To calculate the curl of a 2D vector field in polar coordinates $(r, \theta)$, we need to first understand the form of the vector field in polar coordinates and the corresponding definition of the curl.

1. Vector Field in Polar Coordinates:

In polar coordinates, a vector field $\vec{F}$ is expressed as: $\vec{F}(r, \theta) = F_r(r, \theta) \hat{e}_r + F_\theta(r, \theta) \hat{e}_\theta$ where:

• $F_r(r, \theta)$ is the radial component.
• $F_\theta(r, \theta)$ is the angular (or tangential) component.
• $\hat{e}_r$ and $\hat{e}_\theta$ are the unit vectors in the radial and angular directions, respectively.

2. Definition of Curl in 2D:

In two dimensions, the curl of a vector field $\vec{F} = F_r \hat{e}_r + F_\theta \hat{e}_\theta$ is a scalar (representing the rotation in the plane). In polar coordinates, the curl is given by: $\text{Curl}(\vec{F}) = \frac{1}{r} \left( \frac{\partial}{\partial r}(rF_\theta) - \frac{\partial F_r}{\partial \theta} \right)$ This measures how much the field "rotates" around a point in the plane.

3. Steps to Calculate Curl in Polar Coordinates:

For a given vector field $\vec{F}(r, \theta) = F_r(r, \theta) \hat{e}_r + F_\theta(r, \theta) \hat{e}_\theta$:

1. Compute the partial derivative of $r F_\theta$ with respect to $r$: $\frac{\partial}{\partial r}(r F_\theta)$
2. Compute the partial derivative of $F_r$ with respect to $\theta$: $\frac{\partial F_r}{\partial \theta}$
3. Plug these into the formula for curl: $\text{Curl}(\vec{F}) = \frac{1}{r} \left( \frac{\partial}{\partial r}(rF_\theta) - \frac{\partial F_r}{\partial \theta} \right)$

4. Geometrical Meaning of Curl in 2D:

In 2D, the curl tells us how much the vector field is rotating around a point in the plane. If the curl is positive, it indicates counterclockwise rotation, and if it's negative, it indicates clockwise rotation.

Would you like me to go over a specific example, or do you have more questions?

Relative Questions:

1. Can you explain the difference between curl and divergence in 2D vector fields?
2. How would this process differ for a 3D vector field in cylindrical coordinates?
3. Can the curl of a vector field in polar coordinates ever be zero? Under what conditions?
4. What is the physical significance of the curl in fluid dynamics?
5. How do polar and Cartesian coordinates relate in vector calculus?

Tip:

When working with vector fields in polar coordinates, always carefully distinguish between radial and angular components, as their partial derivatives behave differently than in Cartesian coordinates.