00:01
In multiple dimensions, there are what's the equivalent of a derivative.
00:07
There are at least three different derivative operators that you could use.
00:12
And they are known as div, curl, and gradient.
00:16
It will exclude the directional derivative from that.
00:20
And we're going to demonstrate their use with a scalar field and two given factor fields, v &w, and a scalar field omega.
00:32
We will look at several different operators, and it's a good thing to keep in mind whether you are expecting a vector, and also be aware of any parentheses that are shown in the operator with the idea that what happens in the parentheses happens first, and you work outwards from there.
00:54
So let's take a look at some examples of using these operators.
00:59
Our first example, example a, is a curl of a vector cross product.
01:10
This is going to be very tedious, but what we need to understand is those parentheses mean, first of all, find the cross product between v and w.
01:23
So that's our first step, v -cross -w comes about.
01:28
I usually like to set up a determinant of kind of like a tensor thing, ijk, and then the second row is the first vector in the cross product, and then the last row is the second vector.
01:54
And what you're doing is finding the determinant, which means you work off of sub -determinants.
02:00
The first sub -determinant comes from marking off the i column in row, then you flip.
02:07
The sign for the second column minus j, and then plus k.
02:14
And then what goes next to these is a sub -determinant.
02:18
I'll just show one of these for the i, and you'll get the idea as you're working off of a sub -determinate where you've blocked off the i row and column.
02:30
So that is c plus x.
02:32
Whoops, let's not do that in red.
02:39
C plus x.
02:41
Okay, this is going to get a little bit.
02:47
Xy minus xy plus y times x plus y times z x and we do a similar thing for the j column except you mark off the j column and find a subdeterminate there some people instead of putting the negative sign there will do the determinate backwards the subdeterminate but that usually gets me messed up so i don't do it that way and then finally, the same thing with the case of determinant on that row and column.
03:42
By the way, there's a neater way to handle these types of cross products and curls, etc.
03:48
And it's called einstein notation.
03:51
But usually people get very confused when they first start working in it, and even after many years.
03:57
And it's a way of replacing these actual functions with indices and working out which components last.
04:14
Okay, and i'll just spare the details.
04:19
And if i've done this wrong, i apologize.
04:21
So the first component is x squared parentheses, y minus c, and then they kind of permute.
04:34
Yeah, permutations.
04:35
If you like to do permutations, you'll like cross products a whole lot.
04:42
I've never liked permutations a whole lot, which is why i suspect myself of messing up royally when i do these types of things.
05:04
And let me make sure did i do my sign correct? i don't think i have my sign correct in that one.
05:24
Yeah.
05:26
That should be z minus x, i believe.
05:30
I said i hate to do permutations.
05:44
And this should look fairly good.
05:55
Ok, so i think i have it.
06:00
All right.
06:03
Ok.
06:05
And then the next step is to do a curl on that vector field.
06:17
So we're going to take the output of the last step.
06:23
And we're going to do another matrix.
06:28
In a similar fashion.
06:34
But the first row in the vector, not the ijk, but yeah, the second row of the matrix will be d by dx, d by d, y, y, and d by dz.
06:48
And these are partial derivatives, which means you hold the other variables constant.
06:56
Yeah, that's a nice thing to do.
07:03
I've always enjoyed partial derivatives.
07:08
I like to hold things constant.
07:12
It makes life a whole lot easier.
07:16
Okay, so we'll do the same thing, work on sub -matrices, i minus j and k -hat.
07:29
Okay, so first sub -determinant, d -by -d -y, the first term is, yeah, we're doing d -by -d -y, of z -squared x minus y, and that is simply minus z -squared, and then minus y squared from the switching the sign and going the other way.
07:58
And now we're going to block out the j row and column.
08:03
Put a negative sign in front of it.
08:07
And we have z squared plus x squared, but there's a negative sign in front.
08:13
So yeah, the permutations seem to be coming out the way you expect.
08:17
So notice we have a negative sign in front of each component, and then a sum of squares of the quantities that are not the component.
08:30
So let's see, minus y squared minus x squared.
08:38
So that's a fairly nice looking thing.
08:51
Free components, they're all negative, and they all look like a sum of two squares of the non -component variables.
09:02
It's kind of pretty, lots of symmetry there.
09:08
Nice permuting going on.
09:11
Okay, so those are tedious, but not too bad.
09:17
So our next operator is we're first of all going to take the dot product between v and w, and that is going to produce a scalar field.
09:30
So the way you do that is simply take the x components, multiply them together, take the y components, multiply them together, and same with the z, and then add up all those terms.
09:47
So we wind up with what's called a scalar field.
09:54
Yeah, and this will look a little bit messy, of course.
10:05
I never know why they pick such ugly -looking things to work out.
10:14
But i suppose it's an exercise in patience and trying not to lose little pieces here and there, which i very well may do.
10:29
Okay, so yeah, what we're after is the gradient of this...