Let $\Gamma \subset \mathbb{R}^2$ be a curve. Define for a smooth function $f$, $\nabla_\Gamma f = \nabla f - (\nabla f \cdot N)N$ where $N$ is the unit normal.
Let $X:S \to \Gamma$ be a smooth regular parameterisation with $|\partial_s X(s) | > 0$.
Let $\tilde{f}(s) = f(X(s))$. How do I show that $\nabla_\Gamma f = \frac{1}{|\partial_s X|} \partial_s \tilde{f}\frac{\partial_s X}{|\partial_s X(s)|}$ ?
I don't know where to start. The notation is confusing..