Prove that curl of a vector is a vector

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To prove that the curl of a vector is indeed a vector, one must demonstrate that its value changes with the coordinate system used. The discussion highlights the importance of using the Levi-Civita symbol in the definition of the cross product. Confusion arises regarding index notation and the proper placement of indices in the calculations. A clear demonstration of the transformation properties of the curl under coordinate changes is essential. Providing detailed steps and calculations is crucial for receiving effective assistance.
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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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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.
 

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