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

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

• 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:

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:

## 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 $\frac{\partial Q}{\partial x} > 0$ :
• 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 $\overrightarrow{v}(x,y)$
• 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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