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Evaluate the Jacobians $J(u, v, w)$ for the following transformations. $u=x-y, v=x-z, w=y+z$ (Hint: Solve for $x, y,$ and $z$ first.)

All right if this promise was to find Jacoby in for the given equations in Gold X equals U plus v MSW in U minus fee plus W and negative U plus V plus w So we're just going Teoh Teik, determine of the partial derivatives of Are listed. So for X respect to you, we have one. He's one w b negative one for why we would have one negative 11 I received a negative 111 And so that means air determinate is equal Teoh. So I vector so one times crisscross so negative one times one is negative one minus one times one is one minus j components of one ties crisscross want is one is one minus Want is negative one It's negative once that he said a plus one plus k vector is actually gonna be a negative one and a crisscross So one negative one to the ones hot one So one minus negative natives Positive So one minus positively. So this becomes equal to making sure my compositions director looks fine is getting nervous. Yeah, so we have a negative too. Then minus one plus one. So one the negative too bad struggling, Um, and then my zero. But when we do talk openly, do the absolute value and so this just becomes equal to four and we're done.

Okay. So I want to show how to create the following products. And in the first one we're starting with a three carbon alcohol with the metro on carbon to I'm going to create the structure to to metal super panel and we basically want to dehydrate this too. Put the alcohol on the more substitute carbon. So your first gonna treat this with a strong acid like sulfuric acid and we don't overheat to aid in dehydration. And this will give us not keen and then reacting this using oxen regulation, oops so with water and then this is over sodium hydroxide and peroxide. And it's gonna add the alcohol that we hold on should be a structure. So it's going to add the alcohol to more substantive carbon which is this one Yes sir products. And in the second one we're starting with this five carbon out keen and we want to create the structure. Okay, so let's first get an alcohol and then we can dehydrate the alcohol. So I'm gonna put the alcohol on the more stable carbon or secondary are the more substitute carbon. So we're gonna use the same reaction as above. And you can also use water and sulfuric acid to that's gonna give us a secondary alcohol and then dehydrating the secondary alcohol with sulfuric acid and heat. Well give us more. Stay walking. So it's gonna grab this part to be is italy to more subsidy aachen. It's not gonna go at this one and the next one we want to we're starting with um the same seems Sergeant cereal. Going to create this key tune this five carbon key tune and it's not carbon through. So we want to put a functional group on the little carbon which is the carbon adjacent to the A keen. So we're gonna react this with NBS and this is over a peroxide. Or this is with them. Light or heat and it's gonna put the burger meat on the little carbon, it's about the structure. And then if we turn the alcohol it into an alcohol, we're also only the are keen as well. So our whatever reduce the akin's and cocaine using hydrogen gas and a metal catalyst and then turn the alcohol out into an alcohol. So then get a key tune rent, treat this with sodium hydroxide and this could be a negative reaction. So occurring on set, it's going to attack there, that's gonna leave and then we want to oxidize the alcohol. I already use other stronger with oxidizing agent and then I give us the key tune. So we have one, two, three, four steps x one is going from a four carbon. Are keen to get a hit tune. So the first one to get alcohol to oxides to a key tune and um, oh wait, hold on. So this should be right here. It's the same thing. The media's here look better. Okay, so I want to get alcohol first. We're going to use actually not, let's use water and sulfuric acid to the alcohol. So now we have the structure the oxidizing alcohol with Sudan document at this time. Give us the products and rocks One is a shot corporate ring with um, it's alcohol section. We want to get a car bus of acid. So if we treat a primary alcohol with a strong oxidizing agent will give us a a Carlson acid. Some new stadium decorated again and also with sulfuric acid. Sir X one is um structure. So we want to dehydrate the alcohol to get Naquin and then reduce the akin to McCain using sulfuric acid heat for exaggeration and then reducing the arcane with hydrogen gas over palladium or some other catalyst. That's it

So on. This problem we had was to determine in the Jacoby in of the given equations X, y and Z Um, no Jacoby in is the determine its That's not a good determine it. Is it the determine it of those partial derivatives? So that's gonna be equal. Teoh, what's you the determinants oven? Let's see what we got here. So we got extra. Expect to use zero, respectively. Is one w is one. Why would he was one thes zero w is one z is one v is one w zero and we do determine like this. Go. So, uh, zero times zero minus one, uh minus. So that was the I factor. Now do Dre vector. So one times zero minus one again because you go criss cross. Uh, Plus now the cave factor, which is one times one minus Syria. What is that equal? Well, looks like we got a zero plus one. That's one. Yeah, and we were taking that. So ticket. That's the value of it. So because so that's we have the apse value science around our q B in. And so it's a value which types of I have a positive two is still just to and, uh,

I just finished Kobe anywhere except V you wise guy. Do you? W and c console you squared minus b squared, uh, Jacoby and retake the determined at the partial derivatives that I have already written up right there. So it's gonna go through him. So x, with respect to you is zero. The would be W W would be v Jay. Why, for you would be w for w would be you. And for you. Uh, hold on my set up for you would be w for her, the would be zero. And for w would be you que would be so for you. For z inspector, you would have to you the negative to be it zero. And then if you do that, Germany s Kinkel, the German. And so what is the determiner of that? So the I component zero times well, zero plus to UV. But the doubts p zero minus the J component, which is w times zero plus two the squared plus the cake opponents or to you, times zero. Hold on. Wait a minute. I messed up again. W you find a serial and so weighing up within a with negative to w V squared, plus to you squared W um and it only it's the absolute value of that. It will be done.