R}^{2}</math> sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.]]
For a general [[differentiable function]], much of the above carries over by considering the [[Jacobian matrix]] of ''f''. For
:<math>f: \mathbf R^n \rightarrow \mathbf R^n,</math>

the Jacobian matrix is the {{math|''n'' × ''n''}} matrix whose entries are given by the [[partial derivative]]s
:<math>D(f) = \left(\frac {\partial f_i}{\partial x_j}\right)_{1 \leq i, j \leq n}.</math>

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'''<sup>''n''</sup> (the domain of ''f''), the integral over ''f''(''U'') of some other function {{math|''φ'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} is given by
:<math>\int_{f(U)} \phi(\mathbf{v})\, d\mathbf{v} = \int_U \phi(f(\mathbf{u})) \left|\det(\operatorname{D}f)(\mathbf{u})\right| \,d\mathbf{u}.</math>

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.<ref>{{Cite book|last=Lay|first=David|title=Linear Algebra and Its Applications 6th Edition|publisher=Pearson|year=2021|pages=172|language=English}}</ref>