Writing these index notation equations in vector form

Question

How can I write the following two equations in vector form: $$u_k\epsilon_{kji}\frac{\partial v_i}{\partial x_j}$$ $$u_jv_k\frac{\partial f}{\partial x_j}$$

Thanks.

Answer 1

I suppose that you are using the Einstein notation, so we have: $$ u_k\epsilon_{kji}\frac{\partial v_i}{\partial x_j}=u_k\left(\epsilon_{kji}\frac{\partial }{\partial x_j}v_i \right)=u_k\left(\nabla\times \vec v \right)^k=\vec u \cdot (\nabla \times \vec v) $$ $$ u_jv_k\frac{\partial f}{\partial x_j}=u_jv_k\left(\nabla f \right)_j=u_j(\nabla f)_j v_k=\left(\vec u \cdot (\nabla f) \right)\vec v $$