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
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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.
 

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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