- [Voiceover] So I'm explaining the formula for

three-dimensional curl and where we left off,

we have this determinant of a three-by-three matrix,

which looks absurd because

none of the individual components

are actual numbers, but nevertheless,

I'm about show how when you kind of

go through the motions of taking a determinant,

you get a vector-valued function

that corresponds to the curl.

So let me show you what I mean by that.

If you're computing the determinant,

of the guy that we have pictured

there in the upper-right,

you start by taking this upper-left component,

and then multiplying it by the determinant

of the sub-matrix, the sub-matrix whose

rows are not the row of I and

whose columns are not the row of I.

So what that looks like over here,

is we're gonna take that unit vector I,

and then multiple it by a certain little determinant,

and what this sub-determinant involves

is multiplying this partial-partial Y by R,

which means taking the partial derivative

with respect to Y of the multi-variable function R

and then subtracting off the partial derivative

with respect to Z of Q.

So we're subtracting off partial derivative

with respect to Z of the multi-variable function Q,

and then that,

so that's the first thing that we do,

and then as a second part,

we take this J and we're gonna subtract off,

so you're kind of thinking plus, minus, plus,

for the elements in this top row,

so we're gonna subtract off J,

multiplied by another sub-determinant,

and then this one is gonna involve,

you know, this column that it's not part of

and this column that it's not part of,

and you imagine those guys as a two-by-two matrix,

and its determinant involves taking the

partial derivative with respect to X of R,

so that's kind of the diagonal,

partial-partial X of R,

and then subtracting off the partial derivative

with respect to Z of P,

so partial-partial Z of P,

and then that's just two out of three

of the things we need to do for our overall determinant,

because the last part we're gonna add,

we're gonna add that top-right component, K,

multiplied by the sub-matrix whose

columns involve the column it's not part of

and whose rows involve the rows that it's not part of,

so K multiplied by the determinant of this guy

is going to be, let's see,

partial-partial X of Q,

so that's partial-partial X of Q

minus partial-partial Y of P,

so partial derivative with respect to Y

of the multi-variable function P,

and that entire expression is the

three-dimensional curl of the function

whose components are P, Q, and R.

So here we have our vector function,

vector-valued function V whose

components are P, Q, and R,

and when you go through this whole process

of imagining the cross-product between

the Del operator, this Nabla symbol,

and the vector output P, Q, and R,

what you get is this whole expression,

and, you know, here we're writing with I, J, K notation,

if you're writing it as a column vector,

I guess I didn't erase some of these guys,

but if you're writing this as a column vector,

it would look like saying the curl

of your vector valued function V

as a function of X, Y, and Z

is equal to,

and then what I'd put in for this first component

would be what's up there,

so that would be your partial with respect to Y of R

minus partial of Q with respect to Z,

so partial of Q with respect to Z,

and I won't copy it down for all of the other ones

but in principle, you know,

you'd kind of, whatever this J component is,

and I guess we're subtracting it

so you'd subtract there,

you'd copy that as the next component and then over here.

But often times when you're computing curl,

you kinda switch to using this IJK notation.

My personal preference, I typically default

to column vectors and other people

will write in terms of I, J, and K,

it doesn't really matter as long as

you know how to go back and forth between the two.

One really quick thing that I wanna highlight

before doing an example of this

is that the K-component here,

the Z-component of the output,

is exactly the two-dimensional curl formula.

If you kind of look back to the videos on 2-D curl

and what its formula is,

that is what we have here.

And in fact all the other components

kind of look like mirrors of that

but you're using slightly different operators

and slightly different functions

but if you think about rotation

that happens purely in the XY plane,

just two-dimensional rotation,

and how in three dimensions that's

described with a vector in the K direction

and again, if that doesn't quite seem clear,

maybe look back at the video on

describing rotation with a three-dimensional vector

and the right-hand rule,

but vector is pointing in the pure Z direction,

describe rotation in the XY plane,

and what's happening with these other guys

is kind of similar, right?

Rotation that happens purely in the XZ plane

is gonna correspond with a rotation vector

in the Y direction, the direction

perpendicular to the X, let's see, so

the XZ plane over here.

And then similarly this first component

kind of tells you all the rotation happening

in the YZ plane and the vectors

in the I direction, the X direction of the output,

kind of corresponds to rotation in that plane.

Now when you compute it,

you're not always thinking about

oh, you know, this corresponds to

rotation in that plane and this

corresponds to rotation in that plane,

you're just kind of computing it to

get a formula out,

but I think it's kind of nice to

recognize that all the intuition

that we put into the two-dimensional curl

does show up here

and another thing I wanna emphasize

is this is not a formula to be memorized.

I would not, if I were you, try to

sit down and memorize this long expression.

The only thing that you need to remember,

the only thing, is that curl is

represented as this Del cross V,

this Nabla symbol cross product with

a vector-valued function V,

because from there, whatever your components are,

you can kind of go through the process

that I just did and the more you do it,

the quicker it becomes,

it's kind of long but it doesn't take that long,

and it's certainly much more fault-tolerant

than trying to remember something that

has as many moving parts as the formula

that you see here,

and in the next video I'll go through

an actual example of that.

I'll have functions for P, Q, and R

and walk through that process in

a more concrete context.

I'll see you then!