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## Multivariable calculus

### Course: Multivariable calculus>Unit 5

Lesson 1: Formal definitions of div and curl (optional reading)

# Formal definition of curl in three dimensions

After learning how two-dimensional curl is defined, you are ready to learn about the formal definition of three-dimensional curl.  This is advanced, so be prepared to take things slowly.

## Background

Understanding this article really does require these two prerequisites. The definition of curl in three dimensions has so many moving parts that having a solid mental grasp of the two-dimensional analogy, as well as the three-dimensional concept we are trying to capture, is crucial.
In particular, if you did not just come from reading the article giving the formal definition of curl in two dimensions, I would highly recommend taking a quick look at it right now, even if you've seen it before, and even if it's just the summary.

## What we're building to

• We define three-dimensional curl one component at a time, looking at the components of fluid rotation which are parallel to the y, z-plane, the x, z-plane, and the x, y-plane.
• You can capture all three of these coordinate-by-coordinate definitions of start text, c, u, r, l, end text, start bold text, F, end bold text by defining what the dot product between start text, c, u, r, l, end text, start bold text, F, end bold text and any arbitrary unit vector start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f should be.
\begin{aligned} \big( \text{curl}\; \blueE{\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{n}}} = \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{n}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{n}}} \right)}\right|} \oint_\redD{C} \blueE{\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}

## Limit our view to one plane

Curl in three-dimensions is a rather complicated thing to think about. For example, let start bold text, F, end bold text, left parenthesis, x, comma, y, comma, z, right parenthesis be a three-dimensional vector field:
\begin{aligned} \textbf{F}(x, y, z) = \left[ \begin{array}{c} F_1(x, y, z) \\ F_2(x, y, z) \\ F_3(x, y, z) \end{array} \right] \end{aligned}
An example of what this could look like is shown in the following video.
Khan Academy video wrapper
Now imagine the three-dimensional fluid flow that start bold text, F, end bold text could represent. As you know, start text, c, u, r, l, end text, start bold text, F, end bold text, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis is a way to measure rotation in that fluid flow near the point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis, but this is a tricky concept to quantify rigorously.
There are some good analogies out there to gain an intuition for curl. One of my favorites is to think of a tiny tennis ball centered at the point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis, and how the surrounding fluid flow would cause it to rotate. In this analogy, start text, c, u, r, l, end text, start bold text, F, end bold text, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis gives the vector of the tennis ball's resulting rotation.
However, these descriptions can only go so far when the goal is to formally define what curl is; to capture this intuition with mathematical rigour.
Our basic strategy moving forward will be to limit our view to the rotation in a specific plane. For example, the following video shows a plane representing a constant x value, x, equals, 1, point, 6 to be specific, as well as the vectors from start bold text, F, end bold text which stem from this plane.
Khan Academy video wrapper
In formulas, you might describe this as all the vectors of the form
start bold text, F, end bold text, left parenthesis, 1, point, 6, comma, y, comma, z, right parenthesis
Here, y and z range freely. When we project those vectors onto the plane and lay it out flat as a picture, we'd get something like this:
Note, the axes are labeled "y" and "z" because this plane was originally parallel to the y, z-plane in three-dimensional space. We could describe this two-dimensional vector field with a new two-dimensional function start bold text, F, end bold text, start subscript, 1, point, 6, end subscript, left parenthesis, y, comma, z, right parenthesis defined as follows:
\begin{aligned} \textbf{F}_{1.6}(y, z) = \left[ \begin{array}{c} F_2(1.6, y, z) \\ F_3(1.6, y, z) \end{array} \right] \end{aligned}
More generally, if we slice the vector field with any plane of the form x, equals, x, start subscript, 0, end subscript for some constant x, start subscript, 0, end subscript, then project the vectors stemming from that plane onto the plane itself, we will get a two-dimensional vector field described by a function that looks like this:
\begin{aligned} \textbf{F}_{x_0}(y, z) = \left[ \begin{array}{c} F_2(x_0, y, z) \\ F_3(x_0, y, z) \end{array} \right] \end{aligned}
Concept check: Why does the definition of start bold text, F, end bold text, start subscript, x, start subscript, 0, end subscript, end subscript, left parenthesis, y, comma, z, right parenthesis not include F, start subscript, 1, end subscript, the x-component of start bold text, F, end bold text, left parenthesis, x, comma, y, comma, z, right parenthesis?
Choose 1 answer:

Concept check: For a given point left parenthesis, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis in the plane above, what does start text, 2, d, negative, c, u, r, l, end text, start bold text, F, end bold text, start subscript, x, start subscript, 0, end subscript, end subscript, left parenthesis, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis represent?
Choose all answers that apply:

## A component-wise definition

So why am I talking about projecting vectors and trajectories in three dimensions onto a two-dimensional plane? Basically, it's hard to think about three dimensions, so it's worth doing everything you can to frame things two-dimensions at a time.
The importance of this last concept check is that we can describe the x-component of the three-dimensional curl of start bold text, F, end bold text purely in terms of the two-dimensional curl of the function start bold text, F, end bold text, start subscript, x, end subscript:
\begin{aligned} \text{x-component of } \underbrace{ \text{curl}\,\textbf{F}(x, y, z) }_{\text{3d vector}} = \underbrace{ \text{2d-curl}\,\textbf{F}_x(y, z) }_{\text{Scalar value}} \end{aligned}
We can also more elegantly pull out the x-component of start text, c, u, r, l, end text, start bold text, F, end bold text by dotting it with the unit vector in the x-direction,
$\hat{\textbf{i}} = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]$
This means our expression looks like this:
\begin{aligned} \big(\text{curl}\,\textbf{F}(x, y, z)\big) \cdot \hat{\textbf{i}} = \text{2d-curl}\,\textbf{F}_{x}(y, z) \end{aligned}
In terms of the formula you already know, this explains why the x-component of start text, c, u, r, l, end text, start bold text, F, end bold text has the form that it does,
\begin{aligned} \; \text{curl}\,\textbf{F}(x, y, z) = \overbrace{\left( \dfrac{\partial F_3}{\partial y} - \dfrac{\partial F_2}{\partial z} \right)}^{ \text{2d-curl}\,\textbf{F}_{x}(y, z) }\hat{\textbf{i}} + \small \left( \dfrac{\partial F_1}{\partial z} - \dfrac{\partial F_3}{\partial x} \right)\hat{\textbf{j}} + \left( \dfrac{\partial F_2}{\partial x} - \dfrac{\partial F_1}{\partial y} \right)\hat{\textbf{k}} \end{aligned}
But remember, the whole point of this article is that curl is one of those funny operations where the formula we use to compute it is not its definition. Our goal is to find a definition of curl by directly representing fluid rotation. With that in mind, the significance of representing the x-component of start text, c, u, r, l, end text, start bold text, F, end bold text using a two-dimensional curl is that we can take the line-integral-limit definition of start text, 2, d, negative, c, u, r, l, end text found in the last article, and use it to define the x-component of start text, c, u, r, l, end text, start bold text, F, end bold text.
\begin{aligned} \big( \text{curl}\; {\textbf{F}} {(x, y, z)} \big) \cdot {\hat{\textbf{i}}} \overbrace{=}^{\text{definition}} \lim_{\redE{A} \to 0} \left( \dfrac{1}{|\redE{A}|} \oint_\redD{C} {\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
• start color #bc2612, A, end color #bc2612 is some two-dimensional region in the plane perpendicular to start bold text, i, end bold text, with, hat, on top, and passing through the point left parenthesis, x, comma, y, comma, z, right parenthesis.
• start color #e84d39, C, end color #e84d39 is the boundary of start color #bc2612, A, end color #bc2612.
• The orientation of start color #e84d39, C, end color #e84d39 is determined based on the right-hand rule: Stick the thumb of your right hand in the direction of start bold text, i, end bold text, with, hat, on top, and curl your fingers. The direction your fingers point as they wrap around start color #e84d39, C, end color #e84d39 is the direction of integration.
• vertical bar, start color #bc2612, A, end color #bc2612, vertical bar represents the area of start color #bc2612, A, end color #bc2612.
• limit, start subscript, vertical bar, start color #bc2612, A, end color #bc2612, vertical bar, \to, 0, end subscript indicates that we are considering the limit as start color #bc2612, A, end color #bc2612 shrinks to the point left parenthesis, x, comma, y, comma, z, right parenthesis on the plane where x is constant.
Although it will clutter things up, for clarity's sake it will help if our formula expresses the fact that the region start color #bc2612, A, end color #bc2612 must always include the point start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, and that it is perpendicular to start color #0d923f, start bold text, i, end bold text, with, hat, on top, end color #0d923f. To do this, I'll write start color #bc2612, A, end color #bc2612 with subscripts, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, comma, start color #0d923f, start bold text, i, end bold text, with, hat, on top, end color #0d923f, end subscript
This means our full definition looks like this:
\begin{aligned} \big( \text{curl}\; {\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{i}}} \overbrace{=}^{\text{definition}} \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{i}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{i}}} \right)}\right|} \oint_\redD{C} {\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
This is a very heavy definition, which assumes a lot of prior knowledge from the reader. And that's just for one component! The key to understanding this is to:
• Make sure you have a full grasp of the definition of curl in two-dimensions.
• Understand how this definition is applying that same concept to a plane sitting in three-dimensional space.
• Make sure you understand why two-dimensional curl of start bold text, F, end bold text, start subscript, x, start subscript, 0, end subscript, end subscript should represent the x-component of the curl of start bold text, F, end bold text.

## Full definition

There is of course nothing special about the x-direction, we can also define the other two coordinates of start text, c, u, r, l, end text, start bold text, F, end bold text similarly:
\begin{aligned} \big( \text{curl}\; {\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{j}}} = \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{j}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{j}}} \right)}\right|} \oint_\redD{C} {\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
\begin{aligned} \big( \text{curl}\; {\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{k}}} = \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{k}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{k}}} \right)}\right|} \oint_\redD{C} {\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
Concept check: What does start color #bc2612, A, end color #bc2612, start subscript, left parenthesis, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, comma, start color #0d923f, start bold text, j, end bold text, with, hat, on top, end color #0d923f, right parenthesis, end subscript represent?
Choose 1 answer:

This gives a full definition, since each component of start text, c, u, r, l, end text, start bold text, F, end bold text is accounted for.

## Arbitrary unit normal vectors

However, it is a little inelegant to define curl with three separate formulas. Also, when curl is used in practice, it is common to find yourself taking the dot product between the vector start text, c, u, r, l, end text, start bold text, F, end bold text and some other vector, so it is handy to have a definition suited to interpreting the dot product between start text, c, u, r, l, end text, start bold text, F, end bold text and any vector, not just start bold text, i, end bold text, with, hat, on top, start bold text, j, end bold text, with, hat, on top and start bold text, k, end bold text, with, hat, on top.
Think about an arbitrary plane cutting through the vector field start bold text, F, end bold text, left parenthesis, x, comma, y, comma, z, right parenthesis:
Khan Academy video wrapper
Suppose that this plane is defined to be perpendicular to some unit vector start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, such as
\begin{aligned} \greenE{\hat{\textbf{n}}} = \left[ \begin{array}{c} 1/\sqrt{3} \\ 1/\sqrt{3} \\ 1/\sqrt{3} \\ \end{array} \right] \end{aligned}
Now imagine mimicking everything we did before with the plane x, equals, x, start subscript, 0, end subscript.
• Considering the vectors which stem from points on this plane.
• Project them onto the plane
• Measure the resulting two-dimensional curl on that plane.
This will let us define the component of three-dimensional curl in the start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f-direction:
\begin{aligned} \big( \text{curl}\; \blueE{\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{n}}} = \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{n}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{n}}} \right)}\right|} \oint_\redD{C} \blueE{\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
This is the definition of curl that you might come across in other texts. It's a weird definition, since instead of defining the vector start text, c, u, r, l, end text, start bold text, F, end bold text itself, it only defines what the dot product between this vector and anything else would be.
But here's why it kind of makes sense to do things this way, even if it feels convoluted: Rotation is an inherently two-dimensional idea, and when we try to talk about rotation in three dimensions (e.g. rotation of the earth) we are somewhat awkwardly forced to use vectors. A given rotation vector is saying "the rotation is really happening in some two-dimensional plane, and I'm just telling you what plane that is."
When it comes to fluid rotation, what we really want is a way of taking any possible rotation vector (which is the same as saying any possible plane in which rotation occurs), and asking "how much does the fluid rotation near a given point look like this vector?" The curl gives us a way to answer this question. For a given vector, representing some rotation, when you dot that vector against the curl of a fluid flow, it tells you how much the fluid rotation resembles the rotation represented by that vector.

## Summary

• To define curl in three dimensions, we take it two dimensions at a time. Project the fluid flow onto a single plane and measure the two-dimensional curl in that plane.
• Using the formal definition of curl in two dimensions, this gives us a way to define each component of three-dimensional curl. For example, the x-component is defined like this:
\begin{aligned} \big( \text{curl}\; {\textbf{F}} \goldE{(x, y, z)} \big) \cdot \greenE{\hat{\textbf{i}}} \overbrace{=}^{\text{definition}} \lim_{|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{i}}} \right)}| \to 0} \left( \dfrac{1}{\left|\redE{A}_{\left( \goldE{(x, y, z)}, \greenE{\hat{\textbf{i}}} \right)}\right|} \oint_\redD{C} {\textbf{F}} \cdot d\textbf{r} \right) \end{aligned}
• You can replace start color #0d923f, start bold text, i, end bold text, with, hat, on top, end color #0d923f with any unit vector, thus defining what the component of start text, c, u, r, l, end text, start bold text, F, end bold text should be in any direction.

## Congrats!

Understanding this complicated definition fully is a sign that you have fully grasped both curl and line integrals, each of which are formidable concepts in their own right.
Also, this definition will prepare you very well for understanding Stokes' theorem, a topic which stands at the very pinnacle of multivariable calculus.

## Want to join the conversation?

• It says here that "if you follow the three-dimensional motion of particles starting in the plane x=x0 according to the velocity vectors given by F, then project those trajectories onto the plane x=x0, you get the two-dimensional fluid motion described by F_x0". But the velocity field might change when the particles move in the x direction, so the projection there shouldn't be the same as the trajectory described by F_x0. Am I wrong?
Of course, this intuition is correct when you think about infinitesimal distances (very small neighborhoods), but not if you just let the particles flow freely an indefinite distance.
(2 votes)
• I think I understand what you are picking up on here, and you're right that the velocity field will, in general, be dependent on x and so change as the particles move through different planes.

If you were to look 'down' the x-axis towards the yz plane, you would 'see' (if you could somehow ignore the particles passing through all other planes) that the trajectories of the particles passing (instantaneously) through the plane x=x0 is exactly as those described by F_x0. But this is, of course, only valid for that particular plane! Change your choice of x0 and you'll get a different projection.
(5 votes)
• On the projection of vector onto a plane, keeping x constant 1.6, Doesn't the points in that plane x=1.6 have vectors pointing in all the three direction. Doesn't points in the plane F(1.6,y,z) have components in all direction as (F1,F2,F3). If however can be projected on that plane, how is that their compenets take the neat value [F291.6,y,z),F3(1.6,y,z)].....
I think I quite kinda get the answer, but feel free to correct me in line of thought and intuition.
(3 votes)
• One thing I don't understand about the final definition: If the dot product is the definition of curl, for arbitrary vector n, what ensures that all those infinitely many definitions for a particular point in a particular vector field result in a consistent curl vector?

My intuitive guess would be something along the lines of, dot product with an arbitrary vector is just a linear combination of the definitions for i, j and k, and projecting the vector field onto the perpendicular plane is an equivalent linear combination, but I'm really struggling to come up with anything rigorous.
(2 votes)