“Change of Variables Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Change of Variables Theorem

释义

Change of Variables Theorem

A theorem which effectively describes how lengths, areas, volumes, and generalized -dimensional volumes(Contents) are distorted by Differentiable Functions. In particular, thechange of variables theorem reduces the whole problem of figuring out the distortion of the content to understanding theinfinitesimal distortion, i.e., the distortion of the Derivative (a linear Map), which is given by the linearMap's Determinant. So is an Area-Preserving linearMap Iff , and in more generality, if is any subset of , theContent of its image is given by times the Content of the original. The change ofvariables theorem takes this infinitesimal knowledge, and applies Calculus by breaking up the Domain intosmall pieces and adds up the change in Area, bit by bit.

The change of variable formula persists to the generality of Differential Formson Manifolds, giving the formula

under the conditions that

and

are compact connected oriented Manifolds with nonemptyboundaries,

is a smooth map which is an orientation-preserving Diffeomorphism of the boundaries.

In 2-D, the explicit statement of the theorem is

and in 3-D, it is

where is the image of the original region ,

is the Jacobian, and

is a global orientation-preserving Diffeomorphism of

and

(which are opensubsets of

The change of variables theorem is a simple consequence of the Curl Theorem and a little de Rham Cohomology.The generalization to -D requires no additional assumptions other than the regularity conditions on the boundary.

See also Implicit Function Theorem, Jacobian

References

Kaplan, W. ``Change of Variables in Integrals.'' §4.6 in Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 238-245, 1984.

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