Why is the expression $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$?

Question

It says in my lecture notes that the index notation $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ is equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$, but does not explain why. Could someone please show me why? Thanks

Answer 1

The chain (or product) rule states:$$\frac{d}{dx}\left(\color{red}{f(x)}\color{blue}{g(x)}\right)=\color{red}{f(x)}\frac{d\color{blue}{g(x)}}{dx}+\color{blue}{g(x)}\frac{d\color{red}{f(x)}}{dx}$$ $$\therefore \frac{\partial}{\partial x_{i}}(\color{red}{u_{j}}\color{blue}{u_{j}})=\color{red}{u_{j}}\frac{\partial \color{blue}{u_{j}}}{\partial x_{i}}+\color{blue}{u_{j}}\frac{\partial \color{red}{u_{j}}}{\partial x_{i}}=2u_j\frac{\partial u_{j}}{\partial x_{i}}$$