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Problem 6 Easy Difficulty

Find the Jacobian of the transformation.

$ x = u + vw $, $ y = v + wu $, $ z = w + uv $

Answer

Jacobian $=1+2 u v w-u^{2}-v^{2}-w^{2}$

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Video Transcript

So for this problem, let's borrow from a previous problem because this will help us save some time. So right here is the formula for the Jacoby in. We're gonna be taking the determinant of all these partial derivatives. So first, let's look at the partial derivatives of X. We'll do it with respect to you first, that will be just one then, with respect to V, it's w and with respect to W it's V um And then what will end up getting is why you w why V is one. Alright, w is you and then z you is V Z V is you nz w is one. So now we can set this up. We'll have one W v w one v or w one you and then the u one. We take the determinant of this We get one times one minus you squared minus w times w minus UV plus the times you w minus to be. When we combine all these, we end up getting one plus two u v w minus You squared minus V squared minus w squared. And that will be our final answer for the Chicopee in

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