# Understanding Curl in 2D/3D Space

• Harry Mason
In summary, the conversation discusses the concept of curl in 2D and 3D space and how it represents local rotation in a vector field. The examples given include a 2D vector field with no rotation and a 3D vector field with global rotation on the z-axis. The conversation also brings up the issue of a vector field that does not cause rotation on an object's center of mass, but still has a non-zero curl. The conversation concludes with a suggestion to consult a resource for a better understanding of the concept.
Harry Mason
Hello everybody,

i have some troubles with the interpretation of curl in 2D/3D space.

I was looking for a better understanding of Curl, watching this video

generally curl represents the 'amount' of local rotation in a vector field, point by point.
If we think at the 2D vector field described by the function (x,y) ----> (0,x) we see that vectors in the field don't rotate at all, but if we put a 'solid' object in the field, it will experience an amount of rotation due to the fact that 'torque' is different by zero. You can see this example on the english page of wikipedia concerning the definition of curl. - (Cit. We might not see any rotation initially, but if we closely look at the right, we see a larger field at, say, x=4 than at x=3. Intuitively, if we placed a small paddle wheel there, the larger "current" on its right side would cause the paddlewheel to rotate clockwise, which corresponds to a curl in the negative z direction. )

Considering now the field (x,y) ----> (-y,x) [A], described in that page too, we can see a 'global' rotation on the z axis. Now if we put an object in the field (avoiding (0,0) it will experience an amount of rotation on his center of mass due to the torque (the vector's absolute values decrease as 1/r, so it experience torque as before) plus an amount of rotation around the z-axis due to the 'global' rotation of the field.

But if we imagine another vector field described by (x,y) ----> (-y,x)/sqrt(x^2 + y^2) , the field does not provoke any rotation on the object's center of mass, due to the costancy of vector's absolute values, and it seemed to me like the example of the 'flow outside the vortex' explained by the video at [ 7:36 ] .

The problem is that even if the object does not rotate on itself, if we calculate the curl of the expression it's different by zero.

Should I consider the explanation of the video as wrong?

## 1. What is Curl in 2D/3D Space?

Curl is a mathematical concept that describes the rotation or circulation of a vector field in a given space. In 2D space, it is represented by a scalar value and in 3D space, it is represented by a vector.

## 2. How is Curl calculated?

In 2D space, Curl is calculated using the partial derivatives of a vector field in the x and y direction. In 3D space, it is calculated using the cross product of the vector field and the unit vector in the direction of rotation.

## 3. What is the significance of Curl in physics?

Curl is important in physics as it helps us understand the behavior of fluid flow, electromagnetism, and other physical phenomena. It is also used to analyze the stability of a system and to calculate the work done by a force.

## 4. How is Curl visualized in 2D/3D space?

In 2D space, Curl can be visualized as the rotation of a vector field around a point. In 3D space, it can be visualized as the rotation of a vector field around an axis. This can also be represented graphically using arrows to show the direction and magnitude of the Curl.

## 5. What are some real-world applications of understanding Curl in 2D/3D space?

Understanding Curl is important in various fields such as fluid dynamics, aerodynamics, electromagnetism, and weather forecasting. It is also used in computer graphics to create realistic animations and simulations. Additionally, it has applications in engineering, geology, and oceanography.

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