3d curl computation example Khan Academy English (auto-generated) English

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so let's go ahead and work through an

- [Voiceover] So let's go ahead and work through

actual curl computation let's say our

an actual curl computation.

vector value function V which is a

Let's say our vector valued function V,

function of XY and Z this is going to be

which is a function of X, Y, and Z,

three-dimensional is defined by the

this is gonna be three-dimensional,

functions and I don't know let's say the

is defined by the functions, I don't know,

first component is x times y the second

let's say the first component is X times Y.

one is cosine of Z and then the last

The second one is cosine of Z,

component is Z squared plus y let's say

and then the last component is Z squared plus Y.

so if you take this guy how do you

Let's say. So if you take this guy,

compute the curl curl of that vector

how do you compute the curl

valued function so what do you do as I

of that vector valued function?

mentioned in the last video as you

So what you do, as I mentioned in the last video,

imagine taking this del operator and

is you imagine taking this del operator

taking the cross-product between that

and taking the cross product between that

and your vector valued function and what

and your vector valued function.

that means when you expand it is that

And what that means when you expand it

del operator you just kind of fill it

is that del operator, you just kinda fill it

with partial differential operators you

with partial differential operators, you could say,

could say but really is just the symbol

but really it's just the symbol partial partial X,

partial partial X partial partial Y partial palster partial partial Z and

partial partial Y, partial partial Z.

these are things that are just waiting

And these are things that are just waiting

to take in some kind of function so

to take in some kind of function.

we're going to take the cross-product

So we're gonna take the cross product between that

between that and the the function that

and the function that we have defined here.

we have defined here so let me just actually copy it over copy it over here

So I'm gonna just actually copy it over,

and little residue and to compute this

copy it over here. And a little residue.

cross-product we take a certain

And to compute this cross product,

determinant so I'm going to write over

we take a certain determinant.

here determinant and it's going to be of

So I'm gonna write over here determinant,

a 3x3 matrix but really it's kind of

and it's gonna be of a three-by-three matrix,

like a quote-unquote matrix because each

but really it's kinda like a quote-unquote matrix,

component has something funky so the top

because each component has something funky.

row just like we would have with any

So the top row, just like we would have

other cross product that we're computing

with any other cross product that we're computing

is going to have I J and K these unit

is gonna have i, j, and k, these unit vectors

vectors in three-dimensional space and

in three-dimensional space.

the second row here is going to have all

And the second row here is gonna have

of these partial differential operators

all of these partial differential operators,

since that's the first vector in our

since that's the first vector in our cross product.

cross-product so that's partial partial X partial

So that's partial partial X,

partial Y and again all of these are

partial partial Y,

just kind of waiting to be given a

and again, all of these are just kind of

function that they can they can take the

waiting to be given a function that they can take the derivative of.

derivative of and then that third row is

And then that third row is gonna be

going to be the functions that we have

the functions that we have.

so the first component here is X Y the

So the first component here is XY,

second component is cosine of Z cosine

the second component is cosine of Z,

of Z and then that final component is Z squared plus y Z squared plus y now so

cosine of Z, and then that final component is Z squared plus Y. Z squared

I'll give some room here maybe make it

plus Y. Now let's all get some room here,

maybe make it more visible.

so this is the determinant we need to

So this is the determinant we need to compute.

compute and this is going to be broken up into three different parts the first

And this is gonna be broken up into three different parts.

one we take this top part I and multiply

The first one, we take this top part, i,

it by the determinant of this sub matrix

and multiply it by the determinant of this sub-matrix.

so when we do that this sub determinant

So when we do that, this sub-determinant,

we're taking partial derivative with

we're taking partial derivative with respect to Y

respect to Y of Z squared plus y now as

of Z squared plus Y.

far as Y is concerned Z looks like a

Now as far as Y is concerned,

constant so Z squared is a constant and

Z looks like a constant.

the partial derivative of this entire

So Z squared is a constant,

guy is just 1 so that'll look like 1

and the partial derivative of this entire guy is just one. So that'll look like one.

and then we're subtracting off the

And then we're subtracting off the partial derivative

partial derivative with respect to Z of cosine of Z and that just looks the same

with respect to Z of cosine of Z,

as you know a derivative of cosine Z

and that just looks the same as a derivative

which is negative sine so that's

of cosine Z, which is negative sine,

negative sine of Z so that's the that's

so that's negative sine of Z.

the first part and then as the next part

So that's the first part.

we were gonna take J but we're

And then as the next part, we're gonna take j,

subtracting because you're always kind

but we're subtracting, 'cause you're always kinda thinking

of thinking plus minus plus when you're

plus, minus, plus, when you're doing these determinants.

doing these determinants so we're going to subtract off J multiply it by its own

So we're gonna subtract off j,

little sub determinant and this time the

multiplied by its own little sub-determinant,

sub determinant is going to involve the

and this time the sub-determinant is gonna involve

2 columns that it's not part of so

the two columns that it's not part of.

you're imagining this first column and

So you're imagining this first column

the second column is being part of a

and this second column as being part of a matrix.

matrix so the first thing you do is you

So the first thing you do is you take

take this partial derivative with

this partial derivative with respect to X

respect to X of Z squared plus y well no

of Z squared plus Y.

X is show up there right that's Z

Well no Xs show up there, right?

squared and Y each look like constants

That's Z squared and Y.

as far as X is concerned so that's 0

Each look like constants as far as X is concerned.

then we take the partial with respect to

So that's zero.

Z of x times y and again there's there's

Then we take the partial with respect to Z of X times Y.

no Z that shows up there so that's also

And again, there's no Z that shows up there,

0 so it's kind of subtracting off 0 and

so that's also zero, so we're kinda subtracting off zero.

then finally we're adding this last

And then finally, we're adding this last component.

components we're going to add that last component K multiplied by the

We're gonna add that last component, k,

determinant of this sub matrix of the

multiplied by the determinant of this sub-matrix

columns that it's not part of so this

of the columns that it's not part of.

involves partial derivative with respect

So this involves partial derivative

to X of cosine Z well no X's show up

with respect to X of cosine Z.

there so that's just 0 so that's just a

Well no Xs show up there, so that's just zero.

0 and then we're subtracting off the

So that's just a zero.

partial with respect to Y of x times y

And then we're subtracting off the partial with respect to Y of X times Y.

well X looks like a constant Y looks

Well X looks like a constant, Y looks like the variable,

like the variable so that partial derivative is just X so we're

so that partial derivative is just X.

subtracting off X which means if we

So we're subtracting off X.

simplify this so the curl of our vector

Which means if we simplify this,

field curl of our vector field as a

so the curl of our vector field, curl of our vector field as a whole,

whole as this function of XY and Z is equal to

as this function of X, Y, and Z,

and that first component the the I

component we've got 1 minus negative

and that first component, the i component, we've got one minus negative sine of Z,

sine of Z so minus minus sine of Z

so minus minus sine of Z.

that's 1 plus sine of Z and then the J

That's one plus sine of Z.

component we're subtracting off but but

And then the j component, we're subtracting off,

at 0 usually if you're subtracting off

you'd have to make sure to remember to

Usually if you're subtracting off,

flip those but both of those are 0 so

you'd have to make sure to remember to flip those,

the entire J component here or that the

but both of those are zero, so the entire j component here,

Y component of the output is 0 and then

or the Y component of the output, is zero.

finally we're adding the the K component is 0 minus X so that entire thing is

And then finally, the k component is zero minus X,

just negative X and that's the curl of

so that entire thing is just negative X.

the function and in general that's how

And that's the curl of the function.

you do it you would you would take a

And in general, that's how you do it. You would take a look at the way

look at the way that your function is defined and each component there and

that your function is defined in each component there,

imagine taking the cross product between

and imaging taking the cross product

this Dell symbol this partial partial X

between this del symbol, this partial partial X,

partial partial Y partial partial Z and

partial partial Y, partial partial Z,

you take the cross product between that

and you take the cross product

and your function and it involves taking six different partial derivatives and

between that and your function, and it involves taking six different partial derivatives.

you just mainly it's a matter of

And mainly it's a matter of bookkeeping

bookkeeping to make sure you do it right

to make sure you do it right.

and you'll end up with something like this

And you'll end up with something like this.

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