I have some uncertainty when it comes to manipulating differentials and the frequent use of identifications. I understand we make a lot of identifications between spaces when doing computations, but I still feel uneasy about some of their justification even in simple settings. For example, following a paragraph in LeeRM, suppose $\mathbb{R}^n$ has the Euclidean metric $\bar g$ and $V$ is an $n$-dimensional real inner product space endowed with metric $g_p(v,w) = \langle v, w\rangle$ at every $p\in V$. I understand there is a canonical isomorphism between $T_pV$ and $V$ suppressed in this notation in order to make sense of the right-hand side since $v,w\in T_pV$ while the inner product is defined on $V$.
Let $(b_1,\dots,b_n)$ be an orthonormal basis for $V$, I want to show that the map $F \colon\mathbb{R}^n \to V$ defined by $x\mapsto x^ib_i$ is an isometry. I can see that it is a diffeomorphism (in coordinates it is the identity), so I check the pullback condition. Letting $x^1,\dots, x^n$ be global coordinates on $\mathbb{R}^n$ and $y^1,\dots, y^n$ be the global coordinates on $V$ that pick out the components of our orthonormal basis, I can think in coordinates and write $v,w\in T_{x^ib_i}\mathbb{R}^n$ as $v = v^j\partial_j|_{x^ib_i}, w = w^k\partial_k|_{x^ib_i}$. $$(F^*g)_x(v,w) = g_{x^ib_i}(dF_x(v),dF_x(w)) = \langle dF_x(v),dF_x(w)\rangle = v^jw^k\langle \partial y_j|_{x^ib_i},\partial y_k|_{x^ib_i}\rangle$$ Here I used the fact that the Jacobian of the coordinate representation of $F$ is the identity. Now to make sense of the expression on the right-hand side, I have to recall the canonical identification between vector spaces and their tangent spaces defined by LeeSM, where we identify $v$ with a particular derivation $D_v|_a$ which takes directional derivatives. Unraveling the definition of $\partial y_j|_{x^ib_i}$ we can actually see that it equals $D_{b^j}|_{x^ib_i}$. Hence $$v^jw^k\langle \partial y_j|_{x^ib_i},\partial y_k|_{x^ib_i}\rangle = v^jw^k\langle b^j,b^k\rangle = v^jw^k\delta^j_k = \sum_i v^iw^i$$ Then undoing the isomorphism between $v,w\in T_{x^ib_i}\mathbb{R}^n$ and their coefficients in the canonical basis $v = v^j\partial_j|_{x^ib_i}, w = w^k\partial_k|_{x^ib_i}$, this is just $v\cdot w = g_x(v,w)$ in the sense of the Euclidean metric.
But this is certainly not how any person who values their time would think of these things. Instead I would observe that if $v,w\in T_x\mathbb{R}^n \cong \mathbb{R}^n$ then I might think something like $$(F^*g)_x(v,w) = g_{x^ib_i}(dF_x(v),dF_x(w)) = \langle dF_x(v),dF_x(w)\rangle = \langle v^jb_j,w^kb_k\rangle = \sum_i v^iw^i = v \cdot w = g_x(v,w)$$ Here I reasoned that $F(x) = Bx$ so $dF_x = B$ and so $dF_x(v) = v^ib_i$ etc., where I am basically always viewing $v,w$ as just lists of numbers throughout, and I am a bit more fast and loose with the differential because of my familiarity with calculus in $\mathbb{R}^n$. Is there a happy medium between these two approaches, or is this just something you get used to? Any thoughts/insights on these ideas is appreciated.
