Mathematics

2D Curl

[latexpage]

Background

Two versions/ types of curl:

  1. 2D curl
  2. 3D curl

Curl can best be understoon using the “fluid flow” interpretation of a vector field

  • Assume the input space is occupied by particles (e.g., water molecules)
  • Since vector fields associatae a point in the input space with a vector, we can think of each particle in the input space as moving over time in a way that corresponds with the outputs in our vector field
    • Specifically, the vector associated with a given input point indicates the direction and velocity at which a particle passing through that point will travel
  • Note: curl doesn’t only apply in the context of a vector field representing fluid flow
    • Fluid flow is just a really intuitive interpretation when visualizing curl

2D Curl Formula

Consider the 2D vector field,

\overrightarrow{v}(x,y) = \begin{bmatrix} P(x,y) \\ Q(x,y)  \end{bmatrix}

  • Aside: for the purposes of this article, let’s denote our curl operation on a point in a 2D vector field as \mathrm{2D-curl} \overrightarrow{v}(x,y)
  • The curl of a 2D vector field retuns a scalar-valued function 2D-curl system diagram.jpg

Now consider the following quintessential 2D curl scenario where \mathrm{2D-curl} \overrightarrow{v}(x_0,y_0) is positive:

positive 2D-curl, quinessential example

  • For the above diagram,
    • As y in the input space increases, the x-component in the output decreaes
      • Thus, \frac{\partial P}{\partial y} < 0
        • ⇒ a negative \frac{\partial P}{\partial y} seems to correspond with positive 2D-curl
    • As x in the output space increawses, the y-component in the output increases
      • Thus \frac{\partial Q}{\partial x} > 0
        • ⇒ a positive \frac{\partial Q}{\partial x} seems to correspond with positive 2D-curl

Conclusion: the 2D curl of a 2D vector field involves - \frac{\partial P}{\partial y} and \frac{\partial Q}{\partial x}

  • As it turns out,

\mathrm{2D-curl} \overrightarrow{v}(x,y) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}

2D Curl Example 1

Consider the following 2D vector field:

\overrightarrow{v}(x,y) = \begin{bmatrix} P(x,y) \\ Q(x,y)  \end{bmatrix} = \begin{bmatrix} y^3 - 9y \\ x^3 - 9x \end{bmatrix}

Find the 2D curl for \overrightarrow{v}(x,y) :

  • \frac{\partial Q}{\partial x} = 3x^2 - 9
  • \frac{\partial p}{\partial y} = 3y^2 - 9

\mathrm{2D-curl} \overrightarrow{v}(x,y) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3x^2 -9 - (3y^2 -9) = 3x^2 - 3y^2

Evaluate the curl at (3,0):

  • \mathrm{2D-curl} \overrightarrow{v}(3,0) = 3(3)^2 - 3(0)^2 = 3(9) - 0 = 27
    • \mathrm{2D-curl} \overrightarrow{v}(3,0) > 0
      • Indeed, if we look at a plot of our vector field, we can see that the vectors surrounding (3,0) form a counterclockwise trajectory: 2Dcurlexample_a

Evaluate the curl at (0,3):

  • \mathrm{2D-curl} \overrightarrow{v}(0,3) = 3(0)^2 - 3(7)^2 =0 - 3(9) = -27
    • \mathrm{2D-curl} \overrightarrow{v}(0,3) < 0
      • Indeed, if we look at a plot of our vector field, we can see that the vectors surrounding (3,0) form a clockwise trajectory: 2Dcurlexample_b

2D Curl Nuance

  • Our quiessential example for positive 2D curl (in the beginning of the post) is a little oversimplified
    • Consider the \frac{\partial Q}{\partial x} component in our 2D curl formula
      • Some cases where \frac{\partial Q}{\partial x} > 0 :
        1. 2D-curl positive y component a
        2. 2D-curl positive y component b
      • According to our formula for 2D curl, both of  these cases should contribute approximately the same amount to curl
        • Even though this seems a little counterintuitive, as far as 2D-curl is concerned, this does indeed hold

Note: curl wasn’t developed by mathematicians and physicists for trying to understand fluid flow

  • They found this formula to be significant for things like electromagnetism
  • Our fluid flow interpretation just gives a really good intuition as to what 2D curl represents

2D Curl Example 2:

Consider the following 2D vector field:

\overrightarrow{v}(x,y) = \begin{bmatrix} P(x,y) \\ Q(x,y)  \end{bmatrix} = \begin{bmatrix} -y \\ x \end{bmatrix}

  • Find the 2D curl:
    • \frac{\partial Q}{\partial x} = 1
    • \frac{\partial P}{\partial y} = -1
    • \mathrm{2D-curl} \overrightarrow{v}(x,y) = 1 - (-1) = 2
      • Here, we have a constant 2D curl everywhere in the vector field
  • Plot of \overrightarrow{v}(x,y)2Dcurlexample_c
  • Since we have a constant curl everywhere, our interpretation for curl being the rotation of fluid around a particulaszr point should bethe same forthe origin as for everywhere else in the vector field
    • This isn’t depicted in the actual vector field though
    • ***Conclusion: curl represents more than just rotation around a point

 

References

 

 

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