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University of North Texas

# Find the Jacobian of the transformation.$x = u^2 + uv$, $y = uv^2$

## Jacobian $=4 u^{2} v+u v^{2}$

#### Topics

Multiple Integrals

### Discussion

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### Video Transcript

to find the Jacoby in of this. We just need to come over here and find these partials that we have listed in this formula. So let's go ahead and place them where we need to over here first. So we're first going to take the partial effects with respect to you. Um, so that's where they're going to be. So the derivative of U squared with respect to you is going to be to you. The derivative of U V will be treaties a constant. So just take the directive, you. So that would just be P. Now we take the derivative of this with respective V. So you square is just going to be zero because it's constant, and then you be well, just give us you. Now we do the same thing for why so take the derivative of this with respect to you. So that would give us b squared. And then we take the direction of this with respect to be. So now we treat V or you as a constant. So just take the derivative of the square. So that would be to you the now, in order for us to find the determine of this, we're going to do the downs downs minus the cups. At least this is one way you can do it for two by two matrix. And doing this would give us, um, so to you, plus a V times to U V and then minus you times eastward. Uh, now, let's just go ahead and distribute, and then we can combine any like terms before use where to be plus two U V squared minus U V squared so we can combine those you've eastwards. So before you square TV plus you, you the square And so this here should be our Jacoby in.