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Formal definition of curl in two dimensions

Learn how curl is really defined, which involves mathematically capturing the intuition of fluid rotation.  This is good preparation for Green's theorem.

Background

If you haven't already, you may also want to read "Why care about the formal definitions of divergence and curl" for motivation.

What we're building to

In two dimensions, curl is formally defined as the following limit of a line integral:
start text, 2, d, negative, c, u, r, l, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, equals, limit, start subscript, vertical bar, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, vertical bar, \to, 0, end subscript, left parenthesis, start fraction, 1, divided by, vertical bar, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, vertical bar, end fraction, \oint, start subscript, start color #e84d39, C, end color #e84d39, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, right parenthesis
This is complicated, but it will make sense as we build up to it one piece at a time.

Formalizing fluid rotation

Suppose you have a flowing fluid whose velocity is given by a vector field start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, left parenthesis, x, comma, y, right parenthesis, such as the one we looked at in the two-dimensional curl article.
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If you didn't already know about curl, but you did just learn about line integrals through a vector field, how would you measure fluid rotation in a region?
To take a relatively simple example, consider the vector field
F(x,y)=[yx]\begin{aligned} \blueE{\textbf{F}}(x, y) = \left[ \begin{array}{c} -y \\ x \end{array} \right] \end{aligned}
This is the quintessential counterclockwise rotation vector field.
How can we make the idea of fluid rotation mathematical (before knowing about curl)? One way to do this is to imagine walking around the perimeter of some region, like a unit circle centered at the origin, and measuring if the fluid seems to flow with you or against you at each point.
Concept check: Let C represent the circumference of a unit circle centered at the origin, oriented counterclockwise. Given the picture of the vector field start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 above, consider the following line integral:
CFdr\begin{aligned} \oint_C \blueE{\textbf{F}} \cdot d\textbf{r} \end{aligned}
Without calculating it, what is the sign of this integral? (Recall that the symbol \oint just emphasizes the fact that the line integral is being done over a closed loop, but it's computed the same way as any other line integral).
Choose 1 answer:
Choose 1 answer:

More generally, if a fluid tends to flow counterclockwise around a region, you would expect that the line integral of that fluid's velocity vector field around the perimeter of the region would be positive (when it's oriented counterclockwise).
You could also imagine a more complicated vector field, in which the fluid flows with you at some points on your counterclockwise walk around the circle, but against you at others.
The value start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text will be positive while the flow is with you, and negative when it's against you. In a way, the integral \oint, start subscript, C, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text is like a voting system that counts up how much these different directions cancel each other out and which one wins overall.

Letting the size of the region change

So, after mathematically expressing the idea of fluid rotation around a region, you might want to capture the more elusive idea of fluid rotation at a point. How might you go about that?
You could start by considering smaller and smaller regions around that point, such as circles of smaller and smaller radii, and seeing what the fluid flow around those regions looks like.
Concept check: Back to our vector field F=[yx]\blueE{\textbf{F}} = \left[\begin{array}{c} -y \\ x \end{array} \right], rather than just looking at the unit circle, let C, start subscript, R, end subscript represent​ a circle centered at the origin with radius R. This circle will still be oriented counterclockwise.
Compute the line integral of start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 around this circle as a function of R.
CRFdr=\begin{aligned} \oint_{C_R} \blueE{\textbf{F}} \cdot d\textbf{r} = \end{aligned}

How does this value relate to the circle C, start subscript, R, end subscript?
Choose 1 answer:
Choose 1 answer:

Average rotation per unit area

The answer to this last question suggests something interesting. The rotation around a region seems to be proportional to the area of that region. Of course, you've only shown this for circles centered at the origin, not all possible regions, but it is nevertheless suggestive. This might give you an idea.
Key idea: Maybe if you take \oint, start subscript, C, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, which measures the fluid flow around a region, and divide it by the area of that region, it can give you some notion of the average rotation per unit area.
The idea of "average rotation per unit area" might feel a bit strange, but if you think back to the interpretation of curl, that's kind of what we want curl to represent. Rather than thinking about fluid rotation in a large region, curl is supposed to measure how fluid tends to rotate near a point.
Concept check: The vector field from the previous example is a little bit special in that the "rotation-per-unit-area" of circles around the origin is the same value for all circles. What is that value?

Concept check: Recall that the formula for start text, 2, d, negative, c, u, r, l, end text is
2d-curl  F=F2xF1y\begin{aligned} \text{2d-curl}\; \blueE{\textbf{F}} = \dfrac{\partial F_2}{\partial x} - \dfrac{\partial F_1}{\partial y} \end{aligned}
where F, start subscript, 1, end subscript and F, start subscript, 2, end subscript are the components of start color #0c7f99, start bold text, F, end bold text, end color #0c7f99. Given the definition
F(x,y)=[yx]\begin{aligned} \blueE{\textbf{F}}(x, y) = \left[ \begin{array}{c} -y \\ x \end{array} \right] \end{aligned}
compute the curl of start color #0c7f99, start bold text, F, end bold text, end color #0c7f99.
2d-curl  F=\begin{aligned} \text{2d-curl}\; \blueE{\textbf{F}} = \end{aligned}

Defining two-dimensional curl

Those last two questions show that the "average rotation per unit area" in circles centered at the origin happens to be the same as the curl of the function, at least for our specific example. This turns out to apply more broadly. In fact, the way we define the curl of a vector field start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 at a point left parenthesis, x, comma, y, right parenthesis is to be the limit of this average rotation per unit area in smaller and smaller regions around the point left parenthesis, x, comma, y, right parenthesis.
Specifically, (drumroll please), Here's the formula defining two-dimensional curl:
2d-curlF(x,y)=limA(x,y)0(1A(x,y)CFdr)Average rotation per unit area\begin{aligned} \text{2d-curl}\,\blueE{\textbf{F}}\goldE{(x, y)} = \lim_{|\redE{A}_{\goldE{(x, y)}}| \to 0} \underbrace{\left( \dfrac{1}{|\redE{A}_{\goldE{(x, y)}}|} \oint_\redD{C} \blueE{\textbf{F}} \cdot d\textbf{r} \right)}_{ \text{Average rotation per unit area} } \end{aligned}
where
  • start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 is a two-dimensional vector field.
  • start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05 is some specific point in the plane.
  • start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript represents some region around the point start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05. For instance, a circle centered at start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05.
  • vertical bar, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, vertical bar indicates the area of start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript.
  • limit, start subscript, vertical bar, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, vertical bar, \to, 0, end subscript indicates we are considering the limit as the area of start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript goes to 0, meaning this region is shrinking around start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05.
  • start color #e84d39, C, end color #e84d39 is the boundary of start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, oriented counterclockwise.
  • \oint, start subscript, start color #e84d39, C, end color #e84d39, end subscript is the line integral around start color #e84d39, C, end color #e84d39, written as \oint instead of integral to emphasize that start color #e84d39, C, end color #e84d39 is a closed curve.
This formula is impractical for computation, but the connection between this and fluid rotation is very clear once you wrap your mind around it. It is a very beautiful fact that this definition gives the same thing as the formula used to compute two-dimensional curl.
2d-curl  F=F2xF1y\begin{aligned} \text{2d-curl}\; \blueE{\textbf{F}} = \dfrac{\partial F_2}{\partial x} - \dfrac{\partial F_1}{\partial y} \end{aligned}

One more feature of conservative vector fields

If start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, left parenthesis, x, comma, y, right parenthesis is a conservative vector field, all line integrals over closed loops are 0. Looking at the integral above, what does this imply?
Choose 1 answer:
Choose 1 answer:

This gives an important fact: If a vector field is conservative, it is irrotational, meaning the curl is zero everywhere.
In particular, since gradient fields are always conservative, the curl of the gradient is always zero. That is a fact you could find just by chugging through the formulas. However, I think it gives much more insight to understand it using the definition of curl together with the intuition for why gradient fields are conservative.
What about the converse? If a vector field has zero curl everywhere, does that mean it must be conservative?

Summary

  • If a vector field represents fluid flow, you can quantify "fluid rotation in a region" by taking the line integral of that vector field along the border of that region.
  • To go from the idea of fluid rotation in a region to fluid flow around a point (which is what curl measures), we introduce the idea of "average rotation per unit area" in a region. Then consider what this value approaches as your region shrinks around a point.
  • In formulas, this gives us the definition of two-dimensional curl as follows:
    start text, 2, d, negative, c, u, r, l, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, equals, limit, start subscript, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, \to, 0, end subscript, start underbrace, left parenthesis, start fraction, 1, divided by, vertical bar, start color #bc2612, A, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, right parenthesis, end color #a75a05, end subscript, vertical bar, end fraction, \oint, start subscript, start color #e84d39, C, end color #e84d39, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, right parenthesis, end underbrace, start subscript, start text, A, v, e, r, a, g, e, space, r, o, t, a, t, i, o, n, space, p, e, r, space, u, n, i, t, space, a, r, e, a, end text, end subscript
  • This relationship between curl and closed-loop line integrals implies that irrotational fields and conservative fields are one and the same.

Want to join the conversation?

  • blobby green style avatar for user Yingtong
    Could you please explain how this formal definition equals the del operator cross F?
    (12 votes)
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  • blobby green style avatar for user parsanoori79
    Why the integral of F.dr is proportional to the area ?
    (4 votes)
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    • blobby green style avatar for user guilhem.escudero
      I think if I'm right it is because the integral of F.dr depend on dr which is equal to to a tiny step along the curve which is equal to the derivative of the parametrize function times dt. Since you add all tiny steps of the curve, you wander all the curve. The bigger the curve is, the bigger the area in the curve is, so the integral of F.dr is proportional to the area
      (3 votes)
  • blobby green style avatar for user Sergii Sergii
    I guess the diameter of the region should tend to zero in the defining limit, not area, as area can tend to zero without shrinking of the region to a point, e.g. by flattening it into a line.
    (2 votes)
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  • blobby green style avatar for user Andrea Menozzi
    where i can find the connection between the formal definition to the "practical" definition? do wo I go from one to the other?
    (2 votes)
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    • blobby green style avatar for user Michael Gorbunov
      Imagine a point that is at the center of a counter clockwise curl. On the right of that center point, the vector field points up, while on the left the vector field field points down. Above, the vector field points left, and below it points right.

      Let's call this vector field F = <f(x,y), g(x,y)>

      Speaking in derivatives, as we go left to right (dx), the vertical component of the vector field (f) should increase. On the left, vectors point down (negative) while on the right, vectors point up (positive). This is equivalent to a positive dg/dx.

      As you move vertically, bottom to top, the horizontal component should decrease. Below vectors point right (positive) and above the point left (negative).

      This is why curl (F) = dg/dx - df/dy makes sense.

      Another interesting bit is this seems connected to the idea that 90 degree rotations in 2D are (x, y) -> (-y, x), however I'm not 100% sure if there's some simple and nice connection or it's just happen chance.
      (1 vote)
  • blobby green style avatar for user Mehmet Anil ASLIHAK
    This definition of curl only gives a scalar, should not it be a vector? I mean what comes out from the integral in the definition is just a number, but mustn't curl be represented by a vector?
    (1 vote)
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    • leaf green style avatar for user 𝜏 Is Better Than 𝝅
      Technically, curl should be a vector quantity, but the vectorial aspect of curl only starts to matter in 3 dimensions, so when you're just looking at 2d-curl, the scalar quantity that you're mentioning is really the magnitude of the curl vector. The curl vector will always be perpendicular to the instantaneous plane of rotation, but in 2 dimensions it's implicit that the plane of rotation is the x-y plane so you don't really bother with the vectorial nature of curl until you get to 3 dimensional space. Then it starts to matter.
      (2 votes)
  • purple pi purple style avatar for user Aneesh Vasudev
    OMG!! So, 2D divergence and curl are the same?? And the line integral around a closed curve of a vector function, = to 2D Flux?? which is used to interpret divergence and curl?
    (0 votes)
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    • leaf green style avatar for user Alexander Wu
      Divergence and curl are not the same. (The following assumes we are talking about 2D.) Curl is a line integral and divergence is a flux integral. For curl, we want to see how much of the vector field flows along the path, tangent to it, while for divergence we want to see how much flow is through the path, perpendicular to it.

      Line integrals and flux are different for the same reason. But yes, they are used to interpret divergence and curl, as a limit.
      (9 votes)