Thread: Valid way of proving vector differential identities

Hello again!

I'm making my way through proving many of the Vector differential identities. Even though it's hardcore math juggling on some of the larger ones, proving the whole thing is tedious work at best.

The Question:

For any of these larger vector differential identities; for example nabla x (F x G) = (nabla * G)F.... . Is it generally accepted to prove only one of the components, say i, and say that the other two (j and k) will turn out the same? Is there any rule in calculus that says we can do that? If yes, could someone tell me the name of that rule?

Generally i'm always thinking in exam terms, so would this be acceptable as a proof during an exam.

Originally Posted by Hanga

Hello again!

I'm making my way through proving many of the Vector differential identities. Even though it's hardcore math juggling on some of the larger ones, proving the whole thing is tedious work at best.

The Question:

For any of these larger vector differential identities; for example nabla x (F x G) = (nabla * G)F.... . Is it generally accepted to prove only one of the components, say i, and say that the other two (j and k) will turn out the same? Is there any rule in calculus that says we can do that? If yes, could someone tell me the name of that rule?

Generally i'm always thinking in exam terms, so would this be acceptable as a proof during an exam.

Yes, if there's enough symmetry that you can validly say that the other components will be the same. With cross products, I'm not so sure you've got that level of obvious symmetry.

Incidentally, you can prove identities such as the one you mentioned with the Levi-Civita notation coupled with the Kronecker Delta.

Yes I agree that symmetry in cross products is hard to call.
I've checked Levi-Civita notation and Kronecker Delta and they seem to be what I need in order to prove this without using numbers. I won't use them yet, but they seem like a good next step to take. I'm going to check around for steps how to use these notations in ordinary nabla calulation and If I wont find anything i'll PM you

Thx!

You're welcome! If you have any further questions, please don't PM me. If the questions are related to the question in the OP, then post them here. Otherwise, start a new thread.

Thank you.