1. Derivative Trouble

Let $S\subseteq\mathbb{R}^3$ be a regular surface and let $p\in S$. then (U,F,V) is a local parametrisation of S at p. Let $u = F^{-1}(p)$
Define $\gamma: (-\delta,\delta) \rightarrow U$, with $\gamma(0)=u$. Then $\{\partial_1F(\gamma(t)), \partial_2 F(\gamma(t))\}$ is a basis of the tangent plane $T_{F(\gamma(t))}S$

Here $\partial_i F(\gamma(t)) = DF(u)e_i$ , i.e the jacobian then applied to $e_i$
I am having a lot of trouble trying to show that

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

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

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

2. Originally Posted by slevvio
$
\frac{d}{dt} \partial_i F(u+t e_j) = \partial_{ji} F(u)
$
This part should read $\frac{d}{dt} \partial_i F(u+t e_j)|_{t=0} = \partial_{ji} F(u)$ otherwise it makes no sense, since they don't have the same domain. To solve this just apply the chain rule to the left side (maybe writing out all the partials of the components of F could help you see this).

3. Sorry for my mistake. YEs thanks for your reply but I do not know how to apply the chain rule

Lets take $F_1$ to be the first component of $F$.

then I guess I am trying to work out $\frac{d}{dt} (\frac{\partial F_1}{\partial x_i} \circ \gamma )(t)$ But how do I do this? I cannot use the normal chain rule since there is an intermediate step in $\mathbb{R}^2$ , so how do I do this? I have never used a higher dimensional chain rule before. Thanks for any help

4. Okay, so $F:U \subset \mathbb{R}^2 \rightarrow \mathbb{R} ^3$ so $\partial _i F(v)= (\partial _iF_1(v),\partial _iF_2(v), \partial _iF_3(v))$ (why?). This is a maping $g_i : U\subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Use of the chain rule gives $\frac{d}{dt} g_i(u+te_j)=Dg_i(u+te_j)e_j=(\partial _{ji} F_1(u+te_j),\partial _{ji}F_2(u+te_j),\partial _{ji}F_3(u+te_j))$

5. Thank you very much, I looked on wikipedia as well for the definition of the chain rule and used the jacobian matrices to work it out rigourously, thanks alot !!