I'm back again with another question. So I'm reading a proof of: $$u\times\omega = \nabla\ (\frac{ u\cdot\ u}{2}) - u\cdot\nabla\ u$$
In this proof, it says: $$ u_j\frac{\partial}{\partial x_i}u_j=\frac{1}{2}\frac{\partial}{\partial x_i}(u_j^2) $$ It might be trivial, but I cannot get my head around this later equality. Does anyone know how to prove/motivate it (using Einstein notation)? Would be really helpful! Thanks on beforehand.
You can do this for fixed $j$ by either the chain rule or the product rule, and then sum over $j$. It may be easier to start on the right-hand side and evaluate it. By the chain rule:$$\frac{\partial}{\partial x_i}(u_j^2)=\frac{\partial}{\partial x_i}(u_ju_j)=\frac{\partial}{\partial u_k}(u_ju_j)\frac{\partial u_k}{\partial x_i}=2u_j\delta_{jk}\frac{\partial u_k}{\partial x_i}=2u_j\frac{\partial u_j}{\partial x_i}.$$By the product rule:$$\frac{\partial}{\partial x_i}(u_j^2)=\frac{\partial}{\partial x_i}(u_ju_j)=\frac{\partial u_j}{\partial x_i}u_j+u_j\frac{\partial u_j}{\partial x_i}=2u_j\frac{\partial u_j}{\partial x_i}.$$