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Let $B_1\subset \mathbb R^4$ and $B_2\subset \mathbb R^3$ be closed balls of radius $1$ centered at the origin. I would like to prove:

$$\operatorname{vol}B_1=2\int_{B_2}\sqrt{1-(x^2+y^2+z^2)}dxdydz$$

In order to prove this I'm trying to use change of variables. So I'm looking for a diffeomorphism $f:\mathbb R^3 \to \mathbb R^4$ such that $f(B_2)=B_1$. I couldn't found out any such function.

Is my reasoning correct?

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    There is no diffeomorphism between $\Bbb R^3$ and $\Bbb R^4$ because the derivative of such a map would be an isomorphism between the two. – 2017-01-28
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    @OpenBall you're right!!! so do you have any ideas how to proceed? – 2017-01-28
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    Not quite, as I'm not into these stuff. I just wanted to point that out so you can stop searching for such a function. – 2017-01-28
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    Thank you any way. That's weird because the chapter of the book I'm studying is about change of variables. – 2017-01-28

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