Main content

## Multivariable calculus

- 2d curl intuition
- Visual curl
- 2d curl formula
- 2d curl example
- Finding curl in 2D
- 2d curl nuance
- Describing rotation in 3d with a vector
- 3d curl intuition, part 1
- 3d curl intuition, part 2
- 3d curl formula, part 1
- 3d curl formula, part 2
- 3d curl computation example
- Finding curl in 3D
- Symbols practice: The gradient

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# 3d curl computation example

A worked example of a three-dimensional curl computation. Created by Grant Sanderson.

## Want to join the conversation?

- What about 4-dimensions?(9 votes)
- Thats very intuitive. So when some vecotor quantity signify curl, its direction implies axis of rotation, being counterclockwise if the value turns out positive and cloclwise if the value turns out negative. So does the magnitude imply its itensity.

For instance, suppose we have curl = (ai+bi+ck), direction simply implies the axis about which it is rotating, and what does the magnitude √a²+b²+c² imply.

since the vecotr looks like any old position vector, is there some symbolic representaion implicitly defining that the vector is the curl, and not any other forms of vector like position vector, or a gradient vector, or a field vector.(3 votes) - so how would one interpret the resulting curl and what does the 0 in the j component mean?(2 votes)
- By my current understanding (so someone correct me if i'm wrong), 0 in the j component would mean your curl vector at any point (x, y, z) would have no y component. This in turn means that if you imagine a vector field of v(x, y, z) and a bunch of particles in that field, they won't rotate along the y-axis (rotate in the xz plane), since their axis of rotation has no y component.(1 vote)

## Video transcript

- [Voiceover] So let's
go ahead and work through an actual curl computation. Let's say our vector valued function V, which is a function of X, Y, and Z, this is gonna be three-dimensional, is defined by the functions, I don't know, let's say the first
component is X times Y. The second one is cosine of Z, and then the last component
is Z squared plus Y. Let's say. So if you take this guy, how do you compute the curl of that vector valued function? So what you do, as I
mentioned in the last video, is you imagine taking this del operator and taking the cross product between that and your vector valued function. And what that means when you expand it is that del operator,
you just kinda fill it with partial differential
operators, you could say, but really it's just the
symbol partial partial X, partial partial Y, partial partial Z. And these are things that are just waiting to take in some kind of function. So we're gonna take the
cross product between that and the function that
we have defined here. So I'm gonna just actually copy it over, copy it over here. And a little residue. And to compute this cross product, we take a certain determinant. So I'm gonna write over here determinant, and it's gonna be of a
three-by-three matrix, but really it's kinda like
a quote-unquote matrix, because each component
has something funky. So the top row, just like we would have with any other cross
product that we're computing is gonna have i, j, and
k, these unit vectors in three-dimensional space. And the second row here is gonna have all of these partial
differential operators, since that's the first
vector in our cross product. So that's partial partial X, partial partial Y, and again, all of these are just kind of waiting to be given a function that they can take the derivative of. And then that third row is gonna be the functions that we have. So the first component here is XY, the second component is cosine of Z, cosine of Z, and then that final component
is Z squared plus Y. Z squared plus Y. Now let's all get some room here, maybe make it more visible. So this is the determinant
we need to compute. And this is gonna be broken
up into three different parts. The first one, we take this top part, i, and multiply it by the
determinant of this sub-matrix. So when we do that, this sub-determinant, we're taking partial
derivative with respect to Y of Z squared plus Y. Now as far as Y is concerned, Z looks like a constant. So Z squared is a constant, and the partial derivative of
this entire guy is just one. So that'll look like one. And then we're subtracting
off the partial derivative with respect to Z of cosine of Z, and that just looks the
same as a derivative of cosine Z, which is negative sine, so that's negative sine of Z. So that's the first part. And then as the next
part, we're gonna take j, but we're subtracting, 'cause
you're always kinda thinking plus, minus, plus, when you're
doing these determinants. So we're gonna subtract off j, multiplied by its own
little sub-determinant, and this time the
sub-determinant is gonna involve the two columns that it's not part of. So you're imagining this first column and this second column as
being part of a matrix. So the first thing you do is you take this partial derivative with respect to X of Z squared plus Y. Well no Xs show up there, right? That's Z squared and Y. Each look like constants
as far as X is concerned. So that's zero. Then we take the partial with respect to Z of X times Y. And again, there's no
Z that shows up there, so that's also zero, so we're
kinda subtracting off zero. And then finally, we're
adding this last component. We're gonna add that last component, k, multiplied by the determinant
of this sub-matrix of the columns that it's not part of. So this involves partial derivative with respect to X of cosine Z. Well no Xs show up there,
so that's just zero. So that's just a zero. And then we're subtracting off the partial with respect to Y of X times Y. Well X looks like a constant,
Y looks like the variable, so that partial derivative is just X. So we're subtracting off X. Which means if we simplify this, so the curl of our vector field, curl of our vector field as a whole, as this function of X, Y, and Z, is equal to, and that first component, the i component, we've got one minus negative sine of Z, so minus minus sine of Z. That's one plus sine of Z. And then the j component,
we're subtracting off, but it's zero. Usually if you're subtracting off, you'd have to make sure
to remember to flip those, but both of those are zero, so
the entire j component here, or the Y component of the output, is zero. And then finally, the k
component is zero minus X, so that entire thing is just negative X. And that's the curl of the function. And in general, that's how you do it. You would take a look at the way that your function is defined
in each component there, and imaging taking the cross product between this del symbol,
this partial partial X, partial partial Y, partial partial Z, and you take the cross product between that and your function, and it involves taking six
different partial derivatives. And mainly it's a matter of bookkeeping to make sure you do it right. And you'll end up with
something like this.