0:00 All right, let's look at this.
00:01 So i know a couple of things.
00:03 I know that x equals e to the u cosine of theta.
00:09 And i know that y equals, let me move the board a little bit, e to the u, right, yeah, sine of theta.
00:18 And i need to prove, let's put this in some sort of color, that partial, second partial of phi with respect to u plus second partial of phi with respect to u, to theta equals x squared plus y squared, second partial of f, second partial of x, second partial of f, second partial of y.
00:54 And i know that f of x, y equals phi of, what was it, u theta, okay.
01:07 So what we're gonna ultimately do is sort of set up a differential operator.
01:15 So that's gonna look like this, right.
01:18 So just a general operator that we can use to find the partial with respect to u.
01:22 And this is gonna help us do our change variable, kind of going from x and y land cartesian to u and theta, which is typically gonna be seen as polar or spherical or something like that.
01:36 So when we do that, we're just gonna kind of write this very generally.
01:42 So we're ultimately trying to get rid of our partials with respect to x and y using the chain rule.
01:47 So i'm saying this partial with respect to u must be partial of x.
01:53 I'm sorry, that's not.
01:56 Yes, no, that's right.
01:59 No, that's not, sorry, my bad.
02:02 Okay, so right, we want the partial of x with respect to u.
02:08 And then we can sort of think of this as a chain rule -y kind of fraction or a partial of something with respect to x.
02:16 And again, the fraction thinking is i know these cancel out and i would just get partial partial u equals partial partial u, which is exactly what i want.
02:27 So i'm gonna keep going because we still need to do the same with y.
02:31 So partial y with respect to u is partial y, right? so they eliminate and you sort of get, like i said, it's kind of hand wavy to think of this as a fraction and kind of just cancel terms out.
02:47 But that's a good way to just sort of look at this and intuitively understand what's going on.
02:52 So let's work on that now.
02:53 So let's do our partials of y with respect to u and x with respect to u.
02:58 Well, i look at my x equation up here.
03:00 That's a pretty easy partial.
03:03 I just take partial with respect to u.
03:05 So that's just gonna give you e to the u cosine of theta, partial partial x, that's easy, plus partial of y with respect to u, which is gonna be very similar.
03:19 E to the u sine of theta now, partial, whoa, partial partial y.
03:31 So this actually becomes very simple because the partial of x with respect to u is just x.
03:36 That's just x.
03:37 This is just y.
03:38 So it worked out pretty simply.
03:40 So partial partial u is gonna be x, partial partial x, plus y, partial partial y.
03:50 Pretty simple.
03:51 So now we have to do the same with theta because we're gonna need that.
03:56 So partial partial theta.
03:59 So we're using the exact same logic here.
04:01 So we're gonna say partial of y with respect to theta, times partial partial y, plus, oh, i'm sorry.
04:12 Why did i do y first? i meant x.
04:15 Oh, i'm sorry.
04:16 I did something silly here.
04:22 This is partial with respect to theta.
04:24 I think i said that, but i wrote it wrong.
04:26 So this is partial partial, partial x partial theta, times partial partial x.
04:30 Sorry, a lot of words flying around.
04:33 So then we have a similar thing.
04:35 Partial y partial theta, times partial partial y.
04:42 Okay, so same thing.
04:44 Now let's take the partial of x with respect to theta.
04:48 So that will give me negative e to the u sine theta, right? so we're gonna get negative e to the u sine theta, partial partial x.
05:02 Okay, now let's look at what partial of y with respect to theta.
05:04 That'll give me e to the u cosine theta.
05:09 E to the u cosine theta, partial partial y.
05:14 And something funny happens here.
05:16 This now is not x.
05:18 It's actually, if we look up at y, it's negative y.
05:24 And this is x...
