Editing Jacobian matrix and determinant (section)

{{Short description|Matrix of partial derivatives of a vector-valued function}}
{{Redirect|Jacobian matrix|the operator|Jacobi matrix (operator)}}
{{Calculus |Multivariable}}

In [[vector calculus]], the '''Jacobian matrix''' ({{IPAc-en|dʒ|ə|ˈ|k|əʊ|b|i|ə|n}},<ref>{{cite web|url=https://en.oxforddictionaries.com/definition/jacobian|title=Jacobian - Definition of Jacobian in English by Oxford Dictionaries|website=Oxford Dictionaries - English|access-date=2 May 2018|url-status=dead|archive-url=https://web.archive.org/web/20171201043633/https://en.oxforddictionaries.com/definition/jacobian|archive-date=1 December 2017}}</ref><ref>{{cite web|url=http://www.dictionary.com/browse/jacobian|title=the definition of jacobian|website=Dictionary.com|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20171201040801/http://www.dictionary.com/browse/jacobian|archive-date=1 December 2017}}</ref><ref>{{cite web|url=https://forvo.com/word/jacobian/|title=Jacobian pronunciation: How to pronounce Jacobian in English|first=Forvo|last=Team|website=forvo.com|access-date=2 May 2018}}</ref> {{IPAc-en|dʒ|ᵻ|-|,_|j|ᵻ|-}}) of a [[vector-valued function]] of several variables is the [[matrix (mathematics)|matrix]] of all its first-order [[partial derivative]]s. If this matrix is [[square matrix|square]], that is, if the number of variables equals the number of [[Euclidean_vector#Decomposition|components]] of function values, then its [[determinant]] is called the '''Jacobian determinant'''. Both the matrix and (if applicable) the determinant are often referred to simply as the '''Jacobian'''.<ref>{{cite web|url=http://mathworld.wolfram.com/Jacobian.html|title=Jacobian|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20171103144419/http://mathworld.wolfram.com/Jacobian.html|archive-date=3 November 2017}}</ref> They are named after [[Carl Gustav Jacob Jacobi]].

The Jacobian matrix is the natural generalization to vector valued functions of several variables of the [[derivative]] and the [[differential of a function|differential]] of a usual function. This generalization includes generalizations of the [[inverse function theorem]] and the [[implicit function theorem]], where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the [[multiplicative inverse]] of the derivative is replaced by the [[inverse of a matrix|inverse]] of the Jacobian matrix. 

The Jacobian determinant is fundamentally used for changes of variables in [[multiple integral]]s.