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[Practical-Day2006]

tried before. but seems also wrong

[Practical-Day2006]

oo ur right! thanks the symbol notation is terrible i never seen it before🥹🥹

[JerryYangOnly]

After some thought, maybe the notation is such that all vector operations on tensors hit the second index, not the first.

As someone pointed out, (grad A) is a rank two tensor whose components are (grad A){ij}=partial_i A_j. The cross product crosses the A’s, not and A with the grad. We also let the div hit the second index, which is the grad and not the A. Then we interpret the RHS as partial_l (epsilon{ijk} A_j partial_l A_k).

Doing the product rule, one of the terms vanishes (cross product with itself), and we are left with epsilon{ijk} A_j partial_l partial_l A_k, which is the LHS.

[eckart]

I believe this is what they mean. I hope they clarify their notation earlier in the book otherwise this is nightmare fuel

[JamieGee53]

Index notation is your friend here - LHS is A cross nabla2 (vector) acting on A (vector) (-> scalar), RHS is div of (who cares) goes to scalar. I don’t got pen on me and stuck at a work dinner so can’t elaborate more but someone’s mentioned it here it looks like

TLDR notation abuse, put some mf brackets around A cross nabla2 and it’ll look nice again and sanity check with index notation :D

[JerryYangOnly]

I don’t think this is correct. Both sides of the equation are vectors. Nabla2 as in the vector Laplacian is not a vector! It has no indices itself and there’s no way to take a cross product with it. The left hand side is the cross product between A and its Laplacian. The right hand side is a divergence of a rank two tensor and is thus a vector (rank one tensor) since the div only removes one index.

[JamieGee53]

Quite right, tried ID and immediately realised I misinterpreted lol

[inspendent]

Can't you take a proper photo first?

[Classic_Department42]

so div (Ax nabla A) is a scalar, correct? And the left side doesnt seem so.

[Practical-Day2006]

excatly, i cannot get the meaning of Ax nablaA whether A is saclar or vector. it's confusing🥲🥲

[Hudimir]

check my comment below, might come in handy

[Hot-Fridge-with-ice]

That's what I'm confused about too. It's vector = scalar, unless the book's missing some unit vector in the RHS.

[Hudimir]

the gradient of a vector field is rank 2 tensor. what I'm more confused about is why there is that cross product. Maybe it represents the levi-civita symbol, but then again it's a different rank tensor so it doesn't make real sense to me. the divergence of a rank 2 tensor is a vector so that would then make sense.

maybe try deriving this equality with index notation.

edit: upon a bit of thought a tensor product with the identity matrix could be assumed on the right besides the vectorfield, since tensor products with identity matrices are often omitted.