Curl measures the rotation in a fluid flowing along a vector field.  Formally, curl only applies to three dimensions, but here we cover the concept in two dimensions to warmup.

Kontekst

Note: Throughout this article I will use the following convention:
  • i^\hat{\textbf{i}} represents the unit vector in the x-direction.
  • j^\hat{\textbf{j}} represents the unit vector in the y-direction.

Do czego zmierzamy

  • Curl measures the "rotation" in a vector field.
  • In two dimensions, if a vector field is given by a function v(x,y)=v1(x,y)i^+v2(x,y)j^\vec{\textbf{v}}(\blueE{x}, \redE{y}) = \blueE{v_1}(x, y)\hat{\textbf{i}} + \redE{v_2}(x, y)\hat{\textbf{j}}, this rotation is given by the formula
    2d-curlv=v2xv1y \text{2d-curl}\;\vec{\textbf{v}} = \dfrac{\partial \redE{v_2}}{\partial \blueE{x}} - \dfrac{\partial \blueE{v_1}}{\partial \redE{y}}

Rotation in fluid flow

Have yourself a nice swirly vector field:
This particular vector field is defined with the following function:
v(x,y)=[y39yx39x]=(y39y)i^+(x39x)j^\begin{aligned} \quad \vec{\textbf{v}}(x, y) &= \left[ \begin{array}{c} y^3 - 9y \\ x^3 - 9x \end{array} \right] \\\\ &= (y^3 - 9y) \hat{\textbf{i}} + (x^3 - 9x)\hat{\textbf{j}} \end{aligned}
Now I want you to imagine that this vector field describes a fluid flow, perhaps in a chaotic part of a river. The following video shows a simulation of what this might look like. A sample of fluid particles, shown as blue dots, will flow along the vector field. This means that at any given moment, each dot moves along the arrow it is closest to. Focus in particular on what happens in the four circled regions.
Amidst all the chaos, you might notice that the fluid is rotating within the circled regions. In the left and right circles, the rotation is counterclockwise, and in the top and bottom circles, the rotation is clockwise.
  • Key Question: If we are given a function v, with, vector, on top, left parenthesis, x, comma, y, right parenthesis that defines a vector field, along with some specific point in space, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, how much does a fluid flowing along the vector field rotate at the point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?
The vector calculus operation curl answer this question by turning this idea of fluid rotation into a formula. It is an operator which takes in a function defining a vector field and spits out a function that describes the fluid rotation given by that vector field at each point.
Technically, the curl operation only applies to three dimensions. You can see what that means and how it is computed in the next article, but in this article, we warm up by describing fluid rotation in two dimensions with a formula.

Capturing two-dimensional rotation with a formula

One of the simplest examples of a vector field which describes a rotating fluid is
v(x,y)=[yx]=yi^+xj^.\begin{aligned} \quad \vec{\textbf{v}}(x, y) = \left[ \begin{array}{c} -y \\ x \end{array} \right] = -y \hat{\textbf{i}} + x\hat{\textbf{j}}. \end{aligned}
Here's what it looks like.
Rotational vector field
Animated, all the fluid particles just go in circles.
In some sense, this is the most perfect example of counterclockwise rotation, and you can understand the general formula for rotation in a two-dimensional vector field just by understanding why the function v(x,y)=yi^+xj^\vec{\textbf{v}}(x, y) = -y \hat{\textbf{i}} + x\hat{\textbf{j}} gives counterclockwise rotation.

The i^\hat{\textbf{i}}-component

First, let's understand why the yi^-y\hat{\textbf{i}} component suggests counterclockwise rotation. Imagine a small twig sitting in our fluid, oriented parallel to the y-axis. More specifically, let's say one end is at the origin left parenthesis, 0, comma, 0, right parenthesis, and the other is at the point left parenthesis, 0, comma, 2, right parenthesis. What does the yi^-y\hat{\textbf{i}} component of the vector field imply for the fluid velocity at points on this twig?
This means the velocity at the top of the twig is 2i^-2\hat{\textbf{i}}, a leftward vector, while the velocity at the bottom of the twig is 0.
For the twig, this means the important factor for counterclockwise rotation is that vectors point more to the left as we move up the vector field. Said with a few more symbols, the important point here is that the i^\hat{\textbf{i}}-component of a vector attached to a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis decreases as y, start subscript, 0, end subscript increases.
Said with even more symbols,
y(y)=1<0\dfrac{\partial}{\partial y}(-y) = -1 < 0
Let's generalize this idea a bit.
  • Question: Consider a more general vector field.
v(x,y)=v1(x,y)i^+v2(x,y)j^\vec{\textbf{v}}(\blueE{x}, \redE{y}) = \blueE{v_1}(x, y)\hat{\textbf{i}} + \redE{v_2}(x, y)\hat{\textbf{j}}
The components start color blueE, v, start subscript, 1, end subscript, end color blueE and start color redE, v, start subscript, 2, end subscript, end color redE are any scalar-valued functions. If you place a small twig at some point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, oriented parallel to the y-axis, how can you tell if the twig will rotate just by looking at start color blueE, v, start subscript, 1, end subscript, end color blueE, comma, start color redE, v, start subscript, 2, end subscript, end color redE and left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?
  • Answer: Look at the rate of change of start color blueE, v, start subscript, 1, end subscript, end color blueE as start color redE, y, end color redE varies near the point of interest, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis:
    v1y(x0,y0) Suggests counterclockwise rotation if negative  \dfrac{\partial \blueE{v_1}}{\partial \redE{y}}(x_0, y_0) \quad \leftarrow \small{\gray{\text{ Suggests counterclockwise rotation if negative }}}
    If this is negative, it indicates that vectors point more to the left as y, start subscript, 0, end subscript increases, so rotation would be counterclockwise. If it is positive, vectors point more to the right as y, start subscript, 0, end subscript increases, indicating a clockwise rotation.

The j^\hat{\textbf{j}}-component

Next, let's see why the xj^x\hat{\textbf{j}} component of the original vector field suggests counterclockwise rotation as well. This time, imagine a twig which is parallel to the x-axis. Specifically, put one end of the twig at the origin left parenthesis, 0, comma, 0, right parenthesis, and put the other at the point left parenthesis, 2, comma, 0, right parenthesis.
The vector attached to the origin is 0, but the vector attached to the other end at left parenthesis, 2, comma, 0, right parenthesis is 2j^2\hat{\textbf{j}}, an upward vector. Therefore, the fluid pushes the right end of the stick upwards, and the left end experiences no force, so there will be a counterclockwise rotation.
For this second twig, the vertical component of vectors increases as we move right, suggesting counterclockwise rotation. That is to say, the y component of a vector attached to a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis increases as x, start subscript, 0, end subscript increases.
In the case of a more general vector field function,
v(x,y)=v1(x,y)i^+v2(x,y)j^\vec{\textbf{v}}(\blueE{x}, \redE{y}) = \blueE{v_1}(x, y)\hat{\textbf{i}} + \redE{v_2}(x, y)\hat{\textbf{j}}
we can measure this effect near a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis by looking at the change in start color redE, v, start subscript, 2, end subscript, end color redE as start color blueE, x, end color blueE changes.
v2x Suggests counterclockwise rotation if positive  \dfrac{\partial \redE{v_2}}{\partial \blueE{x}} \quad \leftarrow \small{\gray{\text{ Suggests counterclockwise rotation if positive }}}

Combining both components

Putting these two components together, the rotation of a fluid flowing along a vector field v, with, vector, on top near a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis can be measured using the following quantity:
v2x(x0,y0)v1y(x0,y0) \dfrac{\partial \redE{v_2}}{\partial \blueE{x}}(x_0, y_0) - \dfrac{\partial \blueE{v_1}}{\partial \redE{y}}(x_0, y_0)
When you evaluate this, a positive number will indicate a general tendency to rotate counterclockwise around left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, a negative quantity indicates the opposite, clockwise rotation. If it equals 0, there is no rotation in the fluid around left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis. If you are curious about the specifics, this formula gives precisely twice the angular velocity of the fluid near left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis.
Some authors will call this the "two-dimensional curl" of v, with, vector, on top. This isn't standard, but let's write this formula as if "2d-curl" was an operator.
2d-curlv=v2xv1y \text{2d-curl}\;\vec{\textbf{v}} = \dfrac{\partial \redE{v_2}}{\partial \blueE{x}} - \dfrac{\partial \blueE{v_1}}{\partial \redE{y}}

Example: Analyzing rotation in a 2d vector field using curl

Problem: Consider the vector field defined by the function
v(x,y)=[cos(x+y)sin(xy)]\begin{aligned} \quad \vec{\textbf{v}}(x, y) = \left[ \begin{array}{c} \cos(x+y) \\ \sin(xy) \end{array} \right] \end{aligned}
Determine whether a fluid flowing according to this vector field has clockwise or counterclockwise rotation at the point
p=(0,π2)\begin{aligned} \quad p &= \left(0, \dfrac{\pi}{2} \right) \\ \end{aligned}
Step 1: Compute the 2, d, negative, c, u, r, l of this function.
2, d, negative, c, u, r, l, space, v, with, vector, on top, equals

We apply the 2d-curl formula we just found to this function. This will give us a new, scalar-valued function that indicates the rotation near each point. Then, we'll plug in the point left parenthesis, 0, comma, pi, slash, 2, right parenthesis to see whether the rotation there is positive (counterclockwise), zero, or negative (clockwise).
Step 2: Plug in the point left parenthesis, 0, comma, pi, slash, 2, right parenthesis.
2, d, negative, c, u, r, l, space, v, with, vector, on top, left parenthesis, 0, comma, pi, slash, 2, right parenthesis, equals

p=(0,π2)cos(0π2)π2+sin(0+π2)=1π2+1=π+22\begin{aligned} \quad p = \left(0, \dfrac{\pi}{2} \right) &\rightarrow \cos\left(0\cdot\dfrac{\pi}{2}\right)\dfrac{\pi}{2} + \sin\left(0 +\dfrac{\pi}{2}\right) \\ &= 1 \cdot \dfrac{\pi}{2} + 1 \\ &= \dfrac{\pi + 2}{2} \end{aligned}
Step 3: Interpret. How does the fluid tend to rotate near this point?
Wybierz 1 odpowiedź:
Wybierz 1 odpowiedź:

Because this is a positive value, the fluid will tend to flow counterclockwise at the point p.
Let's watch a sample of particles in this fluid flow:
The point towards the top where all the particles congregate corresponds with p, equals, left parenthesis, 0, comma, start fraction, pi, divided by, 2, end fraction, right parenthesis. Particles rotate counterclockwise in this region, which should be consistent with your 2, d, negative, c, u, r, l calculations.

Podsumowanie

  • Curl measures the "rotation" in a vector field.
  • In two dimensions, if a vector field is given by a function v(x,y)=v1(x,y)i^+v2(x,y)j^\vec{\textbf{v}}(\blueE{x}, \redE{y}) = \blueE{v_1}(x, y)\hat{\textbf{i}} + \redE{v_2}(x, y)\hat{\textbf{j}}, this rotation is given by the formula
    2d-curlv=v2xv1y \text{2d-curl}\;\vec{\textbf{v}} = \dfrac{\partial \redE{v_2}}{\partial \blueE{x}} - \dfrac{\partial \blueE{v_1}}{\partial \redE{y}}

On to the third dimension!

The true curl operation, covered in the next article, extends this idea and this formula to three dimensions.