- #1

Hiero

- 322

- 68

- Homework Statement
- Simplify ##\nabla \times ( a\nabla b)##

- Relevant Equations
- ##\nabla\times \vec V = \epsilon_{ijk}\frac{\partial V_k}{\partial x_j}\hat e_i##

##\nabla s = \frac{\partial s}{\partial x_i}\hat e_i##

**Attempt:**

$$\nabla \times ( a\nabla b) = \epsilon_{ijk}\frac{\partial}{\partial x_j}(a\frac{\partial b}{\partial x_k})\hat e_i$$ $$ = \epsilon_{ijk}\big(\frac{\partial a}{\partial x_j}\frac{\partial b}{\partial x_k}+a\frac{\partial b}{\partial x_j\partial x_k}\big)\hat e_i$$ $$= \nabla a \times \nabla b + \text{(final term)}$$

That “final term” (a triple sum) should be the zero vector, but I cannot see how. Maybe I messed up elsewhere.

Thanks.