Editing Jacobian matrix and determinant (section)

== Jacobian determinant ==

[[File:Jacobian_determinant_and_distortion.svg|thumb|400px|A nonlinear map <math>f \colon \mathbb{R}^{2} \to \mathbb{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.]]

If {{math|1=''m'' = ''n''}}, then {{math|'''f'''}} is a function from {{math|'''R'''<sup>''n''</sup>}} to itself and the Jacobian matrix is a [[square matrix]]. We can then form its [[determinant]], known as the '''Jacobian determinant'''. The Jacobian determinant is sometimes simply referred to as "the Jacobian".

The Jacobian determinant at a given point gives important information about the behavior of {{math|'''f'''}} near that point. For instance, the [[continuously differentiable function]] {{math|'''f'''}} is [[invertible]] near a point {{math|'''p''' ∈ '''R'''<sup>''n''</sup>}} if the Jacobian determinant at {{math|'''p'''}} is non-zero. This is the [[inverse function theorem]]. Furthermore, if the Jacobian determinant at {{math|'''p'''}} is [[positive number|positive]], then {{math|'''f'''}} preserves [[Orientation (vector space)|orientation]] near {{math|'''p'''}}; if it is [[negative number|negative]], {{math|'''f'''}} reverses orientation. The [[absolute value]] of the Jacobian determinant at {{math|'''p'''}} gives us the factor by which the function {{math|'''f'''}} expands or shrinks [[volume]]s near {{math|'''p'''}}; this is why it occurs in the general [[substitution rule]].

The Jacobian determinant is used when making a [[Integration by substitution#Substitution for multiple variables|change of variables]] when evaluating a [[multiple integral]] of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. This is because the {{math|''n''}}-dimensional {{math|''dV''}} element is in general a [[parallelepiped]] in the new coordinate system, and the {{math|''n''}}-volume of a parallelepiped is the determinant of its edge vectors.

The Jacobian can also be used to determine the stability of [[equilibrium point|equilibria]] for [[matrix differential equation|systems of differential equations]] by approximating behavior near an equilibrium point.