nLab
Jacobian

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


Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

For Jacobian in the sense of Jacobian variety (of an algebraic curve), see there (also more general intermediate Jacobians).

Definition

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.

Properties

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)}

Revised on July 18, 2015 04:52:59 by Urs Schreiber (94.118.161.144)