Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to prove $$\vec{A}\times(\nabla\times\vec{A})=\frac{1}{2}\nabla(A \cdot A)-(\vec{A}\cdot\nabla)\vec{A}$$

This looks strikingly similar to the BAC CAB formula


However, because there is some differentiation involved, I understand I cannot simply move the 'Del' operator around willy nilly. I'm thinking I need to use (Product Rule + Levi Civita + Clairaut's Theorem) to prove this, but when I get to the step:


Up until this I have have NOT seperated the $\partial_l(A_m)$ but I'm unsure how to proceed.

share|cite|improve this question
up vote 3 down vote accepted

Hint: You are very close. Note, for example, that $$\delta_{il}\delta_{jm}A_{j}\partial_l A_m = A_j \partial_i A_j = \frac{1}{2} \partial_i A_j^2.$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.