Editing Jacobian matrix and determinant (section)

== Definition ==
Let <math display="inline">\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m</math> be a function such that each of its first-order partial derivatives exists on <math display="inline>\mathbb{R}^n</math>. This function takes a point {{tmath|1=\mathbf x =(x_1,\ldots,x_n)\in \mathbb{R}^n}} as input and produces the vector {{tmath|1=\mathbf f(\mathbf x) = (f_1(\mathbf x), \ldots, f_m(\mathbf x)) \in \mathbb{R}^m}}  as output. Then the Jacobian matrix of {{math|'''f'''}}, denoted {{math|'''J<sub>f</sub>'''}}, is the {{tmath|m\times n}} matrix whose {{math|(''i'', ''j'')}} entry is <math display="inline">\frac{\partial f_i}{\partial x_j};</math> explicitly 
<math display="block">\mathbf{J_f} = \begin{bmatrix}
  \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n}
\end{bmatrix}
= \begin{bmatrix}
  \nabla^{\mathsf{T}} f_1 \\  
  \vdots \\
  \nabla^{\mathsf{T}} f_m   
\end{bmatrix}
= \begin{bmatrix}
    \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\
    \vdots                             & \ddots & \vdots\\
    \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n}
\end{bmatrix}</math> where <math>\nabla^{\mathsf{T}} f_i</math> is the transpose (row vector) of the [[gradient]] of the <math>i</math>-th component.

The Jacobian matrix, whose entries are functions of {{math|'''x'''}}, is denoted in various ways; other common notations include {{math|''D'''''f'''}}, <math>\nabla \mathbf{f}</math>, and <math display="inline">\frac{\partial(f_1,\ldots,f_m)}{\partial(x_1,\ldots,x_n)}</math>.<ref>{{Cite book |last1=Holder |first1=Allen |title=An Introduction to computational science |last2=Eichholz |first2=Joseph |date=2019 |publisher=Springer |isbn=978-3-030-15679-4 |series=International Series in Operations Research & Management Science |location=Cham, Switzerland |pages=53}}</ref><ref>{{Cite book |last=Lovett |first=Stephen |url=https://books.google.com/books?id=G1bGDwAAQBAJ |title=Differential Geometry of Manifolds |date=2019-12-16 |publisher=CRC Press |isbn=978-0-429-60782-0 |pages=16 |language=en}}</ref> Some authors define the Jacobian as the [[transpose]] of the form given above.

The Jacobian matrix [[Matrix_(mathematics)#Linear_transformations|represents]] the [[total derivative|differential]] of {{math|'''f'''}} at every point where {{math|'''f'''}} is differentiable. In detail, if {{math|'''h'''}} is a [[displacement vector]] represented by a [[column matrix]], the [[matrix product]] {{math|'''J'''('''x''') ⋅ '''h'''}} is another displacement vector, that is the best linear approximation of the change of {{math|'''f'''}} in a [[neighborhood (mathematics)|neighborhood]] of {{math|'''x'''}}, if {{math|'''f'''('''x''')}} is [[Differentiable function|differentiable]] at {{math|'''x'''}}.{{efn|Differentiability at {{math|'''x'''}} implies, but is not implied by, the existence of all first-order partial derivatives at {{math|'''x'''}}, and hence is a stronger condition.}} This means that the function that maps {{math|'''y'''}} to {{math|'''f'''('''x''') + '''J'''('''x''') ⋅ ('''y''' – '''x''')}} is the best [[linear approximation]] of {{math|'''f'''('''y''')}} for all points {{math|'''y'''}} close to {{math|'''x'''}}. The [[linear map]] {{math|'''h''' → '''J'''('''x''') ⋅ '''h'''}} is known as the ''derivative'' or the [[total derivative|''differential'']] of {{math|'''f'''}} at {{math|'''x'''}}.

When <math display="inline">m=n</math>, the Jacobian matrix is square, so its [[determinant]] is a well-defined function of {{math|'''x'''}}, known as the '''Jacobian determinant''' of {{math|'''f'''}}. It carries important information about the local behavior of {{math|'''f'''}}. In particular, the function {{math|'''f'''}} has a differentiable [[inverse function]] in a neighborhood of a point {{math|'''x'''}} if and only if the Jacobian determinant is nonzero at {{math|'''x'''}} (see [[inverse function theorem]] for an explanation of this and [[Jacobian conjecture]] for a related problem of ''global'' invertibility). The Jacobian determinant also appears when changing the variables in [[multiple integral]]s (see [[Integration_by_substitution#Substitution_for_multiple_variables|substitution rule for multiple variables]]).

When <math display="inline">m=1</math>, that is when <math display="inline"> f: \mathbb{R}^n \to \mathbb{R}</math> is a [[scalar field|scalar-valued function]], the Jacobian matrix reduces to the [[row vector]] <math>\nabla^{\mathsf{T}} f</math>; this row vector of all first-order partial derivatives of {{tmath|f}} is the transpose of the [[gradient]] of {{tmath|f}}, i.e.
<math>\mathbf{J}_{f} = \nabla^{\mathsf{T}} f</math>. Specializing further, when <math display="inline">m=n=1</math>, that is when <math display="inline">f: \mathbb{R} \to \mathbb{R}</math> is a [[scalar field|scalar-valued function]] of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function {{tmath|f}}.

These concepts are named after the [[mathematician]] [[Carl Gustav Jacob Jacobi]] (1804–1851).