# Jacobian

*Jacobi determinant*

A determinant of a matrix of special form whose entries are first-order partial derivatives of functions. Let , , be given functions having first-order partial derivatives with respect to the variables . The Jacobian of these functions is the determinant

which for brevity is denoted by the symbol

The modulus of a Jacobian characterizes the dilatation (contraction) of the volume element in the transition from the variables to . The name is given after C.G.J. Jacobi, who first studied its properties and applications.

#### References

[1] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) |

[2] | L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973) (In Russian) |

[3] | S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian) |

#### Comments

The above Jacobian is also denoted as

The entry at place (-th row, -th column) is . The matrix with these entries is also called a Jacobian matrix (not to be confused with a Jacobi matrix). The Jacobian plays a role in the statement of the inverse function theorem and in change-of-variable formulas for integrals and differential forms.

#### References

[a1] | M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965) |

[a2] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |

**How to Cite This Entry:**

Jacobian.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jacobian&oldid=14954