 #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.
$$\nabla \times ( a\nabla b) = \epsilon_{ijk}\frac{\partial}{\partial x_j}(a\frac{\partial b}{\partial x_k})\hat e_i$$ $$ = \epsilon_{ijk}\big(\frac{\partial a}{\partial x_j}\frac{\partial b}{\partial x_k}+a\frac{\partial b}{\partial x_j\partial x_k}\big)\hat e_i$$ $$= \nabla a \times \nabla b + \text{(final term)}$$
That “final term” (a triple sum) should be the zero vector, but I cannot see how. Maybe I messed up elsewhere.
Thanks.