Relating rigorous and informal definitions of vector components

I am trying to relate the informal and rigorous definitions of tangent vectors on a manifold and am getting stuck on some points.

Let $$M$$ be a differentiable manifold. Informally, we can say that $$x^i(\lambda)$$ are the coordinates of some curve $$\gamma(\lambda)$$ on $$M$$. Let $$\gamma(\lambda_p) = x^i(\lambda_p)$$ be the point $$p\in M$$.

Now it is informally stated that tangent vector components for some $$X\in T_pM$$ are given by \begin{align} X^i = \left.\frac{dx^i}{d\lambda}\right|_{\lambda = \lambda_p}\end{align}.

Now I will try and do this rigorously.

Let $$(U, x)$$ be some chart on $$M$$.

Define a curve \begin{align} \gamma: \mathbb{R} \supseteq I &\longrightarrow M \\ \lambda & \longmapsto \gamma(\lambda) \end{align}

With respect to the chart, the components of $$\gamma$$ are $$(x \circ \gamma)(\lambda) = (x^1(\gamma(\lambda)), ..., x^n(\gamma(\lambda))$$. To write this as $$x^i(\lambda)$$, do I need to redefine $$x^i\circ \gamma \rightarrow x^i$$? This is essentially my question, as the rest would follow from this.

Define a tangent vector $$X_{\gamma, p}$$ at $$p\in M$$ by, for $$f\in C^{\infty}(M)$$, \begin{align} X_{\gamma, p}(f) = (f \circ\gamma)'(\lambda_p) \end{align}.

Inserting the coordinate identity: \begin{align} (f \circ x^{-1} \circ x \circ \gamma)'(\lambda_P) = (\partial_i(f\circ x^{-1}))(x(p))\cdot (x^i \circ \gamma)'(\lambda_p) \end{align}

The $$(\partial_i(f\circ x^{-1}))(x(p))$$ form the coordinate basis of the tangent space, whilst the components are $$(x^i \circ \gamma)'(\lambda_p)$$, which is written in different notation as \begin{align} \left.\frac{d(x^i\circ\gamma)}{d\lambda}\right|_{\lambda = \lambda_p} \end{align}

If the redefinition I asked about above is indeed correct, then these would seem to agree. However, if not, I would appreciate an explanation of how they are equivalent.

• Yes it's a standard (but annoying) abuse of notation that people use the same symbol $x^i$ to mean both the ith component of the chart map, i.e, $\text{pr}_{\Bbb{R}^n}^i\circ x$ and also ith component of the chart representative of the curve, i.e $\text{pr}_{\Bbb{R}^n}^i\circ x \circ\gamma$. Here I'm using $\text{pr}_{\Bbb{R}^n}^i:\Bbb{R}^n\to \Bbb{R}$ to mean $\text{pr}_{\Bbb{R}^n}^i(a)=\text{pr}_{\Bbb{R}^n}^i(a^1,...,a^n)=a^i$ Commented Oct 16, 2021 at 16:38
• Thanks, it seemed like that was the case, appreciate the clarification. Commented Oct 18, 2021 at 16:27