00:01
Let's go to flat circular plate here, taking the shape of the closed disk where x squared plus y squared is less than or equal to 1.
00:11
So let's draw that here.
00:13
This is our plate.
00:16
It's hot so we'll highlight it in red.
00:21
And the temperature at the point t, x, y, the temperature at the point x, y is x squared plus 3y squared minus 2 thirds x.
00:35
I like to find the hottest and coldest points on the plate.
00:39
To that end, let's find the gradient.
00:46
We want to find where the gradient is equal to zero.
00:51
So nabla t is, let's see, tx, ty.
00:59
That's going to be the x derivative here is 2x minus 2 thirds, 2x minus 2 thirds.
01:07
The y derivative is just 6y.
01:13
Let's see where this is equal both to zero.
01:18
So thankfully, we can just solve for these independently.
01:23
2x minus 2 thirds is equal to zero if you multiply by 3.
01:28
6x minus 2 must be zero, divide by 2.
01:32
3x minus 1 is zero.
01:34
3x equals 1, so x equals 1 third.
01:37
We're going to have zero gradient in the x direction all along that area there.
01:48
Let's actually give this a new color, green.
01:53
X equals 1 third is probably a little bit closer to there.
01:59
And then 6y is zero, you see, a bit easier.
02:01
That's wherever y is zero.
02:05
So we're going to have either a min or a max there.
02:10
To figure out which one it is, we look at txx.
02:16
We can figure out what exactly that temperature is first.
02:26
T of, let's see, x is 1 third, y is zero is equal to 1 ninth plus zero minus 2 ninths.
02:38
Right? yeah, that checks out.
02:45
Is equal to negative 1 ninth.
02:49
And we expect that in general to be the coldest temperature, negative 1 ninth, because our paraboloid is opening up.
03:01
So as we get x or y to be larger, it's going to be going up from all of that, which is great.
03:12
And in particular, what that means is that the hottest temperature we actually expect to be on the boundary of the circle as opposed to on the, excuse me, we expect to be on the boundary of the circle.
03:28
And let's see how we can figure that out.
03:31
Well, heuristically, i'm going to notice that the temperature goes up with y a lot more than it goes up with x, right? we want to make y bigger more than we want to make x bigger.
03:42
In fact, we want to make x relatively small because we're subtracting that.
03:49
But i don't think that means we just want to maximize y.
03:53
We also might want to make x a little bit negative because it appears to be, if we make x a little bit negative, we're going to be adding a little bit there.
04:03
So we expect the maximum to be somewhere around here...