“Jacobian”的概念、定义、习题解题方法及翻译-数学辞典

单词

Jacobian

释义

Jacobian

Given a set of equations in variables , ..., , written explicitly as

(1)

or more explicitly as

(2)

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

(3)

The Determinant of

is the Jacobian Determinant (confusingly, often called ``the Jacobian'' as well)and is denoted

(4)

Taking the differential

(5)

shows that

is the Determinant of the Matrix

, and therefore gives the ratios of

-Dvolumes (Contents) in

and

(6)

The concept of the Jacobian can also be applied to

functions in more than

variables. For example, considering

and

, the Jacobians

			(7)
			(8)

can be defined (Kaplan 1984, p. 99).

For the case of variables, the Jacobian takes the special form

(9)

where

is the Dot Product and

is the Cross Product,which can be expanded to give

(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.

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