I encountered a statement in a book:
If we define the manifold as embedded in a higher-dimensional Euclidian space, as a hypersurface $X(x)$, then the metric is given by $$g_{\mu\nu}=\partial_\mu X\cdot \partial_\nu X$$ and the Christoffel symbols are easily derived to be $$\Gamma^\mu_{\nu\lambda}=\partial^\mu X\cdot \partial_\nu\partial_\lambda X=\frac{1}{2}g^{\mu\rho}(\partial_\nu g_{\rho\lambda}+\partial_\lambda g_{\rho\nu}-\partial_\rho g_{\nu\lambda}).$$
Now I can not see how to derive $\Gamma^\mu_{\nu\lambda}=\partial^\mu X\cdot \partial_\nu\partial_\lambda X$ from the point of view of embedding. Can someone explain it? Thank you very much.