Simplify the proof of different vector calculus identities

[Mappe]

Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl and grad etc? I know nothing about geometric algebra, but is that a good way?

 

[Dr. Courtney]

https://en.wikipedia.org/wiki/Vector_calculus_identities

Simplify the proofs compared with what?

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

 

[Mappe]

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

 

[robphy]

"tensor calculus"
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

 

[Fredrik]

You can use the Levi-Civita symbol without knowing anything about tensors other than the convention to not bother to write a summation sigma for an index that appears exactly twice. For example, $\varepsilon_{ijk}A_j(B\times C)_k$ really means $\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}A_j(B\times C)_k$. The proof of the first identity in post #3 goes like this:
\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.