Q1: Suppose B=[0,1]x[0,2]x[0,3]x[0,4] in R(adsbygoogle = window.adsbygoogle || []).push({}); ^{4}, and that C=[0,1]x[0,1]x[0,1]x[0,1]. Given that

∫ ∫ ∫ ∫f(x)=d^{4}x=(2pi)^{4}

B

What is the value of

∫ ∫ ∫ ∫ f(x_{1},2x_{2},3x_{3},4x_{4}) d^{4}x?

C

Solution:

Definex=G(u)=(u_{1},u_{2}/2,u_{3}/3,u_{4}/4)

∫ ∫ ∫ ∫ f(x_{1},2x_{2},3x_{3},4x_{4}) d^{4}x

C

by change of variables theorem,

=(1/24) ∫ ∫ ∫ ∫ f(u_{1},u_{2},u_{3},u_{4}) d^{4}u

------B=G^{-1}(C)

=(1/24)∫ ∫ ∫ ∫ f(u)d^{4}u

---------B

=(2pi)^{4}/24

============================================

I don't understand the last step...

We are only given that

∫ ∫ ∫ ∫f(x)=d^{4}x=(2pi)^{4}

B

NOT that

∫ ∫ ∫ ∫f(u)=d^{4}u=(2pi)^{4}

B

I will tell you what I am thinking about. Here, u=(u_{1},u_{2},u_{3},u_{4}) andx=(x_{1},x_{2},x_{3},x_{4}) CANNOT be treated as dummy variables since they have a relationshipx=G(u) used to define the transformation G, but the last step of the solution seems to treat thatx=u, which makes me feel very uncomfortable...

Can someone explain? I would really appreciate it!

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# Homework Help: Change of variables for multiple integrals (2)

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