Induced metric from Euclidean space

Question

Suppose $F:(M^{n-1},g)\rightarrow R^n$ is an embedding, $g$ is the metric induced by $F$ from $R^n$, I want to show that $$g(\nabla_{\partial_i}\partial_j,\partial_k)=<\partial_i\partial_jF,\partial_kF>$$ where $<,>$ is the standard inner product in $R^n$, $\partial_i=\frac{\partial}{\partial x^i}$ is the coordinate vector field and $\nabla$ is the Levi-Civita connection induced by $g$ on $M$, what i did is $$\begin{eqnarray*}g(\nabla_{\partial_i}\partial_j,\partial_k)&=&F^*<\nabla_{\partial_i}\partial_j,\partial_k>\\ &=&<F_*(\nabla_{\partial_i}\partial_j),F_*(\partial_k)>\\ &=&<D_{F_*(\partial_i)}F_*(\partial_j),F_*(\partial_k)>\\ &=&<D_{\partial_iF}\partial_jF,\partial_kF> \end{eqnarray*}$$ So that means $D_{\partial_iF}\partial_jF=\partial_i\partial_jF$, $D$ is the connection in $R^n$, I guess this should be straightfoward but I just can't see it clearly, any explanation is greatly appreciated.