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- Thread starter Mappe
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- #2

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Simplify the proofs compared with what?

Post a proof or two and you'll get better feedback on whether they can be simplified.

- #3

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Im talking about all these identities, is there a branch of mathematics that simplifies the proofs of these, and lets me avoid expending the vectors and del operators?

- #4

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https://www.google.com/search?q=tensor+calculus+proofs+vector

for some identities,

"differential forms" or "exterior calculus" might be better

https://www.google.com/search?q=differential+forms+proofs+vector+identities

https://www.google.com/search?q=exterior+calculus+vector+identities

"geometric calculus" incorporates all of the above... but might be harder to learn

https://www.google.com/search?q=geometric+calculus+vector+identities

- #5

Fredrik

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\begin{align*}

&(A\times(B\times C))_i =\varepsilon_{ijk}A_j(B\times C)_k =\varepsilon_{ijk}A_j\varepsilon_{klm}B_l C_m =(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})A_jB_lC_m

=A_jB_iC_j -A_jB_jC_i\\

&=B_i(A\cdot C)-C_i(A\cdot B) =(B(A\cdot C)-C(A\cdot B))_i.

\end{align*} The Levi-Civita symbol ##\varepsilon_{ijk}## is defined by saying that ##\varepsilon_{123}=1## and that an exchange of any two indices changes the sign of ##\varepsilon_{ijk}##. (For example ##\varepsilon_{132}=-\varepsilon_{123}=-1##). Note that this implies that ##\varepsilon_{ijk}=0## when two of the indices have the same value. (If ##i=j##, then ##\varepsilon_{ijk}=\varepsilon_{jik}=-\varepsilon_{ijk}##). The third equality in the calculation above involves one of a small number of identities that you have to prove before you can really start working with this notation.

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