I have a problem proving these formulas using Einstein index notation.
The formulas are:
1) $$\nabla(r^n)= nr^{n-2} \vec{r}$$
2) $$\nabla \cdot (\nabla g \times \nabla f)=0$$
3) $$\nabla \times (\nabla \times \vec{D}) = \nabla(\nabla \cdot \vec{D})- \nabla^2 \vec{D}$$ 4) $$\nabla \cdot(\vec{A} \times \vec{D})= \vec{D}(\nabla \times \vec{A})- \vec{A}(\nabla \times \vec{D})$$ 5) $$\nabla \cdot (a\vec{B})=\vec{B}\nabla a + a \nabla \cdot \vec{B}$$
I just cannot put them right into the notation and it gives me all sort of headaches. Could you show me how should I start with this?
I'll talk you through the index notation; the proofs are up to you, as requested.
1) Equate $i$th parts (we also do this for the other vector equation, 3): $$\partial_i(r^n)=nr^{n-2}x_i$$
2) Write curls with Levi-Civita symbols (this also applies to 3, 4): $$\epsilon_{ijk}\partial_i(\partial_jg\partial_kf)=0$$
3) Carefully recycle indices across terms while contracting (we also need this in 4): $$\epsilon_{ijk}\epsilon_{klm}\partial_j\partial_lD_m=\partial_i\partial_mD_m-\partial_j\partial_jD_i$$
4) $\epsilon_{ijk}\partial_i(A_jD_k)=\epsilon_{kij}D_k\partial_iA_j-A_j\epsilon_{jik}\partial_iD_k$
5) $\partial_i(aB_i)=B_i\partial_ia+a\partial_iB_i$