$u \cdot u_y = 0 \implies (u^2)_y = 0$

Question

There is this simple step in one of my differential equation book.

$u \cdot u_y = 0 \implies (u^2)_y = 0$

How is this derived?

Answer 1

Notice that $$\frac{\partial (u^2)}{\partial y} = 2u \frac{\partial u}{\partial y},$$ by applying the chain rule. Of course you can cancel the factor $2$ in your equality.