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\begin{document}

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\section*{Jacobian}

\begin{quote}%
This entry is about the concept in [[differential geometry]]. For the concept of [[Jacobian variety]] see there.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

If $f : \mathbb{R}^n \to \mathbb{R}^m$ is a $C^1$-[[differentiable map]], between [[Cartesian space]]s, its \textbf{Jacobian matrix} is the $(m \times n)$ [[matrix]]

\begin{displaymath}
J(f) \in Mat_{m \times n}(C^0(\mathbb{R}, \mathbb{R}))
\end{displaymath}
of [[partial derivative]]s

\begin{displaymath}
J(f)^i_j := \frac{\partial f^i}{\partial x^j},\,\,\,\,\,\,\,i=1,\ldots,m; j = 1,\ldots,n,
\end{displaymath}
where $x = (x^1,\ldots,x^n)$. Here the convention is that the upper index is a row index and the lower index is the column index; in particular $\mathbf{R}^n$ is the space of real column vectors of length $n$.

In more general situation, if $f = (f^1(x),\ldots,f^m(x))$ is differentiable at a point $x$ (and possibly defined only in a [[neighborhood]] of $x$), we define the Jacobian $J_p f$ of map $f$ at point $x$ as a matrix with real values $(J_p f)^i_j = \frac{\partial f^i}{\partial x^j}|_x$.

That is, the Jacobian is the matrix which describes the [[total derivative]].

If $n=m$ the Jacobian matrix is a square matrix, hence its [[determinant]] $det(J(f))$ is defined and called \textbf{the Jacobian of $f$} (possibly only at a point). Sometimes one refers to Jacobian matrix rather ambigously by Jacobian as well.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The [[chain rule]] may be phrased by saying that the Jacobian matrix of the composition $\mathbf{R}^n\stackrel{f}\to\mathbf{R}^m\stackrel{g}\to\mathbf{R}^r$ is the [[matrix product]] of the Jacobian matrices of $g$ and of $f$ (at appropriate points).

If $g:M\to N$ is a $C^1$-map of $C^1$-manifolds, then the [[tangent bundle|tangent map]] $T g: T M\to T N$ defined point by point abstractly by $(T_p g)(X_p)(f) = X_p(f\circ g)$, for $p\in M$, can in local [[coordinates]] be represented by a Jacobian matrix. Namely, if $(U,\phi)\ni p$ and $(V,\psi)\ni g(p)$ are [[charts]] and $X_p = \sum X^i\frac{\partial}{\partial x^i}|_p$ (i.e. $X_p(f)  = \sum_i X^i_p \frac{\partial (f\circ \phi^{-1})}{\partial x^i}|_{\phi(p)}$ for all germs $f\in \mathcal{F}_p$), then

\begin{displaymath}
(T_p g)(X_p) = \sum_{i,j} J_p(\psi \circ g\circ\phi^{-1})_i^j X^i_p \frac{\partial}{\partial y^j}|_{g(p)}
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differentiation]], [[Hessian matrix]], [[extremum]]


\item [[pullback of a distribution]]



\end{itemize}
[[!redirects Jacobian matrix]] [[!redirects Jacobian determinant]]

[[!redirects Jacobian matrices]] [[!redirects Jacobian determinants]]



\end{document}