# No ideas to collapse $\boldsymbol{(\nabla\times B)\times C-(B\times\nabla)\times C}+\boldsymbol\nabla(\boldsymbol B\bullet\boldsymbol C)$

I v expanded the vector calculus terms and added them :)

$$\dbinom{\boldsymbol{B}}{\boldsymbol{C}}=\dbinom{b_1\boldsymbol{i}+b_2\boldsymbol{j}+b_3\boldsymbol{k}}{c_1\boldsymbol{i}+c_2\boldsymbol{j}+c_3\boldsymbol{k}}$$

$$\boldsymbol{(\nabla\times B)\times C-(B\times\nabla)\times C}+\boldsymbol\nabla(\boldsymbol B\bullet\boldsymbol C)\\=\boldsymbol{i}\left(\frac{\partial(b_1c_1)}{\partial x}+\frac{\partial(b_1c_2)}{\partial y}+\frac{\partial(b_1c_3)}{\partial z}\right)\\+\boldsymbol{j}\left(\frac{\partial(b_2c_1)}{\partial x}+\frac{\partial(b_2c_2)}{\partial y}+\frac{\partial(b_2c_3)}{\partial z}\right)\\+\boldsymbol{k}\left(\frac{\partial(b_3c_1)}{\partial x}+\frac{\partial(b_3c_2)}{\partial y}+\frac{\partial(b_3c_3)}{\partial z}\right)$$ However I v n ideas to re-collapse it into a term in nabla!! Who can help me!! :((

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What is I v n? I suspect it means I have no. This is not Twitter with a 140 character limit-please spell it out. –  Ross Millikan Apr 30 '13 at 13:55

## 1 Answer

Remember the product rule: $$\frac{\partial(ab)}{\partial w}=a\frac{\partial b}{\partial w}+b\frac{\partial a}{\partial w}$$

Use this on your terms, note which terms cancel, and then you should get something you can make sense of.

By the way, if you're interested in a more systematic way of doing these sorts of calculations, you should get a book that explains suffix notation. Here is my attack on the problem using suffix notation:

\begin{align} &(\nabla\times B)\times C-(B\times\nabla)\times C + \nabla(B.C) \\ &=e_i(\varepsilon_{ijk}\epsilon_{jpq}\nabla_p(B_qC_k)-\varepsilon_{ijk}\epsilon_{jpq}B_p\nabla_qC_k+\nabla_i(B_kC_k)) \\ &=e_i((\delta_{kp}\delta_{iq}-\delta_{k_q}\delta_{ip})(B_q\nabla_pC_k+C_k\nabla_pB_q-B_p\nabla_qC_k)+B_k\nabla_iC_k+C_k\nabla_iB_k) \\ &=e_i(B_i\nabla_kC_k-B_k\nabla_iC_k+C_k\nabla_kB_i-C_k\nabla_iB_k+B_k\nabla_iC_k+C_k\nabla_iB_k \\ &=e_i(B_i\nabla_kC_k+C_k\nabla_kB_i) \\ &=B(\nabla.C)+(C.\nabla)B \end{align}

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Suffix notations?? It is quite sticky to me because high schools in HK v n tutored these!! :(( –  aVWu290 Apr 30 '13 at 13:28