Jacobian

Given a set ${\bf y}={\bf f}({\bf x})$ of $n$ equations in $n$ variables $x_1$, ..., $x_n$, written explicitly as

$${\bf y}\equiv\left[{\matrix{f_1({\bf x})\cr f_2({\bf x})\cr \vdots\cr f_n({\bf x})\cr}}\right],$$ (1)

or more explicitly as

$$\cases{ y_1=f_1(x_1, \ldots, x_n)\cr \vdots\cr y_n=f_n(x_1, \ldots, x_n),\cr}$$ (2)

the Jacobian matrix, sometimes simply called ``the Jacobian'' (Simon and Blume 1994) is defined by

$${\hbox{\sf J}}(x_1,\dots,x_n) = \left[{\matrix{ {\partial y... ...n\over\partial x_n}\cr}}\right]. \hrule width 0pt height 6.6pt$$ (3)

The Determinant of ${\hbox{\sf J}}$ is the Jacobian Determinant (confusingly, often called ``the Jacobian'' as well) and is denoted

$$J = \left\vert{\partial(y_1,\ldots,y_n)\over\partial(x_1,\ldots,x_n)}\right\vert.$$ (4)

Taking the differential

$$d{\bf y}={\bf y}_{\bf x}\,d{\bf x}$$ (5)

shows that $J$ is the Determinant of the Matrix ${\bf y}_{\bf x}$, and therefore gives the ratios of $n$-D volumes (Contents) in $y$ and $x$,

$$dy_1\cdots dy_n = \left\vert{\partial(y_1,\ldots,y_n)\over\partial(x_1,\ldots,x_n)}\right\vert\,dx_1\cdots dx_n.$$ (6)

The concept of the Jacobian can also be applied to $n$ functions in more than $n$ variables. For example, considering $f(u,v,w)$ and $g(u,v,w)$, the Jacobians

$${\partial(f,g)\over\partial(u,v)} = \left\vert\begin{array}{cc}f_u & f_v\\ g_u & g_v\end{array}\right\vert$$ (7)

$${\partial(f,g)\over\partial(u,w)} = \left\vert\begin{array}{cc}f_u & f_w\\ g_u & g_w\end{array}\right\vert$$ (8)

can be defined (Kaplan 1984, p. 99).

For the case of $n=3$ variables, the Jacobian takes the special form

$$Jf(x_1,x_2,x_3) \equiv\left\vert{{\partial {\bf y}\over\part... ... x_2} \times {\partial{\bf y}\over\partial x_3}}\right\vert,$$ (9)

where ${\bf a}\cdot{\bf b}$ is the Dot Product and ${\bf b}\times{\bf c}$ is the Cross Product, which can be expanded to give

$$\left\vert\matrix{\partial(y_1,y_2,y_3)\over\partial(x_1,x_2... ...ver\partial x_3}\cr}\right\vert. \hrule width 0pt height 6.3pt$$ (10)

See also Change of Variables Theorem, Curvilinear Coordinates, Implicit Function Theorem

References

Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 98-99, 123, and 238-245, 1984.

Simon, C. P. and Blume, L. E. Mathematics for Economists. New York: W. W. Norton, 1994.