ðŸ’¬ ðŸ‘‹ Weâ€™re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

No Related Subtopics

Oregon State University

University of Nottingham

Idaho State University

Boston College

01:59

Nutan C.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

00:06

Jeffery W.

01:02

Anshu R.

0:00

Lowie T.

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

Create your own quiz

Okay, Cool. So here we have a vector field and we want to show that it is conservative. And then once we show it's conservative, we know that a potential function exists, and then we actually want to compute the potential function. So this will be an important example because, you know, to show that's conservative, we're just gonna take the curl. And so that is zero. So that's relatively straightforward. The real meat of this example is gonna be how do you find the potential function? And more or less, you just take anti derivatives. But we'll see that in a second. Let's start by actually showing that this is conservative by showing that the curl of death is equal to zero everywhere. So what is Thekla Earl? It's going to be Well, I j okay. And then we have partial partial X partial partial. Why partial Partial Z And then our component functions are so let's just call on p que and are there they are. Okay, so what do I do? I take the partial with respect to why of P, which is easy. And then I take the partial with respect to X Q, which is also Z Z minus c. And then I take the partial with respect to Z of p, which is why and then minus the partial with respect to X of our which is also why so far, so good. And then I take the partial with respect to X of Q which is Z? Yep. Okay. And then I take the partial with respect to why of P which is Z and this is exactly zero. No matter what point I pick because it's C minus e. Why, that's why is he meant to see? So that means our vector field is conservative. Cool. So how do we find the potential election defeat? Well, fear supposed to satisfy Grady int of fee, which is Feess X. Be partial with respect. Why be partial with respect to Z that's supposed to equal f okay and f is two x plus y z two y plus xy to Z plus x. Why? But we get three equations here, right? We get that partial partial X of fee is equal to two x plus y Z and partial with hers. Partial partial. Why of fee is to why, plus XY in the impartial partial Z, the feet equals two Z plus x y. So we're gonna have to use all three of these equations to actually find feet or at least find fee up to just a constant. So remember, it's it's basically an anti derivative, so it can differ by an absolute constant. So what does this tell us? Well, this tells us that okay, fee has to be so the part. So basically, fee is an anti derivative of this function. With respect to X, Sophie is going to be what? I'm just gonna take an anti derivative of this function. Well, that's X squared, plus y si X. And now, when I taken into a derivative, I have to add an integration constant. But here, since I'm taking a partial derivative, my integration constant can actually be a function of y and Z because any function of y and Z when I take the partial with respect to X will be zero. So this is plus some arbitrary function that Onley depends on lines e Okay, but this isn't good enough because I need fee to be a function plus a constant that doesn't depend on X y or Z But let's see what happens when I take the partial A fee with respect to why So when I take the partial of feed with respect to why on one hand I know it has to be two y plus Z, but over here I have another representation, right? I could just take the partial with respect to why and I get well, z X plus the partial of why, with respect to sorry, the partial derivative h with respect to why Okay, So I just took the partial derivative of this with respect to why e got this. But these were supposed to be equal and so notice that I can actually cancel a Z X and I get that Well, actually the partial of age with respect to why is to why and again that comes from I'm just setting these two things equal. I canceled the ZX. I get that the partial derivative of H with respect to why is too why so in other words, h of y z is equal to well, why squared an anti derivative with respect to why and now h dependent on y and Z So I have a constant but that constant can depend on sea. Because if I take the partial with respect to why any function that just depends on Z will vanish. So this is plus an arbitrary function of the okay. But now let's take the partial of fee with respect to Z. And now, really fee is X squared plus y z X plus y squared plus Josie. So if I take the partial with respect to Z, well, I'm going to get mhm x times. Why? So the partial here, this is gonna be zero. This the derivative of this is just gonna be g prime of Z. The derivative of this is gonna be y x or X y plus g prime of Z and I notice I could put g prime of Z because G only depends on the functions E okay. And that's supposed to be the partial A fee with respect to see which is to Z plus x y. Okay, so the ex wives can cancel and I actually get the G Prime of Z is equal to two Z. But that means that GMC is an anti derivative of two Z, so it's Z squared. Plus, here we go An absolute constant This constant can't depend on X, y or Z Otherwise it won't vanish when I take the derivative. So it really is just an arbitrary constant See? And so if we fill all this out, following are chain here fee the potential function. Yeah. Has this form X square plus well, why z x plus h of y z, which was y squared plus GMC the GMC was Z squared. And then finally we have our arbitrary integration constant and you should check that. Indeed, when I take the partial derivative of this with respect to any of these variables, I will get exactly this this and this. So that's the idea. It's a little bit involved, but more or less. The key idea is that when you take an anti derivative with respect to a variable X, the constant might depend on the other two variables. So that's why you have to use the other information to really nail down what that arbitrary function is until you get all the way to just plus an arbitrary, constant C

26:52

05:28

06:38

07:48

05:33

28:40

06:44

10:25

08:33

09:29

11:59

07:49

12:44

03:43

04:53