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Suppopse $X:\mathbb{R}\times (0,T) \to \mathbb{R}^2$ is a parametrisation of a smooth curve $\Gamma(t)$ with $X(a + 1, t) = X(a, t)$ for all $a$.

Let $v:\mathbb{R}\times (0,T) \to \mathbb{R}$ be a velocity field ($v = V\nu + v_\tau$, where $V$ is the normal velocity, $\nu$ is a the normal field, and $v_\tau$ is a tangential velocity field).

How do I show that $$ \frac{f(X,t)X_\theta \cdot \frac{d}{d\theta}(v(X,t))}{|X_\theta|} = |X_\theta|f(X,t)\nabla_\Gamma \cdot v(X,t)$$

Any hints/suggestions? I just can't get it to work. I expand the LHS (using chain rule) but the divergence term isn't popping out.

share|improve this question
    
does your $X$ depend on one or on two variables? You should come to a decision wrt to that question. –  user20266 Apr 16 '12 at 19:45
    
@Thomas, sorry, I edited the post. $X$ depends on two variables: $X(\theta, t)$ where $\theta$ is spatial and $t$ is time. –  blahblah Apr 16 '12 at 20:37

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