Levi-Civita proofs for divergence of curls, etc

[theuserman]

I've also posted this in the Math forum as it is math as well.

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I want to know if I'm on the right track here. I'm asked to prove the following.

a) $$\nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})$$
b) $$\nabla \times (f \vec{A}) = f(\nabla \times \vec{A}) - \vec{A} \times (\nabla f)$$ (where f is a scalar function)

And want (read: need to, due to a professor's insistence) to prove these using Levi-Civita notation. I've used the following for reference:
http://www.uoguelph.ca/~thopman/246/indicial.pdf and http://folk.uio.no/patricg/teaching/a112/levi-civita/

Here's my attempts - I need to see if I have this notation down correctly...

a) $$\nabla \cdot (\vec{A} \times \vec{B})$$

= $$\partial_i \hat{u}_i \cdot \epsilon_{jkl} \vec{A}_j \vec{B}_k \hat{u}_l$$

= $$\partial_i \vec{A}_j \vec{B}_k \hat{u}_i \cdot \hat{u}_l \epsilon_{jkl}$$

Now I thought it'd be wise to use the identity that $$\hat{u}_i \cdot \hat{u}_l = \delta_{il}$$.

= $$\partial_i \vec{A}_j \vec{B}_k \delta_{il} \epsilon_{jkl}$$

In which we make i = l (and the $$\delta_{il}$$ goes to 1).

= $$\partial_i \vec{A}_j \vec{B}_k \epsilon_{jki}$$

Then using 'scalar derivative product rules' we get two terms. Now, here's where I get a little mixed up. I'm wondering if we rearrange the terms and then modify the epsilon to go in order the the terms.

= $$\vec{B}_k \partial_i \vec{A}_j \epsilon_{kij} + \vec{A}_j \partial_i \vec{B}_k \epsilon_{jik}$$

Now since the first epsilon is 'even' it remains positive, the other epsilon is 'odd' so that term becomes negative and we end up with the required result.

= $$\vec{B} (\nabla \times \vec{A}) - \vec{A} (\nabla \times \vec{B})$$

b) $$\nabla \times (f \vec{A})$$ (where f is a scalar function)

= $$\partial_i f \vec{A}_j \hat{u}_k \epsilon_{ijk}$$

= $$f \partial_i \vec{A}_j \hat{u}_k \epsilon_{ijk}+ \vec{A}_j \partial_i f \hat{u}_k \epsilon_{jik}$$

Once again, the first epsilon is the positive ('even') while the other is negative ('odd').

= $$f (\nabla \times \vec{A}) - \vec{A}(\nabla f)$$

Man, my hands hurt from all that tex work :P Been awhile for me.
Since my teacher refuses to tell me if this is the correct method (he's only willing to show the concepts, and while I can appreciate that I don't want my mark to go to hell), can anyone help me out?

 

[AndyW]

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Thank you for sharing your work on proving the given equations using Levi-Civita notation. It seems like you have a good grasp on the notation and the steps involved in proving these equations. However, I would like to offer some feedback and suggestions to further improve your understanding and clarity in your proofs.

Firstly, in part a), you have correctly used the identity \hat{u}_i \cdot \hat{u}_l = \delta_{il}. However, you can also use the identity \epsilon_{ijk} = \epsilon_{jki} = \epsilon_{kij}, which is known as the cyclic property of the Levi-Civita symbol. This will help simplify your calculations and reduce the number of terms you have to consider.

Next, when rearranging the terms in your proof, you can also use the identity \epsilon_{jki} = -\epsilon_{ijk} to make the negative term positive and vice versa. This will help you arrive at the required result more easily.

In part b), you have correctly used the product rule for derivatives to expand the expression. However, when using the cyclic property of the Levi-Civita symbol, you will have to consider all possible arrangements of the indices. For example, you can also have \epsilon_{kij} and \epsilon_{ikj} in addition to \epsilon_{ijk}. By considering all these arrangements, you will have the complete expansion of the expression.

Overall, your understanding of Levi-Civita notation seems to be on the right track. I would suggest practicing more examples and familiarizing yourself with the various identities and properties of the Levi-Civita symbol to become more comfortable with using it in your proofs.

Best of luck with your studies!

 

[sir-pinski]

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Your proof for part a) looks correct to me. Using the Levi-Civita notation, you have correctly expanded the dot product and then used the identity \hat{u}_i \cdot \hat{u}_l = \delta_{il} to simplify the expression. Then by rearranging the terms and modifying the epsilon symbol accordingly, you have arrived at the desired result.

For part b), your proof also seems to be correct. You have expanded the cross product using the Levi-Civita notation and then used the product rule for derivatives to arrive at the result. The only thing I would point out is that in the last step, you have a typo where you have written \vec{A}(\nabla f) instead of \vec{A} \times (\nabla f). Other than that, your proof seems to be correct.

Overall, your understanding of the Levi-Civita notation seems to be good and you have used it correctly to prove these identities. Keep up the good work!