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Jacobian


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

 y=[f_1(x); f_2(x); |; f_n(x)],
(1)

or more explicitly as

 {y_1=f_1(x_1,...,x_n); |; y_n=f_n(x_1,...,x_n),
(2)

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

 J(x_1,...,x_n)=[(partialy_1)/(partialx_1) ... (partialy_1)/(partialx_n); | ... |; (partialy_n)/(partialx_1) ... (partialy_n)/(partialx_n)].
(3)

The determinant of J is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted

 J=|(partial(y_1,...,y_n))/(partial(x_1,...,x_n))|.
(4)

The Jacobian matrix and determinant can be computed in the Wolfram Language using

  JacobianMatrix[f_List?VectorQ, x_List] := Outer[D, f, x]
  JacobianDeterminant[f_List?VectorQ, x_List] := Det[JacobianMatrix[f, x]]

Taking the differential

 dy=y_(x)dx
(5)

shows that J is the determinant of the matrix y_(x), and therefore gives the ratios of n-dimensional volumes (contents) in y and x,

 dy_1...dy_n=|(partial(y_1,...,y_n))/(partial(x_1,...,x_n))|dx_1...dx_n.
(6)

It therefore appears, for example, in the change of variables theorem.

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))/(partial(u,v))=|f_u f_v; g_u g_v|
(7)
(partial(f,g))/(partial(u,w))=|f_u f_w; g_u g_w|
(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)=|(partialy)/(partialx_1)·(partialy)/(partialx_2)×(partialy)/(partialx_3)|,
(9)

where a·b is the dot product and b×c is the cross product, which can be expanded to give

 |(partial(y_1,y_2,y_3))/(partial(x_1,x_2,x_3))|=|(partialy_1)/(partialx_1) (partialy_1)/(partialx_2) (partialy_1)/(partialx_3); (partialy_2)/(partialx_1) (partialy_2)/(partialx_2) (partialy_2)/(partialx_3); (partialy_3)/(partialx_1) (partialy_3)/(partialx_2) (partialy_3)/(partialx_3)|.
(10)

See also

Change of Variables Theorem, Curvilinear Coordinates, Hessian, Implicit Function Theorem, Multivariable Calculus, Wronskian Explore this topic in the MathWorld classroom

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References

Gradshteyn, I. S. and Ryzhik, I. M. "Jacobian Determinant." §14.313 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1068-1069, 2000.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.

Referenced on Wolfram|Alpha

Jacobian

Cite this as:

Weisstein, Eric W. "Jacobian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Jacobian.html

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