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 formula, part 2

This finishes the demonstration of how to compute three-dimensional curl using a certain determinant. Created by Grant Sanderson.

## Want to join the conversation?

- Another question: Before I watched these videos I tried to do some 3D curl exercises, and I got the formula wrong: instead of crossing nabla with the vector valued function, I crossed nabla with the divergence of the vector valued function. Now I realize that this is not what 3d curl is, that is, after watching this video, but still: does crossing nabla with the divergence of a 3d vector valued function has any mathematical/physical meaning at all? Thank you for your time.(5 votes)
- crossing nabla with the divergence of a 3d vector valued function is basically taking the curl of the divergence, which I don't think has any physical meaning on its own.

An interesting point: the reverse--taking the divergence of the curl--is always equal to zero. ∇⋅ (∇ x F) = 0(5 votes)

- What are the units of measurement for curl? Is there a way to tell based on the units of measurement for the vector field? For example, what if the vector field measures velocity?(4 votes)
- why are we subtracting j?(3 votes)
- We are subtracting j because of how the determinant of a matrix is computed. You could as well write +(dP/dz-dR/dx)j instead of -(dR/dx-dP/dz)j and it would have the same meaning, it's just that in the second case you grouped a -1 that in the first case is instead distributed inside the parentheses, thus changing the order of the two partial derivatives.

To add to this, the method used in the video to compute the determinant is called the method of Laplace. If instead you used Sarrus' method, that only works for 3x3 matrices, you would find your j already tied up and with the -1 already distributed inside the parentheses. By the way you would get the same result with both methods, apart for the differently distributed -1.(3 votes)

- For the j column, why do dR/dx - dP/dz rather than dP/dz - dR/dx?(2 votes)
- you can distribute the minus sign before the j-hat inside the parenthesis and it's gonna turn into dP/dz-dR/dx(1 vote)

- Is there a difference between a dot product and a cross product?

If yes, then what is the difference?(1 vote)- Yes, there's a big difference. The dot product of two vectors is a scalar. The cross product of two vectors is a vector.

Also, for two nonzero vectors, the dot product is zero if and only if the two vectors are perpendicular, but the cross product is the zero vector if and only if the two vectors are parallel or antiparallel.

Have a blessed, wonderful day!(3 votes)

- I think it's worth pointing out that the reason why i j k terms are similar to the 2d formula is that when we try to represent 3d rotation with a vector, we are actually trying to represent it using a linear combination of rotations in the positive direction of x, y and z. And as we accept the that 3d rotations are very similar to 3d vectors, that we can add rotations the same way we add vectors is also an interesting and potentially puzzling fact.(1 vote)
- How do you extend cross product into higher dimensional Cartesian space? Like, if I wanted to calculate the curl of a function with an R4 output space, what exactly would I do to calculate that? And how exactly does the determinant relate to the theory behind the cross product? Thank you!(1 vote)
- Curl, as defined here, is only defined in 3-dimensional Euclidean Space. It can be generalized to lower dimensions like how Grant did it for 2-dimensional space in a previous video on 2D curl, but it can't be done for higher dimensions.

The determinant is simply a way to easily remember how a cross product is computed.(1 vote)

- What does the magnitude of curl vector represents? And does it follows all the properties of cross product? Please Explain with some Physical example.(1 vote)
- The magnitude of the curl is the magnitude of angular velocity (in a physical instance ). Remember the situation of an airflow in a room with a ping pond "magically suspended" at a point in the flow? The ball would rotate about an axis either slowly of fast. Computing the curl for the motion of the airflow at that point where the ball is suspended gives the "speed" at which the ball would be rotating about its axis (Speed of the airflow at that point). The axis (direction) about which it rotates is given by the x, y ,z components of the curl.

Reiterating his explanation on cross products.

Now the cross product(c) between two vectors in 2d lets say a(on the x-axis) and b(angle theta to the positive x-axis) is given be |c| = |a||b|sin( theta).

c is perpendicular to both a and b so on the 'z' axis.

Intuitively |c| = the 2d area between a and b.

Thus area = area of the parallelogram = base * height

base = |a| and height = bsin(theta)....the y component of b.

Therefor area = |a||b|sin(theta) = |c|.

The components of the a and b were used directly and the outcome is the area (physically speaking).

if a and b are differentiable functions and their derivatives used instead then the cross product is the rate of rotation. (physically it may be the angular velocity of the ping pong)

Things get quite 'messy' in 3d but i hope this 2d intuition was useful to build up to 3d... would be glad if someone could offer some help.

Maybe in 3d, the curl on the x-y plane is on the z plane(as explained above where the vectors only had a and y components) Since they have three components here then the curl for the x-z and y-z planes have to also be computed and then to find the resultant rotation, they are computed arithmetically.(since they are all vectors - cross products are vectors)

Happy learning(1 vote)

## Video transcript

- [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!