00:01
Okay, so let's compute the jacobian.
00:04
So, u is equal to xy.
00:08
This implies partial u over partial x is equal to y.
00:15
Partial u over partial y is equal to x.
00:23
Partial u over partial z is equal to zero.
00:28
And then let's do the same thing with the other ones.
00:31
So y is equal to x z.
00:35
So partial v or partial x is equal to c, partial z is equal to x.
00:49
And we have one more to do, which is w, which is y, z.
00:59
And it's the same thing.
01:04
Partial y is equal to c, partial w or partial z is equal to y.
01:11
So now let's compute the jacobian of this.
01:14
So jacobian of this is putting all these things into a matrix.
01:19
So we get y x 0, z0 x.
01:27
Note that all i'm doing is taking these and then adding them as rows.
01:32
And then i get 0 z y.
01:39
And now i want to calculate the determinant of this.
01:43
So the determinant of this is just, so i get, so let's do co -factor formation.
01:55
So i get y times the determinant of 0x zy minus x times the determinant of zy, minus x times the determinant of zy, y0 x and then the third one is zero so because you're multiplying by zero so this is just y times minus x z and this one is minus x times y z so this in total is going to be minus two times x y z and now we need to write this as a rate this as in terms of uv so in terms of uv well u is equal to x y v v is equal to v is equal to x z but that's not what we want we want w equals y z my z so what does this mean and then v is equal to xv so note that if you do uv w, this is equal to x squared, y, squared, z squared.
03:44
So this implies x, y, z is equal to 1 over square root of uv, uh, not one over, is equal to square root of uv, w.
04:05
So there's that.
04:06
And then the other thing to note is that we did the jaco...