Let $\displaystyle S\subseteq\mathbb{R}^3$ be a regular surface and let $\displaystyle p\in S$. then (U,F,V) is a local parametrisation of S at p. Let $\displaystyle u = F^{-1}(p)$

Define $\displaystyle \gamma: (-\delta,\delta) \rightarrow U$, with $\displaystyle \gamma(0)=u$. Then $\displaystyle \{\partial_1F(\gamma(t)), \partial_2 F(\gamma(t))\}$ is a basis of the tangent plane$\displaystyle T_{F(\gamma(t))}S $

Here $\displaystyle \partial_i F(\gamma(t)) = DF(u)e_i$ , i.e the jacobian then applied to $\displaystyle e_i$

I am having a lot of trouble trying to show that

$\displaystyle

\frac{d}{dt} \partial_i F(u+t e_j) = \partial_{ji} F(u)

$

where $\displaystyle \partial_{ji} F(u) = \frac{\partial^2 F}{\partial x_j \partial x_i }$.

I was wondering if anyone could give me some advice on how to think about this, I am very weak in this type of calculus. Thanks very much