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.
|||V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)|
|||L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973) (In Russian)|
|||S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)|
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.
|[a1]||M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965)|
|[a2]||W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)|
Jacobian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobian&oldid=14954