Jacobian

2020 Mathematics Subject Classification: Primary: 26B10 Secondary: 26B15 [MSN][ZBL]

Jacobian Matrix

Also called Jacobi matrix. Let $U\subset \mathbb R^n$, $f: U\to \mathbb R^m$ and assume that $f$ is differentiable at the point $y\in U$. The Jacobi matrix of $f$ at $y$ is then the matrix \begin{equation}\label{e:Jacobi_matrix} Df|_y := \left( \begin{array}{llll} \frac{\partial f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^1}{\partial x_n} (y)\\ \frac{\partial f^2}{\partial x_1} (y) & \frac{\partial f^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^2}{\partial x_n} (y)\\ \\ \vdots & \vdots & &\vdots\\ \\ \frac{\partial f^m}{\partial x_1} (y) & \frac{\partial f^m}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^m}{\partial x_n} (y) \end{array}\right)\, , \end{equation} where $(f^1, \ldots, f^m)$ are the coordinate functions of $f$ and $x_1,\ldots, x_n$ denote the standard system of coordinates in $\mathbb R^n$.

Jacobian determinant

Also called Jacobi determinant. If $U$, $f$ and $y$ are as above and $m=n$, the Jacobian determinant of $f$ at $y$ is the determinant of the Jacobian matrix \ref{e:Jacobi_matrix}. Some authors use the same name for the absolute value of such determinant. If $U$ is an open set and $f$ a locally invertible $C^1$ map, the absolute value of the Jacobian determinant gives the infinitesimal dilatation of the volume element in passing from the variables $x_1, \ldots, x_n$ to the variables $f_1,\ldots, f_n$. Therefore the Jacobian determinant plays a crucial role when changing variables in integrals, see Sections 3.2 and 3.3 of [EG] (see also Differential form and Integration on manifolds).

Generalizations of the Jacobian determinant

The Jacobian determinant can be generalized also to the case where the dimension of the target differs from that of the domain (see Section 3.2 of [EG]). More precisely, let $f$, $U$, $n$, $m$ and $y$ be as above:

• If $m<n$, the Jacobian of $f$ at $y$ is given by the square root of the determinant of $Df_y\cdot (Df_y)^t$ (where $Df_y^t$ denotes the transpose of the matrix $Df_y$);
• If $m>n$, the Jacobian of $f$ at $y$ is given by the square root of the determinant of $(Df_y)^t\cdot Df_y$.

These generalizations play a key role respectively in the Coarea formula and Area formula.

An important characterization of the Jacobian is then given by the Cauchy Binet formula: $Jf(y)^2$ is the sum of the squares of the determinants of all $n\times n$ minors of $Df|_y$ (cp. with Theorem 4 in Section 3.2.1 of [EG]).

Jacobian variety

See Jacobi variety.

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001
[IP]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) MR0687827 MR0687828
[Ku]	L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973)
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ni]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) MR0796320Zbl 0479.00001
[Ru]	W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Si]	L. Simon, "Lectures on geometric measure theory", Proceedings of the Centre for Mathematical Analysis, 3. Australian National University. Canberra (1983) MR0756417 Zbl 0546.49019
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965) MR0209411 Zbl 0141.05403