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Background
Two versions/ types of curl:
 2D curl
 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,
 Aside: for the purposes of this article, let’s denote our curl operation on a point in a 2D vector field as
 The curl of a 2D vector field retuns a scalarvalued function
Now consider the following quintessential 2D curl scenario where is positive:
 For the above diagram,
 As in the input space increases, the component in the output decreaes
 Thus,
 ⇒ a negative seems to correspond with positive 2Dcurl
 Thus,
 As in the output space increawses, the component in the output increases
 Thus
 ⇒ a positive seems to correspond with positive 2Dcurl
 Thus
 As in the input space increases, the component in the output decreaes
Conclusion: the 2D curl of a 2D vector field involves and
 As it turns out,
2D Curl Example 1
Consider the following 2D vector field:
Find the 2D curl for :
Evaluate the curl at :


 Indeed, if we look at a plot of our vector field, we can see that the vectors surrounding form a counterclockwise trajectory:

Evaluate the curl at :


 Indeed, if we look at a plot of our vector field, we can see that the vectors surrounding form a clockwise trajectory:

2D Curl Nuance
 Our quiessential example for positive 2D curl (in the beginning of the post) is a little oversimplified
 Consider the component in our 2D curl formula
 Some cases where :
 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 2Dcurl is concerned, this does indeed hold
 Some cases where :
 Consider the component in our 2D curl formula
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:
 Find the 2D curl:

 Here, we have a constant 2D curl everywhere in the vector field

 Plot of :
 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
 Khan Academy – Multivariable Calculus (2D Curl Intuition)
 Khan Academy – Multivariable Calculus (2D Curl Formula)
 Khan Acadamy – Multivariable Calculus (2D Curl Example)
 Khan Academy – Multivariable Calculus (2D Curl Nuance)