I'm reading Hsiung's A First Course in Differential Geometry. On page 207, where he is deriving the Weingarten formulas, he starts by taking partial derivatives of $e_3$ where ${e_1,e_2,e_3}$ is an orthonormal frame field on a surface.
He denotes these partials as: $e_{3u}=ax_u+bx_v$ and $e_{3v}=cx_u+dx_v$. So this tells me that these partials live in a tangent space.
My question is: What is a partial derivative of a vector field? There are several notions of the derivative of a vector field. But what do these partials mean? And how do I know that they live in the tangent space?
Firstly, $(e_1,e_2,e_3)$ is an orthonormal frame adapted to the surface, meaning that $e_3$ is normal, and not tangent. The partial derivatives are taken component by component, just like the partial derivatives of the parametrization ${\boldsymbol x}$. Since $\langle e_3,e_3\rangle$ is constant, differentiating with respect to $u$ gives $\langle e_{3u},e_3\rangle=0$, so that $e_{3u}$ is tangent to the surface (in other words, if a vector is orthogonal to $e_3$, which is normal to the surface, then said vector is tangent to the surface). This is why you can write $e_{3u} =a{\boldsymbol x}_u + b{\boldsymbol x}_v$ as that linear combination. Similarly for $e_{3v}$. And then you procceed with the computations needed.