Editing Jacobian matrix and determinant (section)

== Jacobian matrix ==

The Jacobian of a vector-valued function in several variables generalizes the [[gradient]] of a [[scalar (mathematics)|scalar]]-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian matrix of a scalar-valued [[multivariate function|function of several variables]] is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivative.

At each point where a function is differentiable, its Jacobian matrix can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that the function imposes locally near that point. For example, if {{math|(''x''′, ''y''′) {{=}} '''f'''(''x'', ''y'')}} is used to smoothly transform an image, the Jacobian matrix {{math|'''J'''<sub>'''f'''</sub>(''x'', ''y'')}}, describes how the image in the neighborhood of {{math|(''x'', ''y'')}} is transformed.

If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However, a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order [[partial derivative]]s are required to exist.

If {{math|'''f'''}} is [[derivative|differentiable]] at a point {{math|'''p'''}} in {{math|'''R'''<sup>''n''</sup>}}, then its [[Total derivative#The total derivative as a linear map|differential]] is represented by {{math|'''J'''<sub>'''f'''</sub>('''p''')}}. In this case, the [[linear transformation]] represented by {{math|'''J'''<sub>'''f'''</sub>('''p''')}} is the best [[linear approximation]] of {{math|'''f'''}} near the point {{math|'''p'''}}, in the sense that

<math display="block">\mathbf f(\mathbf x) - \mathbf f(\mathbf p) = \mathbf J_{\mathbf f}(\mathbf p)(\mathbf x - \mathbf p) + o(\|\mathbf x - \mathbf p\|) \quad (\text{as } \mathbf{x} \to \mathbf{p}),</math>

where {{math|''o''(‖'''x''' − '''p'''‖)}} is a [[Big O notation#Little-o notation|quantity]] that approaches zero much faster than the [[Euclidean distance|distance]] between {{math|'''x'''}} and {{math|'''p'''}} does as {{math|'''x'''}} approaches {{math|'''p'''}}. This approximation specializes to the approximation of a scalar function of a single variable by its [[Taylor polynomial]] of degree one, namely

<math display="block">f(x) - f(p) = f'(p) (x - p) + o(x - p) \quad (\text{as } x \to p).</math>

In this sense, the Jacobian may be regarded as a kind of "[[derivative|first-order derivative]]" of a vector-valued function of several variables. In particular, this means that the [[gradient]] of a scalar-valued function of several variables may too be regarded as its "first-order derivative".

Composable differentiable functions {{math|'''f''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} and {{math|'''g''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''k''</sup>}} satisfy the [[Chain_rule#General_rule|chain rule]], namely <math>\mathbf{J}_{\mathbf{g} \circ \mathbf{f}}(\mathbf{x}) = \mathbf{J}_{\mathbf{g}}(\mathbf{f}(\mathbf{x})) \mathbf{J}_{\mathbf{f}}(\mathbf{x})</math> for {{math|'''x''' }} in {{math|'''R'''<sup>''n''</sup>}}.

The Jacobian of the gradient of a scalar function of several variables has a special name: the [[Hessian matrix]], which in a sense is the "[[second derivative]]" of the function in question.