Prove that curl of a vector is a vector

In summary, the conversation revolves around proving a result in a specific coordinate system and the confusion regarding the use of indices. The attempt at a solution involves using the definition of cross product and showing that divergence of a vector is a scalar. However, the individual seeking help is unsure of where to place indices in their work. They request for help in showing their work in order for others to assist them effectively.
  • #1
SSDdefragger
5
0

Homework Statement


Proove it

Iam supposed to change coordinate system, and proove that the result depends on coordinate system.

The Attempt at a Solution


My attempt was to start from definition of cross product using levicivita. I already prooved that divergence of a vector is a scalar. But when I am trying the same approach I am getting confused with indices (i don't know where do i have to put the same index where i dont)
 
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  • #2
What does it mean specifically to prove something is a vector?

Please show your work. We can't help you unless we can see what you did. Generic descriptions aren't that useful.
 

1. What is the definition of "curl of a vector"?

The curl of a vector is a mathematical operation that measures the rotation or circulation of a vector field. It is represented by the symbol ∇ x V, where ∇ is the del operator and V is the vector field.

2. How is the curl of a vector calculated?

The curl of a vector can be calculated using the formula:
∇ x V = (∂Vz/∂y - ∂Vy/∂z)i + (∂Vx/∂z - ∂Vz/∂x)j + (∂Vy/∂x - ∂Vx/∂y)k
where ∂Vx, ∂Vy, and ∂Vz are the partial derivatives of the vector components with respect to the coordinates x, y, and z, respectively.

3. Why is the curl of a vector considered a vector?

The curl of a vector is considered a vector because it has both magnitude and direction. It also follows the vector addition and subtraction rules, making it a vector quantity.

4. What is the significance of the curl of a vector in physics?

The curl of a vector is important in physics as it is used to describe the rotation or circulation of a vector field. It is also a fundamental concept in the study of fluid dynamics, electromagnetism, and other areas of physics.

5. Can the curl of a vector ever be zero?

Yes, the curl of a vector can be zero in certain cases. If the vector field is irrotational, meaning it has no rotation or circulation, then the curl of the vector will be zero. This is often seen in cases of conservative forces, such as gravity, where the curl of the force field is zero.

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