What will be the 4th axis of a 3d curl?

In summary, when doing a curl of two 2-dimensional vectors, the resulting answer is the 3rd axis around which it is rotating. This is also the case when doing a curl of two 3-dimensional vectors, where the answer involves rotations in all three axes. However, it is not possible to find a 4th dimension that encompasses the entire volume of rotation for all three axes.
  • #1
malfoy23111995
1
0
Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x-axis and similarly x-z plane rotating wrt to y axis.
My question is, can't we find out the 4th dimension with respect to which all the x-y-z volume is rotating?
 
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  • #2
Your "two 2-d vectors" and "two 3-d vectors" are actually the exact same thing. The two "3d vectors" lie in a common 2d plane, just as do you "2d" vectors, and then the curl is perpendicular to that plane, in the third dimension.
 

Related to What will be the 4th axis of a 3d curl?

1. What is the concept of a 4th axis in a 3D curl?

The concept of a 4th axis in a 3D curl refers to the addition of a fourth dimension to the traditional three-dimensional curl, which is a mathematical operation that describes the rotation or circulation of a vector field. In this case, the fourth axis represents a new direction or dimension in which the vector field is rotating or circulating.

2. How is the 4th axis of a 3D curl calculated?

The calculation of the 4th axis of a 3D curl involves extending the traditional three-dimensional curl to a fourth dimension, which is typically represented by the letter "w". This can be done using mathematical techniques such as quaternions or hypercomplex numbers.

3. What are the applications of a 4th axis in a 3D curl?

The addition of a 4th axis to a 3D curl has various applications in fields such as physics, engineering, and computer graphics. For example, it can be used to describe the rotation of objects in four-dimensional space or to simulate fluid flow in a four-dimensional medium.

4. How does the 4th axis affect the interpretation of a 3D curl?

The addition of a 4th axis to a 3D curl can change the interpretation of the curl, as it introduces a new dimension of rotation or circulation. This can lead to new insights and understanding of physical phenomena, as well as new approaches to problem-solving and analysis.

5. Are there any real-world examples of a 4th axis in a 3D curl?

Yes, there are several real-world examples of the 4th axis in a 3D curl, particularly in the study of fluid dynamics and electromagnetic fields. For instance, the rotation of a vortex in four-dimensional space can be described using a 4D curl, and the behavior of a four-dimensional electromagnetic field can be analyzed using a 4D curl as well.

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