Change of Variables

Summary

12.1 In this section we define derivatives Dμ, Dμ, and D̄μ of a measure μ with respect to Lebesgue measure in V. Next, we analyze the relationships between the Lebesgue decomposition of μ and the properties of its derivatives; for example, if D̄μ is finite everywhere, μ is absolutely continuous with respect to Lebesgue measure.

12.2 First, we study the image of λ under a linear automorphism. The modulus of an automorphism of R^k is the absolute value of its determinant (Theorem 12.2.1).

12.3 In this section we prove the change of variables formula: if T is a differentiable homeomorphism from V onto W, then λ _W is the image of |J(T)|λ _V under T where J(T) is the Jacobian of T (Theorem 12.3.1).

12.4 This section is devoted to polar coordinates in Rⁿ.