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- [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!