nLab Jacobian


This entry is about the concept in differential geometry. For the concept of Jacobian variety see there.


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



If f: n mf : \mathbb{R}^n \to \mathbb{R}^m is a C 1C^1-differentiable map, between Cartesian spaces, its Jacobian matrix is the (m×n)(m \times n) matrix

J(f)Mat m×n(C 0(,)) J(f) \in Mat_{m \times n}(C^0(\mathbb{R}, \mathbb{R}))

of partial derivatives

J(f) j i:=f ix j,i=1,,m;j=1,,n, J(f)^i_j := \frac{\partial f^i}{\partial x^j},\,\,\,\,\,\,\,i=1,\ldots,m; j = 1,\ldots,n,

where x=(x 1,,x n)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 R n\mathbf{R}^n is the space of real column vectors of length nn.

In more general situation, if f=(f 1(x),,f m(x))f = (f^1(x),\ldots,f^m(x)) is differentiable at a point xx (and possibly defined only in a neighborhood of xx), we define the Jacobian J pfJ_p f of map ff at point xx as a matrix with real values (J pf) j i=f ix j| x(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=mn=m the Jacobian matrix is a square matrix, hence its determinant det(J(f))det(J(f)) is defined and called the Jacobian of ff (possibly only at a point). Sometimes one refers to Jacobian matrix rather ambigously by Jacobian as well.


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

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

(T pg)(X p)= i,jJ p(ψgϕ 1) i jX p iy j| g(p) (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)}

Last revised on June 9, 2023 at 13:57:22. See the history of this page for a list of all contributions to it.